Stress Around Mine Openings in Some Simple Geologic Structures - R. D. CAUDLE, G. B. CLARK

8/12/2019 Stress Around Mine Openings in Some Simple Geologic Structures - R. D. CAUDLE, G. B. CLARK

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L L O S

UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN

PRODU TION NOTEUniversity of Illinois at

Urbana-Champaign LibraryLarge-scale Digitization Project 2007


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byR D CaudleG B l rk

UNIVERSITY O ILLINOIS ULLETIN

~

Stresses around Mine Openings nSomeSimple eologic Structures


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Stresses around Mine Openings n SomeSimple eologic tru tures

by

R D CaudleFORMERLY RESEARCH ASSISTANT IN MINING ENGINEERING

G B l rkFORMERLY PROFESSOR OF MININ N IN RIN

ENGINEERING EXPERIMENT STATION BULLETIN NO 430


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UN3072ERSITY3050 3 55 56728 RESS


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CONTENTSPage

I INTRODUCTION 7II. INITIAL STRESSES AND STRESS CONCENTRATIONSIN THE EARTH S CRUST 9

III. STRESSES AROUND OPENINGS IN SOLID HOMOGENEOUSMATERIALS 111 Early Underground Stress Analysis 112 Theory of Elasticity pplied to Underground Mine Structures 23. Photoelasticity pplied to Underground Mine Structures 24 Stress Distribution around a Single Opening 13

a. Circular Openings 13b. Elliptical Openings 16c Ovaloidal Openings 17d. Rectangular Openings 19e. Summary 20

5 Stress Distribution around Multiple Openings 21a. Circular Openings 22b. Ovaloidal Openings 25c Rectangular Openings 27

d. Summary 27IV. STRESSES IN SIMPLE STRATIFIED ROOFS 30

6. Centrifugal Testing to Simulate Stresses Occurring inRock Beams Underground 307 Comparison of Stresses in Rock Beams for Three Types ofLoads by means of the Theory of Elasticity 32

V MATHEMATICAL ANALYSIS OF STRESSES IN SIMPLEROOF STR T8 Simple Beams 33

a. Uniform Load 33b. Loaded by Own Weight 33c Centrifugal Loading 34

9. Restrained Beams 3510. Summary 37

VI SUMMARY AND CONCLUSIONS 41VII BIBLIOGRAPHY 42


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FIGURES1. Assumed States of Stress in the Earth at a Great Distancefrom any Disturbing Influence 92. Pressure Dome and Stress Trajectories around a Drift 123 Tangential Stresses for a Circular Cylindrical Opening in aSemi-Infinite Mass as Affected by Increasing Depth 144. Uniform Compressive Stresses in an Infinite Plate at a GreatDistance from any Disturbing Influences5 Areal Distribution of Radial Stress along the Horizontal andVertical Axes of Symmetry for a Circular Hole in an Infinite Plate6 Areal Distribution of Tangenital Stress along the Horizontal andVertical Axes of Symmetry for a Circular Hole in an Infinite Plate7 Tangential Stress Concentration on the Boundary of a CircularOpening in an Infinite Plate 168 Stress Concentration on the Boundary of an Ellipse at theMajor and Minor Axes as the Height to Width Ratio Varies 179 Tangential Stress Concentration on the Boundary of an OvaloidalOpening Square with Semicircles Attached to Opposite Ends) 1810 Maximum Stress Concentration as a Function of Height to WidthRatio for Ovaloidal Opening Unidirectional Stress Field 1811 Stress Concentration at End of Axis Perpendicular to the Directionof Applied Stress as a Function of Height to Width RatioUnidirectional Stress Field 1812. Maximum Stress Concentration as a Function of Height to WidthRatio for Rectangular Openings Having Slightly Rounded Corners-Unidirectional Stress Field 1913 Tangential Stress Concentration on the Boundary of a Square Openingin an Infinite Plate for the Three Initial States of Stress 2014. Effect of Shape of Opening on Maximum Stress ConcentrationUnidirectional Stress Field 2015 Comparison of Critical Compressive Tangential Stress forRectangles and Ellipse 2116 Comparison of Critical Compressive Tangential Stress forRectangles and Ellipse under Conditions of Hydrostatic Pressure 2217 Stress Concentration as a Function of the Ratio of OpeningWidth to Pillar Width in an Applied Stress FieldPerpendicular to Line of Centers 2318 Distribution of Shear Stress in Pillar Formed by Two CircularHoles Applied Stress Field Perpendicular to Line of Centers 2319 Shear Stress Distribution in Central Pillars Plate ContainingFive Circular Openings in an Applied Stress Field Perpendicular to

ine of enters


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FIGURES Continued)20. Relation between Maximum Stress Concentration and Number ofPillars for Ratio of Opening Width to Pillar Width of 4.0 2521. Maximum Stress Concentration in Pillars Formed by CircularOpenings as a Function of the Ratio Opening Width to PillarWidth in an Applied Stress Field Perpendicular to Line of Centers 2522. Stress Concentration as a Function of Opening Width to PillarWidthRatio in an Applied Stress Field Perpendicular to Line of Centers 26

23. Shear Stress Distribution in Central Pillars Plate ContainingFive Ovaloidal Openings Height-to-Width Ratio=0.5) in anApplied Stress Field Perpendicular to Line of Centers 2724. Maximum Stress Concentration as a Function of Percent ofRecovery for Pillars Formed by Five Openings 2825. Comparison of the Empirical Equation and the Experimental Data 2926. Simple Beams Showing Method of Support Loads andCoordinate Systems 3427 Restrained Beams Showing Loads Restraints and Coordinate Systems 3728. Isoclinics and Stress Trajectories 39

TABLES1. Critical Values of Tangential Stress on an Elliptical Boundary 162. Ovaloidal Openings Unidirectional Stress Field 183. Stress Concentration for Ovaloids Hydrostatic Stress Field 184. Critical Compressive Tangential Stress for a Pair of Circular Holes 225 Stress Concentration for a Plate Containing Two Circular Openings 236. Stress Concentration for a Plate Containing Three Circular Openings 247. Stress Concentration for a Plate Containing Five Circular Openings 248. Stress Concentration for a Plate Containing Two Ovaloids 269. Stress Concentration for a Plate Containing Five Ovaloids 2610. Stress Concentration for a Plate Containing Five Ovaloidal Openings 27

11 Compilation of Stress Equations for Simple Beam Loaded byThree Type Loads 3612. Compilation of Analogous Stress Equations for Simple BeamLoaded by Three Type Loads 3613. Compilation of Stress Equations for Restrained Beam Loaded byThree Type Loads 3814. Compilation of Analogous Stress Equations for Restrained BeamLoaded by Three Type Loads


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I INTRO U TIONThe problem of accurately determining the

stresses which exist in rocks in the earth s crust haslong been of interest to engineers and geologists.Many mining problems are directly concerned withthe stresses which may cause mine openings to col-lapse during the course of their usage. This Bulletinis concerned with two phases of occurrence of rockstresses: 1) the stresses existing in the rock beforethe introduction of mine openings, and 2) thealtered rock stresses due to the introduction of mineopenings. The development of the investigation ofrock stresses has to the present been limited tosimple geological structures.

There have been many hypotheses formulatedto explain the pre-stressed state of the earth s crustand its causes. Unfortunately, no one has been ableto measure the initial earth stresses without chang-ing them in the process, and thus invalidating theresults. At present, only a qualitative evaluationhas been made of the initial earth stresses from ob -servation of the effect of stresses upon actual mineopenings, and upon geologic structures.It has been the general practice to date for thedesign of mine structures (width, height, and con-tour of mine openings, and the size and shape ofpillars) to be determined upon an empirical basis.This is true largely because 1) the effects ofmaking an underground mine opening upon the pre-existing stresses within the surrounding rock havenot been understood, 2) the concept of pre-existingstresses has been expressed in an inexact manner,and 3) the varying effects of complex geologicconditions have not been determined. Because oflack of knowledge of these three important points,mine structures have been designed from formulaefor which there is no complete justification exceptwhether they succeed or fail). To insure the sta-bility of the mine structures designed in this man-ner, it has been necessary to apply safety factorsof such a magnitude that the original method ofsolution becomes of questionable value.

Research concerned with the stresses aroundmine openings may be classified in three generalcategories: 1) theoretical studies of a purely

mathematical character, 2) studies of models in-tended to duplicate the stress conditions existing inthe prototype mine opening, and 3) observationand measurement of the stress conditions in anactual mine opening.

Theoretical studies differ widely in the basic as -sumptions made about the physical properties ofthe rock itself. This controversial issue has yet tobe resolved completely. Thus, some solutions ofproblems in underground stress analysis assumethat rock is elastic, homogeneous, and isotropic incharacter; others assume that rock possessesplastic, viscous, elasticoviscous properties, or acombination of them. Recently there has been atrend toward the application of soil mechanics tounderground mining problems. As yet, there has notbeen conclusive evidence presented to indicatewhich of the particular methods of solution has thegreatest applicability. In this survey a review ismade of the theoretical studies which have beenbased upon the assumption that rock surroundingthe mine opening is el stic and isotropic in charac-ter. Such an approach was chosen since most of thepreliminary stress studies have been from thisviewpoint, and it furnishes a starting point fromwhich studies involving more complex physicalcharacteristics of rock may be considered.

Experimentation with full-scale rock structuressuch as those which are found in underground mineopenings has, with few exceptions, proven too im -practical and too costly to be worthwhile. If exactresults are to be obtained, it is usually more practi-cal to conduct them in a laboratory where test con-ditions can be closely controlled. Inasmuch asfull-scale models of mine structures cannot be con-structed in the laboratory, however, the most feas-ible approach to the problem is to resort to the useof small scale models. In general, model studies aredivided into two categories, those involving photo-elastic principles, and those concerned with testingmodels made of rock from the prototype. In photo-elastic studies a model of the prototype is madefrom a suitable transparent material. This model isplaced in an instrument (polariscope) and examined


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ILLINOIS ENGINEERING EXPERIMENT ST TION

while under a stress similar to th t applied to theprototype. The stress concentrations which are dis-closed by the polarized light are then comparedwith those in the prototype. In the second method,models made of rock from the prototype arestressed in a manner which will give an approxima-tion of the stresses applied to the prototype; thestresses developed within the model are measuredwith strain gages, or the model is stressed to thefailure point. Several methods of applying stress toa rock model have been attempted. One of the mostsuccessful methods has been th t of applying, a cen-trifugal force to the model in order to simulatestresses due to the model s own weight. This methodof model study is explained in Chapter V, whichcontains a theoretical analysis of some of thesimpler problems th t are applicable to centrifugaltesting.

It should be made clear that none of themethods for solution of underground stresses explains all stress phenomena observed because of thelack of accurate knowledge of the physical proper-ties of rocks under field conditions, and the greatcomplexity of these conditions due to inhomogene-ity of the rock, geologic discontinuities, and manyother factors.

It is the purpose of this paper to present ananalysis of underground stresses which may be applied to certain simple geologic structures to clarifysome points of fundamental research which havepreviously been neglected, and to indicate furtherpossible applications of this research. In addition,it is hoped that the review of literature concernedwith underground stress analysis from the viewpoint of elasticity, photoelasticity, and centrifugaltesting will prove beneficial to others as a basis formore research.


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II. INITIAL STR SS S AND STR SS CONCENTRATIONS IN THE EARTH S CRUSTSeveral scientists have made interpretations of

the initial stresses existing underground before amine opening has been introduced. In general, thesestresses are known to be influenced primarily bythe weight of the overlying material, the relation ofthe opening of the rock masses around it depth ofover-burden, etc.), geologic discontinuities fault-ing, bedding planes, etc.), and the physical charac-teristics of the surrounding rock.

A reasonable hypothesis for the initial stressesexisting in underground rock before a mine openinghas been introduced was used by Mindlin in1 9 3 9 . 3)* It was assumed that stresses within theearth at different depths may be approximated byone of three states of pressure, as shown in Fig. 1:

X-7// 7,////. /// /177-I x y

(a) ydrostatic pressure f)

From observation of the phenomena which occurin all types of underground mine workings, it is ap-parent that these three conditions are not alwayssufficient to embrace all possible initial earthstresses. BeylM has pointed out that the effect ofoverbridging beds and the lateral transmission ofstress, which occurs in pulverulent matter, partlyrelieve the excavations of vertical stress caused bythe weight of the rock. In some cases erosion re-lieves some of the vertical stress while the lateralpressure remains constant. Furthermore, wherework is carried on at great depth, lateral pressuresare sometimes observed which are higher than ex-pected for that depth. Even in areas of regularhorizontal stratification, one can notice traces of

x~~71

d Io II

T IT i i rT rT iSLaterally restrained c) No lateral restraint

Fig 7. Assumed States of Stress in the Earth at a Great Distance from any isturbing InfluenceThey are 1) initially hydrostatic stresses acting oneach unit of the solid, the state of materials atdepths greater than those now mined, 2) initiallateral restraint during the application of the gravi-tational field, an approximation of the forces actingat an intermediate depth within the earth, and3) no initial lateral restraint on a unit of the solid,the state of some materials in the immediate vi-cinity of the surface. These cases represent therange of variation of earth stresses. The actualinitial stress condition existing underground beforea mine opening is introduced generally lies betweenthe two extremes. For this reason, these three con-ditions have been widely used in solutions by pho-toelastic and elastic analytical methods.*Parenthesized superscripts refer to correspondingly numberedentries in the Bibliography.

horizontal thrusts. This is an indication of appre-ciable orogenetic mountain building) pressure in ahorizontal direction. Beyl lists another pressurewith a thermic origin. This pressure is multilateraldue to the exothermic transformation of peat intolignite and coal, the intrusion of magma, or the oc-currence of metamorphism.

Beyl obtains the state of stress in the rock massat the surface and at depth by the superposition ofthree fields of pressure: 1) a horizontal force dueto orogenetic pressure which is often the largestcomponent, 2) a vertical force resulting from theweight and representing part of the weight of over-lying deposits, 3) a hydrostatic pressure equal inall directions. He has made use of these conceptsto explain on a qualitative basis some of the con-ditions observed in rock formations under high


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pressures. Unfortunately solutions based upon thesesuppositions require a knowledge of the orogeneticpressure and the vertical forces transferred by thelayers of rock to lower layers (Beyl s theory of theeffect of overbridging beds) which is not availableat the present time. For that reason, in the study ofthis problem use is made of the three conditionspostulated by Mindlin which cover most of the cir-cumstances encountered except when lateral pres-sures are greater than vertical pressures. For this

latter case it is only necessary to superimposehorizontal stress of the desired magnitude upon thestresses given for the case of no lateral restraint.The difficulty remaining is in determining the re-quired magnitude of the horizontal stress. Ifmethod can be devised for measuring the pre-exist-ing stresses in the earth s crust without disturbingthem, the possibilities of expressing those stressesmathematically with a higher degree of accuracywill be greatly increased.


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III STR SS S ROUND OPENINGS IN SOLID HOMOGENEOUS M TERI LSThe immediate purpose of stress analysis of un-

derground mine structures is essentially twofold:1) to obtain a concept of the effect of the size andshape of a single mine opening upon the stressesexisting initially before the opening was intro-duced) within the surrounding rock, and (2) todetermine the effect of a group of mine openingsupon the stresses existing initially within the rockas their size, shape, number, and relative positionsare varied. The ultimate purpose is to apply theresults of these analyses to achieve more economicalmining operations.Stress analyses of underground mine openingsfor many simple cases have been performed by puremathematical analysis and an analysis of models inthe laboratory. The results of these analyses cannotalways be applied directly to obtain quantitativeresults for the general underground mining prob-lem because they are solutions of problems whichwere chosen for the simplicity of the concepts andmathematics involved. They are specific cases, andtheir field of application is limited. It is this aspectof stress analysis which necessitates the develop-ment of a mode of investigation in which thetheoretical and model studies are checked or supple-mented by field studies and experimental labo-ratory work, or vice versa. In this manner, thefullest benefit of the laboratory work as well as thefield work may be obtained.1 Early nderground Stress nalysis

With early investigators the rock surroundingmine openings was assumed to approximate somesolvable, fundamental structural unit, primarilybecause of the manner in which failure was ob-served to occur at the mine opening. Many earlyinvestigations were centered upon the observationthat a dome-shaped space forms around certaincollapsing underground openings. The rock in thetop of the original opening fails, leaving a dome-shaped structure or opening which apparently re-establishes equilibrium. 17) In most instances thisattempt to correlate underground structures withsome structural unit was only an approximation,primarily because it was based on the assumption

that the rock at some distance from the mine open-ing had no effect upon the stresses in its immediatevicinity.

During the period 1881 to 1885, two investi-gators, Fayol and Rizha, 16) proposed theories withsimilar content. These theories were the forerunnersof the dome theories. The mine opening was as -sumed to be surrounded by a roughly spherical shellwithin which the rock was loaded by its ownweight. Thus, the rock within the dome droppedwhen the pull of gravity exceeded the cohesion withthe surrounding rock.In 1935, the dome theory was extended in anarticle by Dinsdale. 4 ) In essence, he assumed anegg-shaped pressure ring surrounding the mineopening, and within this ring the hanging wall wasseparated from the external rock by shearing actionand rested upon the supports within the opening.Figure 2 represents a cross-section of the mineopening, illustrating the so-called dome. The con-clusions reached by Dinsdale were that the pressureon the immediate ribs and supports within theopening is small in comparison to the stresses ashort distance from the sides of the opening. Dins-dale shows, in a simple static analysis, that theheight of the dome increases with depth; thus, thepressure on the supports is proportional to depth.The value of this analysis is doubtful since thebasis for the assumptions leading to its solution aresubject to question.

A general criticism of the dome theory has beengiven by Shoemaker. 7) His objections are: 1) theoccurrence of the pressure dome itself is assumedbut not explained; (2) the theory takes no accountof forces outside of the dome acting on the rockwithin the dome; (3) the magnitude of forces as-sumed is not sufficient to account for the observedeffects; and (4) the theory is based on the assump-tion that the rock is stressed within its elasticrange. This last assumption can be justified inmany cases, however.)

In similar manner, various other theories havebeen presented which treat layers of rock imme-diately overlying the mine opening as beams loaded


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ILLINOIS ENGINEERING EXPERIMENT STATION

I pressure in unaisurDe oPI pressure on supports within mine opening

P pressure immediately to si esSof drift l

Fig 2 Pressure Dome nd Stress Trajectories around a Drift

individually by their own weight. These havediffered only in the manner in which the beamswere considered to be restrained and in the mannerof their failure. One of the most advanced hypoth-eses of this type was published by George S.Rice 16) in 1923; he assumed that, when failure ofthe beam occurs, inward shearing at the ends causesthe formation of a dome-shaped space. These con-cepts are subject to criticism since the beams wereassumed to be loaded only by their own weight,and no consideration was made of an external loadon them.

Theoretical research upon the problem of under-ground mine structures has advanced rapidly inthe last fifteen years, particularly with the appli-cation of the theory of elasticity and photoelasticityto the problem.2 Theory of Elasticity Applied to Underground MineStructures

A solution of the problem of the distribution ofstress around a mine opening by the theory of elas-ticity requires some basic generalizations and as-sumptions; it involves solving a stress function forthe problem which is related to the existing stressconditions by the boundary stresses and stress-strain relationships. The assumptions to affect asolution are (1) the rock is assumed to be of

a homogeneous isotropic nature; 2) the mine open-ing is approximated by a definite geometrical fig-ure; 3) the mine opening is assumed to behorizontal throughout its length and very long incomparison to its cross-section; 4) the stressesalong the length of the opening are assumed to beuniform; 5) the underground mine is assumed toconsist of an opening or a series of openings in aninfinite or semi-infinite (bounded only by theearth s surface) mass; and 6) the stresses en -countered lie within the elastic limits of the ma-terials. These assumptions are necessary both topermit the application of the theory of elasticityto the problem and to simplify the mathematics ofthe analysis. They necessarily cause a loss of gen-erality and narrow the field of application of theresults, but. they give an approximation of thestresses which may be expected under conditionsapproaching the ideal case. In addition, they forma basis for a more advanced theory in which it isnot necessary to make such confining assumptions.A justification of these assumptions has been givenby several authors. Duvall (' ) in particular hasmade a complete study of them.3 Photoelasticity Applied to Underground MineStructures

In the case of simple ideal problems, the theo-retical approach is perhaps the most satisfactorysince it provides an exact solution to the problem.In most instances, however, underground openingsdo not have simple boundaries. They are often oflarge number, and are arranged in a manner whichis difficult to analyze mathematically. It is in theapproximate solution of these more difficult prob-lems that photoelasticity has its best application.

The photoelastic method of stress analysis isbased upon the principle that, when certain trans-parent materials are stressed, their optical proper-ties undergo changes which can be measured andrelated quantitatively to the stress state. Further-more a model stressed in this manner presents apicture of the stresses in a prototype, regardless ofthe difference in the elastic constants possessed bymodel and prototype, if the elastic constants arethe same throughout the body and if the body con-sists of one continuous piece (loading must besimilar, of course).

The assumptions in applying the photoelasticmethod to stresses around mine openings are:(1) those made for the solution by the theory ofelasticity, and 2) the postulate that the stresses


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Bul 430 STRESSES AROUND MINE OPENINGS IN SOME SIMPLE GEOLOGIC STRUCTURESabout a mine opening may be approximated by thestresses about a similarly shaped opening in a plateunder the same load as exists on a cross-section ofthe mine opening. The error involved in making thelatter assumption is very negligible for depthsgreater than about 2.5 times the diameter of thehole, as shown by Panek. 15)

In general, a solution by the photoelasticmethod requires: 1) selecting a suitable trans-parent material and making from it a model of thestructural number under investigation; 2) loadingthe model in a way similar to the loading of theprototype and measuring the resulting optical ef-fects in a polariscope; 3) translating the opticalmeasurements into terms of stress and interpretingthe latter by means of the fundamental theory ofelasticity; and 4) transforming the stress distribu-tion on the model to the analogous stress system forthe prototype. For a more complete discussion of thephotoelastic method, Frocht 7 ) has compiled a two-volume work upon this subject.

Results obtained by use of a photoelastic modelserve as a solution for the prototype, because itcan be shown mathematically that for a prototypeopening at a considerable distance from the surfacethe prototype is approximately in a state of planestress. Thus, the model stresses are for practicalpurposes directly proportional to the correspondingprototype stresses, because the mathematical solu-tion of the two problems is identical if certainnegligible terms are omitted. In model and in pro-totype, the stress distribution depends only uponthe shapes of the openings and their orientation inrespect to the initial stresses. Therefore, the scaleratio (ratio of a linear model dimension to thecorresponding linear prototype dimension) may bechosen at will. The only requirement is that themodel be geometrically similar to its prototype. 15)

The photoelastic method is not an exact methodof solution, since it is subject to the errors inherentin experimental analysis. It has been shown byDuvall that these errors do not cause a differencefrom the theoretical value in the simple cases ofmore than 6 percent. 5>

The primary objective in most early analyseswas the determination of the stress distributionaround a tunnel or shaft, or in pillars or arches.Only recently has attention been directed towarddetermination of the stresses in a mine as a unit,that is, the stress concentrations due to a numberof underground openings.

4 Stress Distribution around Single OpeningAs a logical starting point, the problem of a

single underground opening and its effects upon thestresses in the surrounding rock will be considered.

The effect of making openings of differentshapes upon the stresses existing in rock massesbefore the openings are made is of fundamentalinterest. A series of geometrical shapes has beenchosen by various investigators which are relativelysimple to solve mathematically and which alsoapproximate certain typical underground openings.Horizontal cylindrical mine openings, with circular,elliptical, ovaloidal, and rectangular cross-sectionsare considered in that order.

Solutions of these problems have three imme-diate objectives: 1) to determine the effect of thedifferent shapes upon the stress concentrations atthe boundaries of the openings for different statesof initial stress in the rock; 2) to determine theshape best suited (smallest stress concentration in-duced in the surrounding rock) for each of theinitial stress conditions within the earth; and 3) todetermine approximately the stress which existsaround actual mine openings.a. ircularOpenings

A complete work on the stresses existing arounda tunnel was published by Mindlin in 939 1 ) Bymeans of the theory of elasticity he solved theproblem of stresses around a horizontal cylindricalhole of circular cross-section in a semi-infiniteelastic solid stressed by gravity. He assumed thatstresses within the earth at different depths may beapproximated by three states of pressure whichexisted before the opening was made, as is shownin Figure 1. The problem is one of mathematicalcomplexity. By introduction of a bipolar coordinatesystem, 7 ) it is greatly simplified, and an exactsolution of the classical elasticity equations can beobtained. The length of the tunnel is considered tobe large in comparison to its diameter. This and thefact that the body force is uniform permits thetreatment of the problem as one in plane strain.

Panek 15) has made a further development inMindlin s analysis and its application. The zone ofstress caused by the introduction of an opening isconfined to a small area about the opening, andthe maximum tensile and compressive stresses occuron the boundary of the opening. In general, theback (roof) and floor of the opening are in tensionand the ribs (walls) are in compression, with theexception that when the lateral initial earth pres-


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sure is greater than about one-half the verticalpressure, the tangential stress on the entire bound-ary of the opening is compressive. The criticaltensile stress seldom exceeds in magnitude theinitial vertical pressure and is only slightly affectedby the size of opening, but it is highly sensitive tothe ratio of lateral to vertical initial pressures.Thus, the value of Poisson s ratio for rock is veryimportant in the intermediate case, since it de -termines the lateral pressure.

The critical values on the horizontal andvertical diameters of the circle at the boundary) ofthe tangential stress are shown in Fig. 3 for thethree different pressure states. In these cases theinitial vertical rock pressure is taken to have amagnitude of -1, i.e., it is compressive. It is appar-ent from these sets of curves that there is littlechange in magnitude of the stress concentrationfactor for a depth of hole-to-hole diameter ratio(d/h) of greater than 2.5. This does not mean thatthe actual stress does not increase with depth; itonly indicates that the stress concentration factorremains constant.

It can also be seen from Fig. 3 that, for a holeat a considerable distance from the surface, the sizeof the opening has little effect upon the criticalstresses. The tangential stress is shown as a dimen-sionless ratio and may be converted to a stress inpounds per square inch by multiplying by 1.2d,where d is depth from the surface in feet. This isequivalent to assuming a specific gravity of 2.77for the rock encountered in the homogeneous mass.

In his analysis, Panek has compared the stressesas obtained by Mindlin with stresses in three casesanalogous to those solved by Mindlin. The latterstresses were obtained by assuming the circularopening to exist in a plate in a uniform-stress field.This presents the problem as one of plane stressrather than plane strain, simplifying it consider-ably, as will be shown.

The initial stresses, applied to the edges of aplate, were:

(a) Uniform compressive stresses S, = Sz .(b) Uniform compressive stress S,; a uni-

form compressive stress S. = 1 S.c) Uniform compressive stress S,; S=0.Figure 4 shows the three cases of initial stress

assumed by Panek, in a manner analogous to Fig. 1.

d/h, Ratio of overburdendepth to opening diam0 / 2 3 4 5 6 7

.aa

a

a

2

0

I

u

a) Stresses at rib

No lateral restraintl _

A. /1/4temly restrained

Hydrostatic pressurev 2( /4 Laterally restrained(vo.0 \\ -v= Z

b) Stresses at floor

No lateral restraintR (V -0 | |tery restined V 0Laterally restr ined v =o )

Laterally restrainedAHydrostaticpressure

0--v )n NW-^___

Laterally restrained (V 2)2 3 4 5 6 7

d/h, Ratio of overburden depth to opening diam(C) Stresses at backFig. 3. Tangential Stresses for a Circular Cylindrical Opening

in a Semi-Infinite Mass as Affected by Increasing Depth

- Hydrostatic pressureV '/2) Laterally restr ined

r v o \ V zl 2)

Laterally restrainedA _0) -0 V=/4

o

I r . . .

( rv kVNoateral restraintt

I

3- I I I I I A'

t-

m

U

I


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Bul 430 STRESSES AROUND MINE OPENINGS IN SOME SIMPLE GEOLOGIC STRUCTURES

I

a) Uniform compressive stressesSx ond Sy equal

^ < i

b) Uniform compressive stress Sy withsTx tT,

d

C) niform compressive stress Sywith Sx

Fig 4 Uniform Compressive Stresses in an nfinite Plate at a Great Distance from any Disturbing InfluencesThe solution of the stresses around a circular

opening for these particular stress fields is not diffi-cult. By use of polar coordinates with origin at thecenter of the opening a solution for the radialtangential and shearing stresses may be obtainedfor the three cases. The stresses for these simplifiecases are independent of the size of the hole andare also independent of the elastic moduli of thematerial. Figures 5 and 6 show that the zone of

Stress concentration

4~ x444

ssur

00000

t~tttt t Jtit~aterally restrained No lateralrestraint

Fig. 5 Areal Distribution of Radial Stress along the Horizontal andVertical Axes of Symmetry for a Circular Hole in an Infinite Plate

disturbance is very definitely localized in theneighborhood of the opening that is within a dis-tance of three times the radius of the center of thecircular opening.

As is the case in the analysis by Mindlin thecritical stresses occur on the edge of the hole on

Stress concentration

hole

Sy sy : ~

aterallyrestrained No lateral restraintFig 6 Areal Distribution of Tangential Stress along the Horizontaland Vertical Axes of Symmetry for a Circular Hole in an Infinite Plate


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ILLINOIS ENGINEERING EXPERIMENT ST TIONthe vertical and horizontal diameters, and they ac tparallel to the boundary of the hole. Figure 7illustrates the variation in tangential stress aroundthe perimeter of the opening for the three differentcases. If it is desired to find the shearing stressacting at any point upon the boundary, it is merelyone-half the tangential stress. One can show thatthe maximum shearing stress also varies from pointto point, reaching its greatest value on the edge ofthe hole.

fttifftHydrostaticpressure

Laterally restraibedsy-7

o

/No ateral restraintFig 7 Tangential Stress Concentration on the Boundary of a

Circular Opening in an Infinite Plate

When the opening is far from the surface forlarge d/h), the values of the tangential stress forthe three respective cases, as determined by Mind-lin's analysis, are found to approach the correspond-ing values as given by Panek for the uniform stressfield. Thus, it is apparent that the much simplersolution may be used to determine the stress con-centration due to a circular opening when it is farfrom the surface of the rock mass (when the roofof the hole is at a distance below the surface equalto more than twice the hole diameter). This is avery important conclusion, because it greatly sim-plifies the testing of experimental models, such asplastic plate models for photoelastic studies. Theuse of uniform-stress fields permits the solution ofproblems, such as the introduction of openings ofgeometrical shape other than circles, which wouldbe difficult and tedious to solve by the theory ofelasticity.

The problem of a circular opening in a uniformstress field has also been solved photoelastically by

various authors, and the results have been foundquite comparable to the theoretical results. In mostinstances, the case of a plate in a vertical uniformstress field with no lateral restraint has been solved,and from these, the other two cases have beenobtained by superposition. (7 In addition, numerousproblems of more complex nature have been solvedby use of circular openings in a plate, using photo-elastic methods. An example is the case of a circularopening very close to a free boundary. ( )b. Elliptical Openings

The problem of stresses around an ellipticalopening introduced in a uniform stress field ha sbeen solved by C. E. Inglis.(11' The solution waseffected by the use of curvilinear coordinates, andthe complete method of solution is given in hispaper. Table 1 indicates the variation in the criticalboundary stresses with a change in the ratio of the

Critical Values of TableTangential Stress on an Elliptical Boundary - 1

S 0

0 1.022Y 1.145 1.367Y 1.290 1.00 1.0222 1.2

45 1.567 1.490 1.0

3 0-5.0-4.6-3.7-2.5-2.0-7.0-6.3-4.6-2.7-1.7-9.0-8.0-5.8-3.0-1.5

Critical Value of SeS = 8 30 -2.70.33 -4.70.29 -4.30.13 3 5 -0.25) -2.4 -0.67) -1.70.44 -6.70.43 -6.10.29 -4.7

-0.23) -3.0-1.33 -1.30.50 -8.70.52 -8.00.42 -6.1 -0.14) -3.7-2.00 -1.2)

S= S-2.0-4.0-4.0-4.0-4.0-4.0-6.0-6.0-6.0-6.06 0-8.0-8.0-8.0-8.0-8.0

Parentheses indicate that the value is not actually a critical one.w/h =the ratio of the major to minor axis-=the angle of inclination of the major axis with the horizontalTable 1 is after Panek, Table 1.

major to minor axes w/h), and with a change inthe angle of inclination (8) of the major axis of theellipse with a horizontal plane and with the magni-tude of the initial horizontal stress S.).

A number of important facts may be establishedfrom the data given for an elliptical opening:(1) In the case of hydrostatic pressure, criticalcompressive stress is always on the major axis,being independent of 8, and the entire boundary isin compression. The minimum stress occurs on theminor axis. 2) For initial horizontal pressure S,)less than the initial vertical pressure S,), thecritical compressive stress tends to occur at thesides of the opening and the critical tensile stressat the top and bottom. 3) For a given S and w/h,the critical compressive stress is greatest at 8 = 0deg and smallest at 8 = 90 deg. 4) Critical com-


21/50

Bul 430. STRESSES AROUND MINE OPENINGS IN SOME SIMPLE GEOLOGIC STRUCTURES

pressive stress increases linearly with increasingw/h when 8 is small.

Some interesting results may be obtained byaltering the shape of the ellipse. Starting with acircular shape and elongating the horizontal diam-eter, the rib compression increases, and the backstress becomes more tensile, never exceeding 1however. Starting with a circular shape and elon-gating the vertical diameter, the rib compressiondecreases slightly and the back stress becomes morecompressive for most values of S . These facts areillustrated by Fig. 8.

9 8 / 6 5 4 3 2 / C /-9 -8 -7 -6 -5 -4 -3 -2 -I 0 /Maximum str ss concentration

Fig 8. Stress Concentration on the Boundary of an Ellipseat the Major and Minor Axes as the Height to

Width Ratio Varies

Variation of the lateral pressure S. has littleeffect on the critical rib stress, but is a major factorin determining the critical back stress. The criticalstresses are at a minimum when the maximum ribstress is equal to the maximum back stress. This istrue only when 8 = 90 deg, or when w/h = 1 acircle).( ') By use of the equation determining thetangential stresses around an elliptical opening, andthe fact that the minimum critical stresses occurat 8 = 90 deg, it can be shown that the shape ofthe ellipse for which the critical stresses are aminimum is given by w/h = S /S from which itcan be seen that the larger Sy/SX the more elongatedthe ellipse. When the shape of the ellipse is givenby the above formula, the ellipse having its majoraxis vertical, it can also be shown that the magni-tude of minimum critical stress is given by st =S y S. a compression equal to the sum of theinitial vertical and horizontal earth pressures.c Ovaloidal Openings

The case of an ovaloid hole in a uniformlyloaded plate has been investigated by MartinGreenspan. 9) He considered the case of an ovaloidwhich was a square with a semi-circle erected oneach of two opposite sides. An exact solution of thisproblem was obtained by approximating a trueovaloid with a figure which could be represented

by a much simpler set of parametric equations.Greenspan considered the problem of a plate in astate of generalized plane stress, the stress at pointsremote from the hole having the constant normalcomponents in the horizontal direction o - = S andin the vertical direction r = Sy; the constant shear-ing stress was rT = T y By making use of a curvi-linear coordinate system, an equation for the tan-gential stress about the opening was obtained. Asexamples of this solution, Greenspan gives two com-plete calculations, one considering a tension appliedto the plate as parallel to the long axis of theovaloid, and the other considering a tension appliedparallel to the short axis as ovaloid. To applyGreenspan's results to a mine opening it is neces-sary to compare his solution for the stresses in aplate with an ovaloid-shape hole, which is essen-tially in a state of plane stress, with the case ofstresses around a horizontal cylindrical openingwith an ovaloid cross-section in a semi-infinitemass, which is essentially in a state of plane strain.As discussed previously, this comparison is justified,as the difference in magnitudes of the solutions forplane stress and plane strain is negligible. Green-span's solution for tension may be applied tomining problems where the initial stresses arecompressive by changing the signs of appropriateterms in the stress equation.

It is also possible, by the principle of super-position, to superimpose one of the results obtainedupon the other and obtain the stress conditionsexisting around the ovaloid opening for the threeinitial states of stress which were investigated inthe case of circular and elliptical openings. Figure 9illustrates the tangential stress existing around asquare with semi-circular ends and with the majoraxis vertical or horizontal, respectively, for thethree initial states of stress.

Duvall(0) tested by the photoelastic method aseries of ovaloidal openings with axes parallel andperpendicular to an applied unidirectional stressfield; he determined the maximum stress concen-tration on the boundary, as well as the stress con-centrations at the ends of each axis.

Table 2 gives the results of these tests. As theheight-to-width ratio decreases, the maximum stressconcentration increases without limit. See Figure10.) The experimental points were obtained bymeans of the photoelastic method.

The relation between the height-to-width ratioand the stress concentration on the horizontal axis

Laterally restrained< No I- -- l er l --

I .-Hydrostatic aestralnt- riticalrib str ss . pr ssur - - rri Critical back stress 1 /o Experimental points by Duvall .I/ .

J^_ \ __ _


22/50


Stress on entrf toon3 2 -/ 0 / 2 - -~

t ftHydrostatic pressure

01Laterally restrained

na) Major axis vertical f if

No lateral restraint

Boundary of ovaloid -- __

Hydrostatic pressure

Laterally restrained

b) Major axis horizontal tNo lateral restraintFig. 9 Tangential Stress Concentration on the Boundary

of an Ovaloidal Opening Square with Semicircles Attached to Opposite Ends

of an ovaloid at its boundary is given in Fig. 11as compared to corresponding data for an ellipticalopening. From a comparison of Figs. 10 and 11 itcan be seen that the maximum stress concentrationat the boundary of an ovaloid does not necessarilyexist on the horizontal axis of the ovaloid, and maydiffer considerably from the stress concentrationon the horizontal axis.

Tablevoloidal Openings Unidirectional Stress Field

Height of Width of RatioOpening Opening of h wh in.) w in.)0.388 1.504 0.258.389 1.141 .341.764 1.500 .509.389 .761 .512.391 .391 1.000.767 .767 1.000

1.511 .764 1.982.262 .765 2.962.512 .765 3.293.015 .765 3.94Table 2 is after Duvall, Table 1

Stress ConcentrationAt End of At End of MaximumVertical HorizontalAxis xis-0.93 5.35 5 35-1.07 4.75 4.75-1.01 4.14 4.14 0.92 4.02 4.02-1.09 3.17 3.171.14 3.05 3.05-1.20 1.90 2.64-1.17 1.72 2.401.28 1.65 2.35-1.17 1.65 33

Stress concentrationFig. 11. Stress Concentration t End of Axis Perpendicular to the

Direction of Applied Stress as a Function of Heightto Width Ratio - Unidirectional Stress Field

The stress concentration produced on the endsof the vertical axis of an ovaloid in a unidirectionalstress field is approximately equal in magnitude,but opposite in sign, to the applied stress and ispractically independent of the height-to-widthratio.

From data obtained for ovaloidal openings inthin plates subjected to a unidirectional stress fieldthe boundary stresses for a hydrostatic stress fieldwere obtained by algebraic addition of stresses.Table indicates the stress concentration at theends of the major and minor axes of an ovaloid ina hydrostatic stress field. The stress concentrations

Table 3Stress Concentration for Ovaloids Hydrostatic Stress Field

Ratio of Minor Stress ConcentrationAxis to Major Axis At End of Major Axis t End of Minor Axis1.00 2.10 2.101.00 1.98 1.98.51 2.95 .83.34 3.68 .65.257 4.28 8* Computed from unidirectional stress field data.Table 3 is after Duvall.

oaximnumstress concentrationFig. 10. Maximum Stress Concentration as a Function of

Height to Width Ratio for Ovaloidal OpeningUnidirectional Stress Field


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are compressive at the ends of both axes of theovaloid regardless of the major to minor axis ratio.As the major to minor axis ratio increases, thestress concentration at the end of the major axisincreases, and that at the end of the minor axisdecreases in a like manner.d. RectangularOpenings

A study has been made of the influence of arectangular opening-with a short dimension ha long dimension w and a small fillet of radius ofcurvature r at the corners upon the stresses in aplate subjected to the three aforementioned statesof initial stress.

Panek 1n5 and Duvall 5) have furnished con-siderable information, largely from photoelastic ex-periment, about the stress concentrations existingaround rectangles with side dimensions of varyingproportion. Primarily, interest lies in the effectupon the maximum stress concentration of varyingthe long-to-short dimension ratio w/h), the radiusof curvature-to-short dimension ratio r/h) or theangle 8 which the long dimension w makes with thehorizontal.

Consider an opening of rectangular cross-sectionin a plate situated in a two-dimensional stress field,where Sy is the initial vertical stress and S. is theinitial horizontal stress. S is permitted to possessall values between zero and S5 so that the forms ofinitial stress previously discussed are amply cov-ered. In general, there will be a tensile bound-ary tangential stress on the top and bottom of theopening. With the major dimension of the rectanglehorizontal, the critical tensile stress occurs at thecenter of the back and floor. As the angle of themajor dimension of the rectangle with the hori-zontal increases, the critical compressive stressmoves from the intersection of the fillets with thevertical sides to the lateral fillets, and the criticaltensile stress shifts to the highest and lowest bothoccurring at or close to the ribs. After a 90-degrotation, the critical stresses are again tensile at themiddle of the top and bottom and compressive atthe ribs at the fillets.

In the case of S near zero no lateral restraint)and between 30 and 80 deg, the critical tensile stressbecomes excessively high; under other conditionsit does not exceed +1. The value of the critical ten-sile stress is sensitive to the value of Se decreasingas the ratio of SI/S increases. In general, for valuesof S greater than Sy/2, the stress on the boundary

of the rectangle becomes wholly compressive innature.

The critical compressive stress increases linearlywith increasing w/h, and is not altered to anyextent by increase in the lateral pressure for smallvalues of 8. For values of 8 less than 80 deg, thecritical tensile stress tends to decrease slightly withan increase in w/h.

Figure 12 gives the relation between the maxi-mum stress concentration and the height-to-widthratio h/w) for a rectangular opening in a plate

0 2 3 4 5 6Maximum stress concenlralion

Fig. 12. Maximum Stress Concentration as a Function of Height-to-Width Ratio for Rectangular Openings Having Slightly

Rounded Corners Unidirectional Stress Field

placed in a unidirectional stress field. The maxi-mum stress concentration increases without limit ash/w decreases, but the rate of increase is not asgreat as for ovaloidal or elliptical openings.

As the fillet radius decreases from r/h = 1/ tor h = 112 the critical compressive stress increases;the stress at r/h = 2 is only 1.5 times greaterthan for r/h = 1 The smaller the radius of curva-ture of the fillet, the nearer the critical compressivestress is located toward the center of the fillet.

Figure 13 illustrates the tangential boundarystresses occurring on the perimeter of a square aspecial case of a rectangle) for the three types ofinitial stress, from a mathematical development byGreenspan. 9) It is interesting to note that themaximum compressive stress on the boundary oc-curs at the filleted corner, and that for the case ofhydrostatic pressure the maximum stress concentra-tion has a value of 4.57 as compared to a circleunder the same conditions. As the initial horizontalstress decreases, the back stress becomes more ten-sile but is never greater than about + 0.8.

I I 1I I I I

L~~ ~__I__L_ _L_ _a 4


24/50


No lateral restraint(Poisson s ratio for rock is taken to be 0.25 in this illustration.)

Fig 73 Tangential Stress oncentration on the Boundaryof a Square Opening in an Infinite Plate for the

Three Initial States of Stresse. Summary

In order to apply the results obtained to themining of long single openings in undergroundmines where the rock formations approach ahomogeneous elastic medium, it is necessary tocompare the stress concentrations for the differentshapes of opening for each of the three initial statesof rock pressure.

No Lateral Restraint. In the case of a mineopening introduced in a unidirectional force field no lateral restraint), two important facts can bedetermined: 1) To reduce the maximum stressconcentration around an opening having a height-to-width ratio greater than unity, the openingshould approximate an ellipse. 2) To reduce themaximum stress concentration around an openinghaving height-to-width ratio less than unity, theopening should approximate a rectangle withrounded corners. Figure 14 gives a comparison ofstress concentrations for different shape openingsin a unidirectional field. Stress concentration de -creases with an increase in h/w. It is observed thatwhen h/w = 1, an ellipse becomes a circle and arectangle becomes a square. A rectangle with r/h =1 2 is an ovaloid; and when h/w = 1, this also be-comes a circle. Figure 15 illustrates the shape ofopening which is most favorable for a given condi-tion of major-to-minor axis ratio and angle of in-clination 8) of the major axis with the horizontal

Hydrostatic pressure

s Alm

Laterally restrained

ttttti=

5 6 7 8 9Maximum stress concentration

Fig. 14. Effect of Shape of Opening on Maximum StressConcentration - Unidirectional Stress Field

where S = 1 and S = S 3 st for S = 0 differsvery little from st for Sz = Sy/3 . It can be seenthat this information agrees with that given inFig. 14.

The two important factors that cause highstress concentrations around openings in a unidirec-tional stress field are a height-to-width ratio lessthan unity and sharp corners on the horizontal axisof the hole.

The maximum stress concentration is related tothe maximum compressive stress which exists in thevicinity of the mine opening, in general found insome portion of the immediate ribs. It is necessaryto choose a shape of opening such that the com-pressive stresses induced are not sufficient to causecrushing of the ribs or any other type of failure ofthe opening.

A stress concentration occurs at the middle ofthe back and floor which is tensile in nature. It isapproximately equal in magnitude to the initialvertical pressure for the different shapes of open-ings. This stress may cause roof failures in manyinstances where the strength of the roof as a unitis less than the tensile stress concentration. In thechoice of openings, the shape which induces theleast critical compressive stress has been described.This choice is based not only on the qualificationthat the critical compressive stresses be minimized,but also on the fact that the shape which inducesthe least critical compressive stress generally in-duces the least critical tensile stress. The least


25/50


8

9 3 4

Mayor toFig. 15. Comparison of Critical Compressiv

critical compressive stress induced by an openingin a unidirectional stress field occurs in the case ofan ellipse with h/w greater than 5.

Laterally Restrained In choosing the mostdesirable shape of opening to be introduced intoa laterally restrained plate in a uniform stressfield we assume that S. = Sy/3 This is equivalentto assuming that v= 0.25 for the rock in the vicinityof the hole, since S S in the case of alaterally restrained plate.

Actually, the tangential stress pattern aroundan opening in a laterally restrained plate does notdiffer greatly from the tangential stress for a platewithout lateral restraint, with the exception thatthe magnitude of the tensile stress is decreasedmarkedly.

The trends of the critical compressive stresswith the height-to-width ratio are plotted in Fig.15. Ovaloids rectangles where r/h = 1/2 ave notbeen included. It can be seen by interpolation,noting that an ellipse and ovaloid are equivalentwhen w = h that for height-to-width ratios lessthan about 1 the ovaloids rectangles with maxi-mum radius fillets) are the preferred shape. Forh/w between 1 and 4 the most favorable shape isthat of an ellipse. For h/w greater than 4 ovaloidsare again the most favorable shape. The lowest

5 6 7 8minor axis ratio R/he Tangential Stress for Rectangles and Ellipse

possible critical compressive stress is induced by anellipse h/w = 3.

Hydrostatic Pressure Under conditions of hy-drostatic pressure the ovaloid induces the smallestcritical stress throughout the range of h/w Nextbest shapes are an ellipse for h/w less than 2 anda rectangle with large radius fillet for h/w greaterthan 2. This is shown in Fig. 16 The least pos-sible critical compressive stress is induced by anopening of circular cross section.5 Stress istribution around Multiple Openings

There are many instances underground wheremine openings occur sufficiently close together thatthe introduction of one opening affects the stressconcentrations around another. This condition is ofprimary importance, because stress concentrationsare increased when two or more openings are inclose proximity. In the case of multiple openings,interest is again centered upon the points of maxi-mum stress concentration, as well as the stressdistribution in pillars formed by two or more open-ings the relationships between stress concentra-tions, and the size and shape of pillars.

In a specific application to mining operations,the results from a study of multiple openings maybe used to estimate the stress distribution in pillarsin order to show what factors cause high local


26/50


stresses and how local stresses may be reduced bythe proper design of mine openings.

To date, there have been only a few solutionsof the problems of stress distribution around mul-tiple openings. Solving these problems by the theoryof elasticity involves very complex equations even

Major-lo-minor ox s ratio R/hFig 16 Comparison of Critical Compressive Tangential Stress for

Rectangles and Ellipse under Conditions of Hydrostatic Pressure

for the most simple geometric shapes; thus only afew have been made. The photoelastic method,however, lends itself readily to the solution of theseproblems.

In addition to the simplifying assumptionsnecessary for the solution of single opening prob-lems by the theory of elasticity or by photoelasticmethods, the further assumption is made for mul-tiple openings that each pillar between openings isuniform in width and height and is long in compari-son to its width and height. In these problems, as inthose solved previously, solutions were obtainedassuming a condition of plane stress. This producedresults which are approximately equal to the actualconditions existing around mine openings which arein a state of plane strain. The results differ onlyby infinitesmals.) With these assumptions the stressdistribution in pillars or around one of the openingsin underground mines can be determined by anelastic or photoelastic study of the stresses existingaround two or more openings in wide, thin plates.

Only two states of initial stress pressure)within the earth will be considered here. They arethat of 1) hydrostatic pressure, and 2) no lateralrestraint. These two states represent the usuallimits of stress expected underground and werechosen because the stresses existing within the

earth where mine openings are introduced shouldoccur somewhere between these extremes.a. CircularOpenings

The first problem considered is that of two cir-cular openings and the pillar left between them.Ling 12 ) solved this problem analytically. He de-termined the critical compressive stresses withinthe pillar on horizontal diameters at the boundaryof the holes). For a ratio of pillar width to open-ing height P h 2, it can be seen that thesestresses differ little from those for a single opening.The differences in st between small P h and largeP h is less than 1 see Table 4). The critical com-pressive stresses range between -3.26 and -2.99

Table 4Critical Compressive Tangential Stress for a Pair of Circular Holes

Sy= 1; Solution by Theory of Elasticity)P h Critical iY -3.26 -2 3.02 2 2.99 24 -3.00 -2P h = the ratio of pillar width to opening heightApplied stress field is perpendicular to the line of centers of the holes.Table 4 is after Panek.

.89.41.16.05

for the case of no lateral restraint, and between-2.89 and -2.05 for hydrostatic pressures; Sy-1).

Duvall 6) solved the problem of two circularopenings which were placed in a unidirectionalstress field supplied perpendicular to the line ofcenters of the two holes, as indicated in Fig. 17.The stress concentrations were determined photo-elastically at points A, B and C, as the opening-to-pillar width ratio was varied. The stress concen-tration at position B pillar rib) proves to be themaximum within the plate. The stresses at A andB are tangent to the boundary of the opening andare compressive in nature. The stress at position Cis tangent to the boundary, equal in magnitude tothe applied load S,), and of opposite sign to theapplied load. An increase in the ratio of opening-to-pillar width causes an increase of stress concentra-tion at A and B. The stress concentration is alwaysgreater at B than at A, and the rate of increasewith the increase of opening-to-pillar width is alsogreater at B than at A. Table 5 lists the results ofphotoelastic studies of the plate with two circularopenings. These results compare favorably withthose given by theoretical analysis. In addition,they show that for P h less than 1 2 a width of


27/50


Ratio of opening width to pill r width5

3

2

3

I

/ 2 3 4Ratio of opening width to pillor width

0 / 2 3 4

t t

C OBC

a) Two openings

o Position A* Position B* Position C* Position E E D40

b) Three openings

/1OBCODEO 0

C) ive openings

O

Fig 17 Stress Concentration as a Function of the Ratio ofOpening Width to Pillar Width in an Applied Stress

Field Perpendicular to Line of Centers

Distance from edge of pillar in units of p i ar widthFig 18 Distribution of Shear Stress in Pillar Formed by

Two Circular Holes - Applied Stress FieldPerpendicular to Line of Centers

Table 56 Stress Concentration for a Plate Containing Two Circular Openings

(Solution by Photoelasticity)Ratio of Opening Width Stress Concentration t Positionsto Pillar Width A B C0.530 2.97 2.97 0 97.808 3.03 3.03 0 911.66 3.08 3.20 0 933.01 3.13 3.61 0 96

3 56 3 27 3.97 1.074.74 3.42 4.61 -1.04Applied stress field is perpendicular to the line of centers of the holes.

t Table is after Duvall.

opening-to-pillar width ratio of greater than 2, thec stress concentration at the pillar rib increases veryBOA rapidly.

A plot of the shear stress concentration againstthe distance in from the boundary of the pillar isI I I shown graphically in Fig. 18 for varying ratios ofopening-to-pillar width. The distribution of shearstress in the pillar becomes more nearly uniform asthe opening-to-width ratio becomes large. Thisindicates that for a small pillar formed by twolarge openings, the average stress in the pillar isalmost as large as the maximum stress.

The next problem considered is that of threecircular openings in a plate (centers lying on astraight line). Duvall ) solved this problem usingSt photoelastic methods for the case of no lateral re-straint. A load was applied in the same manner asfor the problem of two circular openings, and the1C0 stress concentrations determined at points A, B, C,D and E see Fig. 17) as the ratio of openingwidth to pillar width was varied. Table 6 gives theI I data pertaining to this series of models. The stressconcentration at position C (inner ribs of pillars)is the maximum which occurs within the plate. Thestresses at A, B and C have the same sign as the

5

v


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ILLINOIS ENGINEERING EXPERIMENT ST TIONapplied stress, that is, compression when the appliedstress is compressive. The stresses at D and E havethe opposite sign to the applied stress, that is,tension when the applied stress is compression.

Figure 17 shows the relationship between theopening-to-pillar width ratio and the stress concen-tration at positions A, B and C for the three cir-cular openings. The stress concentrations at C arenot only largest, but also show the greatest increasewith increase in the opening-to-pillar width ratio.Stress concentrations at C range from 3.21 foropening-to-pillar width ratio of 1.08, to 5.12 for

TableStress oncentration for a Plate ontaining Three ircular Openings

(Solution by Photoelasticity)Ratio of Opening Stress Concentration at PositionsWidth to Pillar A C DWidth

1.08 3.21 3.21 3.212.22 3.47 3.56 3.742.61 3.60 3.88 4.084.76 3.80 4.92 5.12

Applied stress field is perpendicular to the line ofTable 6 is after Duvall.

0 810.930.94-1.05

centers of

-1.02-1.02-1.02-1.05the holes.

opening-to-pillar width ratio of 5.12. This is a size-able increase over the stresses induced by two cir-cular openings.

Capper (3 also studied the problem of three cir-cular holes in a plate subjected to a stress perpen-dicular to their line of centers. His solutions wereobtained by photoelastic methods. The case of aplate whose edge was at a distance of two radiifrom the center of the outside hole was investi-gated. Green1 ) solved a similar problem theoreti-cally in which he assumed an infinitely wide plate.Duvall s photoelastic solutions utilized plateswhere the plate width was such that the distancefrom the edge of the plate to the center of the out-side hole was equal to or greater than four timesthe radius of the outside opening. The three solu-tions agree if the differences in edge conditions aretaken into account.

Another case which has been studied is that offive circular holes forming four pillars see Fig. 17).The problem has been solved by Duvall. ( 6 ) Theload was applied perpendicular to the line ofcenters, and the stress concentrations at positionsA, B, C, D, and E were determined photoelastically.Results are given in Table 7. The stress concen-trations produced on the boundary of the circlesat the ends of the diameters perpendicular to theline of centers of the openings are approximatelyunity; they are of opposite sign to the applied

Table 7Stress Concentration for a Plate Containing Five Circular Openings(Solution by Photoelasticity)

Ratio of Opening Stress Concentration at PositionsWidth to Pillar A C DWidth1.07 3.29 3.29 3.29 3.29 3 292.21 3.63 3.72 3.89 4.03 4.032.96 3.53 4.08 4.22 4.39 4.394.35 3.96 5.12 5.22 5 28 5 28

Applied stress field is perpendicular to the line of centers of the holes.Table 7 is after Duvall.

stress, that is, tension occurs when the appliedstress is compression. Figure 17 shows the relationbetween the opening-to-pillar width ratio and thestress concentrations at positions A, B, and E. Thedifference between the stress concentration for posi-tions E, D, C, and B is small, but between B and Ait is large. Thus for a large number of openingsthe maximum stress concentrations in all but theoutermost pillars is nearly uniform, being less in thepillars near the side walls than in the central pillars.

The distribution of shearing stress through thecentral pillars at the line of centers, Fig. 19, indi-cates that the maximum stress concentration pro-duced in the pillars does not increase as rapidly asthe average pillar stress with an increase in room-to-pillar width. Therefore the average stress in thepillar is very close to the maximum stress for largeratios of room-to-pillar width.

The theoretical stress distribution has beenstudied by Howland for an infinitely wide platecontaining an infinite row of circular holes, theplate being subjected to a uniform stress perpen-dicular to the lines of centers of the holes no

Distance from edge of pillar in units of pillar widthFig 19 Shear Stress Distribution in Central Pillars Plate

Containing Five Circular Openings in an AppliedStress Field Perpendicular to Line of Centers


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Bul 430. STRESSES AROUND MINE OPENINGS IN SOME SIMPLE GEOLOGIC STRUCTURES

lateral restraint). 10 ) Duvall s experimental photo-elastic analysis of the stress distribution for fivecircular holes in a plate with edges a finite distance4 times the radius of a hole) from the center ofthe outside hole, which was done under similarconditions, was found to be in close agreement withHowland s theoretical study. Howland found thestress concentration at the end of the horizontal

:

22 3 4 5Number of PillarsFig 20 Relation between Maximum Stress Concentration

and Number of Pillars for Ratio of Opening Widthto Pillar Width of 4 0

hole diameters to be 3.24 when the ratio of openingwidth to pillar width was 1.0, and Duvall foundthat the stress concentration in the center pillars forfive openings was 3 29 for opening-to-pillar widthratio of 1.07. This agreement indicates that fivecircular holes approach the case of a row of aninfinite number of holes in an infinite plate.

By plotting the maximum stress concentrationagainst the number of pillars for a given ratio ofopening-to-pillar width, a curve is obtained asshown in Fig. 20. From the curve it is apparentthat for five openings the maximum stress concen-tration in the pillars has reached an asymptoticvalue and the addition of more pillars would no tincrease the maximum stress concentration appreci-ably. Figure 21 indicates that the stress concentra-tion in the central pillars for five openings ap -proaches the average stress concentration in thepillars for opening-to-pillar width ratios greaterthan 4.

b Ovaloidal OpeningsThe stresses around two ovaloidal openings

(height-to-width ratio of openings equal to 0.5)were investigated by Duvall. 6 ) A load was appliedperpendicular to the line of centers of the openingsas shown in Fig. 22, and the stress concentrationsproduced were measured by means of the photo-elastic method. Table 8 gives the data for this seriesof tests. The stress at position C is of oppositesign to the applied stress (that is, tension occurs ifthe applied stress is compression) and is somewhatless than the applied stress. The stresses at posi-tions A and B are of the same sign as the appliedstress. The stress concentration at point B is themaximum produced on the boundary of the open-ings. The relation between the opening-to-pillarwidth ratio and the stress concentrations at posi-tions A and B is given in Fig. 22. For small open-

aaaaaa

aa

4Ratio of opening width to pillar widthFig 21 Maximum Stress Concentration in Pillars Formed by

Circular Openings as a Function of the Ratio OpeningWidth to Pillar Width in an pplied Stress Field

Perpendicular to Line of Centers

ing-to-pillar width ratios it can be seen that twocircular openings produce a much smaller stressconcentration than two ovaloidal openings, whereasfor large opening-to-pillar width ratios the stressconcentrations are almost the same for the twoshapes of opening, although circular openings arestill slightly favorable.

With five ovaloidal openings (height-to-widthratio of openings equal to 0.5), the load was applied

o Five circular openings* Three circular openingso Two circular openings

* omputed ver ge stress concentr tion npill rs for infinite number of openings

O0


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atioof opening width to pillar wi th as in the previous experiments, and the stress con-

i0i

i

Ratio of opening width to pillar width circular openings.Fig 22 Stress Concentration as a Function of Opening In the case of a plate containing five ovaloidal

Width to Pillar Width Ratio in an Applied Stress In the case of a plate containing ve ovloidField Perpendicular to Line of Centers openings height-to-width ratio equal to 2.0 , the

centrations produced were evaluated photoelasti-cally. Figure 22 indicates the manner of loading andpositions A, B, C, D and E where stress concentra-tions were determined. Table 9 gives the results ofthe tests. The stresses at positions A, B, C D andE have the same sign as the applied stress. Thestresses on the boundary at the vertical axis of theopenings are of slightly less magnitude and of op-

TableStress oncentration for a Plate ontaining Two Ovaloids

h/w = 0.5; Solution by Photoelasticity)Ratio of Opening Width Stress Concentration at Positionto Pillar Width A C1.01 3.94 4 00 0 812.05 3.94 4 20 0 732.76 4.19 4.43 0 694.27 4.22 4.81 0.70Applied stress field is perpendicular to the line of centers of the holes.Table 8 is after Duvall.

Table 9Stress oncentration for a Plate ontaining Five Ovaloids

h/w = 0.5; Solution by Photoelasticity)Ratio of Opening Stress Concentration at PositionWidth to Pillar A B C D EWidth

1.03 3.90 3.90 3.90 4.05 A 72.09 4.09 4.50 4.61 4.70 4 7 93 40 4.41 5.02 5.40 5.47 5 564 28 4.66 5.67 5.93 6.10 6.10Applied stress field is perpendicular to the line of centers of the holes.Table 9 is after Duvall.

posite sign to the applied stress. Figure 23 illus-trates the relationship between the opening-to-pillar width ratio and the stress concentrations atpositions A, B, and E. The values of stress con-centrations at positions C and D lie between thoseof the stress concentrations at B and E; thus theyhave been omitted. The maximum stress concentra-tion occurs at position E, ranging from 4.17 for anopening-to-pillar width ratio of 1.03, to 6.10 for anopening-to-pillar width ratio of 4.28. This is con-siderably higher than for the. case of five circularholes in the same situation.

The shear stress distribution through the centralpillars along the line of centers of the openings wasdetermined and plotted as a function of the distancefrom the edge of the pillar for the different opening-to-pillar width ratios. These curves Fig. 23 showthat stress distribution in the pillars becomesmore nearly uniform as the opening-to-pillar widthra.in hbenmes larcger this wtss seen in the case of


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Bul 430 STRESSES AROUND MINE OPENINGS IN SOME SIMPLE GEOLOGIC STRUCTURESload was applied as in previous experiments, andthe stress concentrations produced at positions A,B, and C see Fig. 22) were determined photoelas-tically. At position C the stress is tangent to theboundary, it is of slightly less magnitude than theapplied stress, but of opposite sign. The stresses atpoints A and B are tangent to the boundary and ofthe same sign as the applied stress. Maximum

0000000

5~

0a*0

L .1 C . .J. 0Distance from edge of Pillar in units of pillar width

Fig. 23. Shear Stress Distribution in Central Pillars PlateContaining Five Ovaloidal Openings Height-to-Width

Ratio = 0.5) in an Applied Stress Field Perpen-dicular to Line of Centers

stress concentration occurs at A. The relations be-tween the stress concentrations and the opening-to-pillar width ratio are shown in Fig. 22 and in Table10. It is interesting to note that the stress concen-trations for five ovaloidal openings with height-to-width ratio of 2.0 are less than the correspondingstress concentrations for five circular openings whenboth are in a unidirectional stress field.c Rectangular Openings

Panek1 1)

performed an extensive series of ex-periments with plates containing two rectangularopenings with corner fillets. The ratio of the majorto minor axis R/h), the ratio of the radius offillets to the minor axis r/h), the angle of inclina-tion of the major axis with the horizontal 8), andthe ratio of the pillar width to the minor axis P/h)were all varied to determine their effects upon thestress concentrations around the openings; the pho-toelastic method was used.

It was found that the pillar width had com-paratively little effect on the critical tensile stress.Decreasing the pillar width from P/h = o to P h- 1 adds a critical compressive stress concentrationabout 1.5 to the original stress concentration, whenR/h = 3 The greater the R/h, the greater this in-crease in stress. R/h is the most important factorcontrolling the critical pillar stress when P/h isgreater than 1. The stresses in the roof and floor ofthe openings appear to be little affected by eitherR/h or P/h. Hence, if the lateral pressure is smalland the roof fails in tension, there will be practi-cally no advantage to be gained by decreasing the

Table 1Stress Concentration for a Plate Containing Five Ovoloidal Openings

h/w = 2.0; Solution by Photoelasticity)Ratio of Opening Width Stress Concentration at Position

to Pillar Width B C1.04 2.81 2.21 0 722.10 3.41 2.93 0.842.79 3.83 3.41 0 964.20 4.67 4.31 -1.02

Applied stress field is perpendicular to the line of centers of the openings.Table 10 is after Duvall.

roof span. Panek observed that the smaller the P/hthe more rapid the increase of st with R/h. WhenP/h is less than 1, the critical stress is influencedat least as much by the pillar width as by the roomwidth; for pillars several times as high as they arewide, P/h is probably more important than R/h.Rectangles with large radius fillets were more satis-factory than those with small radius fillets.d Summary

It has been shown by investigators that thestress concentrations induced by a number of open-ings are greater than those induced by a singleopening of a similar geometrical shape. In general,the stress concentrations increase with the additionof each hole until about five holes are in the plate.Then the stress concentrations remain almost con-stant with the addition of more holes. It can besaid that the stress conditions existing in the centralpillars approximate those existing in the pillarsformed by an infinite number of holes.

As the opening-to-pillar width ratio increases,the average stress concentration in the pillars in-creases at a faster rate than does the maximumstress concentration. This indicates that the averagestress approaches the value of the maximum stressconcentration within the pillar. At about 75 per-cent recovery removal of material), the stress


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concentration and average stress within the pillarare nearly equal.

In most instances a tensile stress is produced inthe roof and floor of the mine openings, approxi-mately of the magnitude of the applied stress. Withthe application of lateral confining pressures, thesetensile stresses diminish very rapidly.

In an ore body, where pillars are all of the sameheight, the pillars in the center of the stope areindicated (theoretically) as under more stress thanthose near the side walls of the stope. The maxi-mum stress concentration in pillars is on or nearthe rib of the pillar, indicating th t failure shouldoccur first at the surface of the pillar.

For a ratio of pillar width to pillar height notless than one, it is more advantageous to decreasethe long cross-sectional dimension of the openingsrather than to widen the pillars if critical pillarstress is too high. If the ratio of the pillar width topillar height is less than one, pillar width should beincreased and possibly room width decreased ifcritical pillar stress is too high.

Duvall 6) gives a very simple derivation of thestresses existing in the pillars formed by the intro-duction of an infinite number of holes in a plateof infinite extent. Holes were all of equal size (anygeometrical shape), equally spaced. The derivationis given here in its entirety since it illustrates asimple, direct approach to a difficult problem. Ifthe plate is stressed perpendicularly to the line ofcenters of the holes, so th t the average stress inthe plate at a great distance from the row of holesis Sg then the load on any one pillar is given by

L = St W, W,) 1)where

Lp = load supported by one pillarSg = average stress in plate t a distance

from row of holesWo = width of openingWp = width of pillart = thickness of plate

If a uniform stress distribution in each pillar ispresumed so th t the average stress within the pil-lar is given by S. than the load on any pillar isgiven also by

L ntWp 2)From Eqs. 1 and 2 the following relation can

be derivedQ W

= 1 V

S,/S, is the stress concentration for the averagestress in the pillars. This is approximately equalto the maximum stress concentration where theratio of the opening-to-pillar width is large.

Insufficient data were obtained to compare thedifferent shapes of openings for the hydrostaticloading. In the case where the initial earth stresscondition is one of no lateral restraint, a compari-son of the maximum stress concentration produced

0

00c;0

0

Percent o recoveryFig. 24 Maximum Stress Concentration as a Function of Percent

of Recovery for Pillars Formed by Five Openings

in pillars formed by five ovaloids having a height-to-width ratio of 0.5, five ovaloids having a height-to-width ratio of 2.0, and five circles is of value.The maximum stress concentration for each of thesethree shapes has been plotted as a function of thepercent mined area or percent recovery in Fig. 24.Also shown is a plot of Eq. 3 which gives the aver-age stress concentration for an infinite number ofpillars. Figure 24 shows th t for less than 50 per-cent recovery the maximum stress concentration isnot greatly affected by the percent recovery, butfor recoveries greater than 50 percent the maximumstress concentration increases more rapidly with anincrease in the percent recovery. The three curvesfor the circular and ovaloidal openings are similarin shape; thus the effect of percent recovery uponthe maximum stress concentration in pillars is inde-pendent of the shape of the openings. The orderof preference for underground mine openings withregard to stress concentrations is 1) ovaloids withheight-to-width ratio of 2.0, 2) circles, and3) ovaloids with height-to-width ratio of 0.5.

From the experimental data contained in Fig.24, the following equation was derived: (

- / 1 2 -K = S + 0.091 n. - 1 g p


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where Equation 3 gives good results or percent re-coveries greater than 75 percent, but Eq. 4 gives aK maximum stress concentration inpiar better approximation over a wider range.pllannrs

S = maximum stretss concentrati on aroundi m sa osinge op nin

R = percent recoveryEquation 4 can be made to fit data for openings

of any shape by choosing the proper value of themaximum stress concentration. For example, in anunidirectional stress field no lateral restraint

= 3.0 for a circle, S = 3.9 for ovaloid having a1

heig t-to-width ratio of 0.5 a-ldio1 having a neight to widtn ratio o z u eom ercent of recoveryparison of the experimental data and results Fig 25 Comparison of the Empirical Equation and theobtained from the empirical equation are given in Experimental DataFig. 25. Curves are drawn from the equation; pointsare from the experimental data.


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IV STR SS S IN SIMPLE STRATIFIED ROOFSOne of the principal difficulties in determining

the stress distribution around actual mine openingslies in attempting to evaluate the effect of complexgeologic conditions upon the stresses. In the pre-vious chapter, some solutions for the stress con-ditions existing around mine workings in solidhomogeneous materials stressed within their elasticlimit were given. It is apparent, however, thatfissures, faults, bedding planes, intrusions, meta-morphism of the rock, or any features which de-stroy the continuous, homogeneous nature of therock, alter the stresses existing around mine open-ings. Solutions taken from the previous chapter andapplied under these conditions would thus be inerror because of the discontinuities introduced bythese geologic conditions.

In general, these geologic factors are not easilyanalyzed. The strengthening or weakening effectswhich they have upon rock and the included mineopenings have not been determined except for a fewfield correlations which have generally been of localnature. There is a definite need for widespread fieldcorrelations of the different geologic factors to de -termine the effect of each individual factor uponthe strength of rock and the stresses around mineopenings. This would simplify any attempt to ex-press the effect of a specific geologic condition uponthe stress around a mine opening quantitatively.

Some very simple geologic conditions have beenapproximated in the laboratory and their effectupon the stress distribution around mine openingshas been evaluated. One example is the case of mineopenings existing in stratified materials which areuninterrupted by faults or fissures, and the beds ofwhich are homogeneous in character. Solutions ofthis problem have been obtained by the use of rockmodels taken from the site of the actual mine open-ing, and more recently, solutions for some simplespecific cases were obtained using the theory ofelasticity.6 Centrifugal Testing to Simulate Stresses Occurringin Rock Beams Underground

An examination of Fig. 2 which was used toillustrate the dome theory, indicates that the im-

mediate roof of the opening, which is rectangular,may be considered as a beam or group of beamsin layers loaded by the material above it and by itsown weight. The material overlying the roof beamsapplies a variable weight to the beams accordingto its rigidity and other properties. In stratifiedmaterials, the roof of a rectangular mine openingmay be considered as a beam loaded by the over-lying material and restrained between the pillarsand the overlying material degree of restraint isalso dependent upon the amount and character ofoverlying material .One method of determining the stresses in theserock beams has been to make small-scale modelsfrom the rock and apply a load similar to that ap-plied to the prototype. To be of benefit, it must bepossible, by the laws of similitude, to calculate fromthe model what the results will be in the prototype.Thus, if the scale is one to ten, and the roof of themodel deflects .01 in. how much will the prototypedeflect? The pr

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Stress Around Mine Openings in Some Simple Geologic Structures - R. D. CAUDLE, G. B. CLARK