IntroductiontoRock MechanicsSecond EditionRichard E. GoodmanUniversity or California at BerkeleyI]WILEYJohnWiley &SonsNew YorkI ChichesterI Bri baneI TorontoI Singapore ....Dedicated tothe memory ofDanielG. MoyePreface lo lhe Firsl EditionBibliography: p.IncludesindexoISBN 0-471-81200-51. Roek meehanics.Library of Congress Cataloging in Publication Data:Goodman, Riehard E.Introduetion to roek meehaniesIRichard E. Goodman.-2nd ed.p. em.Rock mechanics is a truly interdisciplinary subject, with applications in geol-ogyandgeophysics, mining, petroleum, andcivil engineering. Itrelates toenergyrecoveryanddevelopment, constructionof transportation, waterre-sources and defense facilities, prediction of earthquakes, and many other activ-ities of greatest importance. This book introduces specific aspects of this sub-ject most immediately applicable to civil engineering. Civil engineeringstudents, at the advanced undergraduate and beginning graduate leveI, will findhereaselectionof concepts, techniques, andapplicationspertainingtotheheart oftheir field-for example, how to evaluate the support pressure requiredtoprevent squeezingof c1aystoneintunnels, howtoevaluatetheoptimumangleof arockcutthrougha jointed rockmass, andhowtodeterminethebearing capacity of a piersocketed into rock. Students in other fieldsshouldalsofindthisworkuseful becausetheorganizationisconsistentlythat of atextbook whose primary objective is to provide the background and techniquefor solving practical problems. Excellent reference books cover the fundamen-tal bases for the subject well. What has been lacking is a relativeiy short workto explain how the fundamentals of rock mechanics may be applied in practice.Thebookis organizedintothreeparts. Part I, embracingthefirst sixchapters, provides a survey ofthe methods for describing rock properties. Thisinc1udes index properties for engineering c1assification, rock strength and de-formabilityproperties, thepropertiesandbehavior of joints, andmethods ofcharacterizing thestate of initialstress. Modem fracturemechanics has beenomitted but some attention is given to anisotropy and time dependency. Part 2,consisting of Chapters 7, 8,and 9, discussesspecific applications of rock me-chanics for surface and underground excavations and foundations. Part 3 is aseries of appendices. Oneappendixpresentsderivationsof equations, whichwereomittedfromthechapters tohighlight usableresults. 'Thereisalsoathorough discussion of stresses in two and three dimensions and instructions inthe measurement of strains. Appendix 3 presents a simple scheme for identify-ing rocks and mineraIs. It is assumed that the reader has some familiarity withintroductorygeology; this sectiondistills theterminologyof petrologyandmineralogy to provide a practical naming scheme sufficient for many purposesin rock mechanics. Part 3 also inc1udes answers to alI problems, with elabora-tion of the methods of solution for a selected set. The problems presented at theends of each chapter and the worked out solutions in the answers section are av87-34689CIPI. Title. ..:, .. . . .. .. .Copyright1989, by RiehardE. Goodman.All rightsreserved. Published simultaneously in Canada.Reproduetion or translation of any part ofthis work beyond that permitted by Seetions107 and108 of the1976 United States CopyrightAet without thepermission of the eopyrightowner is unlawfuI. Requests for permissionor further information should beaddressed tothe Permissions Department,John Wiley& Sons.TA706.G65 1989624.1'5132-dcl9..~ : ~ ' ' ' ~ : ' ' ' ' : :'. .. .. .. ..... : .... ' ..vi Prtiface lo lhe Firsl Editionvital part of this book. Most of the problems are not just exercises in filling invaluesfor equations offeredinthetext, but trytoexplorenewmaterial. Ialwaysenjoy learning newmaterial in a practical context and thereforehaveelected to introduce new ideas in thisway.Although this is largely a presentation of results already published in jour-nals and proceedings, previously unpublished materiaIsaresprinkled throughthe text, rounding out the subject matter. In almost alI such cases, the deriva-tions in the appendix provide complete details.Thisbook isused fora one-quarter, three-creditscourse for undergradu-ates and beginning graduate students at the University of California, Berkeley,Department of Civil Engineering. Attention is riveted to the problems with littletimespent on derivationsof equations. Appendices I and2 andalImateriaIsrelatingtotimedependencyareskipped. Inasecondcourse, derivationsofequations aretreatedinclass andthemateriaIs presentedhereare supple-mented with the author's previous book Methods ofGeological Engineering inDiscontinuous Rocks (West PublishingCo.) 1976, as welI as withselectedreferences.Iam deeply indebted to Dr. John Bray ofImperial ColIege for illuminatingand inspiring contributions from which I have drawn freely. A number of indi-viduaIs generously loanedphotographsand other illustrations. These includeK. C. Den Dooven, Ben KelIy, Dr. Wolfgang Wawersik, Professor Tor Brekke,Dr. DougalIMacCreath, ProfessorAlfonsoAlvarez, Dr. TomDoe, DuncanWyllie, ProfessorH. R. Wenk etal., andProfessor A. J. Hendron Jr. ManycolIeagues assisted me in selection of material and criticism of the manuscript.The list includes E. T. Brown, Fred Kulhawy, Tor Brekke,GregoryKorbin,Bezalel Haimson, P. N. Sundaram, WilliamBoyle, K. Jeyapalan, BernardAmadei, J. DavidRogersandRichardNolting. IamparticularlygratefultoProfessorKulhawyfor acquaintingmewithmuchmaterial concerningrockfoundations. Iam also very appreciative of Cindy Steen's devoted typing.Richard E. GoodmanPrefaceSince the publication of the first edition in 1980 we have developed a geometricapproach to rock mechanics calIed "block theory." This theory on.thetype of data that comes most easily and ?aturalIya geologlcal mvestIga-tion namely the orientations and propertIes of the Jomts. Block theory formal-izes'procedures for selecting the wisest shapes and orientations excavationsin hard jointed rock and is expounded in abook by Gen hua Shl andmyself,published in 1985, and in additional articles derived from subsequent researchat Berkeley.In preparing this edition my main objective was to incorporate anintroduction to the principIes of block theory and its application to rock slopesand underground excavations.This has been accomplished in lengthy supple-ments to Chapters 7 and 8,as well as in a series of problems and answers.Anadditional objectiveinpreparing thisnewedition wastoincorporatepreviously omitted subjects that have since proved to be important in practice,or that have appeared subsequent to initial publication. In the former categoryare discussions ofthe Q system ofrock classification and the empirical criterionof joint shear strength, both introduced by Barton and co-workers at the Nor-wegianGeotechnicalInstitute(NGI). In thelatter categoryare fundamental,new contributions by Indian engineers Jethwa and Dube on the interpretationof extensometerdatainsqueezing tunnels; analysisof rockbolting usinganexponential formulation by Lang and Bischoff; properties of weak rocksbrought to light by Dobereiner and deFreitas; representation of thestatisticalfrequencyof jointingbyPriest andHudson; anempirical criterionofrockstrength by Hoek and Brown; and development of a "block reaction curve" asamodel for designofsupports inundergroundopenings (analogous tothegroundreaction curveconcept previously presented in Chapter 7). Addition-aHy, several useful figures presenting derived relationships were updated; thesedeal with the directions of stresses in the continental United States summarizedby Zoback andZoback, and therelationship between the rock massrating ofBieniawski, and the"stand-up time"of tunnels.To present this material, I have elected to develop a series ofnew problemsandworked-outsolutions. Thus, totakefuHadvantage of thisbook youwillneedtostudytheproblems andanswers. The statements oftheproblemssometimes contain important material not previously presented in the chapters.And, of course, if you can take the time to work them through yourself, youwill better understand and appreciate thevalue of the material...::CHAPTER6 Deformability Df Rocks 179CIlAPTER 8 Applications Df Rock Mechanicsto Rock Slope Engineering 293CIlAPTER 9 Applications Df Rock Mechanicsto Foundation Engineering 3411xi1955427389555495101141409475415ContentsAPPENDIX 1 StressesIndexAPPENDIX 3 IdentificationDf Rocks and MineraIsAPPENDIX 2 Strains and StrainRosettesCHAPTER 5 PlanesDf Weaknessin RocksAPPENDIX 4 Derivations Df EquationsAPPENDIX 5 TheUseDf Stereographic ProjectionCHAPTER 7 Applications Df Rock Mechanicsin Engineering forUndergroundOpenings 221cHAPTER 3 Rock Strengthand FailureCriteriaCHAPTER 4 Initial Stressesin Rocks and Their MeasurementAnswersto ProblemsCHAPTER1 IntroductionCHAPTER 2 Classification and Index PropertiesDf RocksSymbols and NotationRichard E. GoodmanToday, manyworkersinrockmechanicstendtousecomprehensivenu-merical modeling to study the complex issues relating to the disposal of nuclearwaste, energy storage and conversion, and defense technology. Although thesemodels are powerful, much headway can also be made with simp1er approachesby using statics with well-selected free-body diagrams, elegant graphical meth-ods like thestereographicprojection, andmodest computations facilitatedbymicrocomputers. Ifthere is an overriding purpose in this book, it is to help yousee thesimple truthsbefore trying to takehold of the big numerical tools.viii PrefaceSymbals and NatalianSymbols are defined where they are introduced. Vectors are indicated by bold-face type, for example, B, with lowercase boldface letters usually reserved forunitvectors. Thesummation convention isnot used. Matrix notation isusedthroughout, with( ) enclosingone-andtwo-dimensional arrays. Occasion-ally, { }are used to enclose a column vector. The notation B(u) means that Bisa function of u. Dimensions of quantities are sometimes given in brackets, withF=force,L= length, and T= time; for example, the units of stress are givenas (FL-2). Adot over a letter or symbol (e.g., &) usually means differentiationwith respect to time. Some ofthe more commonly used symbols are the follow-ing:DiDoddevEgGGPaKI, m, nInMPan, s, Iunit vector parallel to the dipchange in the length of a diameter of a tunnel or boreholesubscript identifying deviatoricstress componentsYoung's modulus (FL-2)acceleration of gravityshear modulus; also, specific gravity103MPaangle of the leading edge of an asperity on a jointinvariants of stressunit vector parallel to the line of intersection of planesi and jused for different purposes as defined locally, including conductiv-ity (LT-I)andstiffness coefficientsused variously for the bulk modulus, the Fisher distribution param-eter, permeability (L2), 12) - 1Figure 3.18 portrays thiscondition graphicalIy. The buildup ofwater pressurein therock near a reservoir or in an aquifer can cause rock failureand earth-quakes, if therockisinitialIystressednearthelimit. However, earthquakesinduced by reservoir construction and by pumping water into deep aquifers arebelieved to originate from rupture along preexisting faults in determined orien-tations. The mechanism is similar but the equations contain the influence of therelativedirectionsof initial stress as discussedinChapter5(comparewithEquation 5.9). -= qu + [tan2(45 + 1) -1] (3.12)Since the differential stress is unaffected by pore pressure, Equation 3.12 mayalso be written(TI,p - (T3 = qu +(T3 - Pw) [tan2(45 + 1) -1]Solving for Pw, we can calculate the water pressure in the pores or fissures of arock required to initiate failure from an initial state of stress defined by(TI and(T3:orS6 Rock Strength and FailureCriteria3.9 Empirical Criteria ofFailure 89

----------------(a)(b)Figure 3.20 An empirical criterion of failure derivedfrom the ring shear testo(After Lundborg, 1966.) (a) Aplot of equation 3.16; (b) graphical determination of /L"_1_Tp -SiL..- 1a(The JL' has beenusedinplaceofLundborg's JL todistinguishthis from the Mohr-Coulomb coefficient of internaI friction, JL = tancP.)EquatlOn 3.16 can lsobewritten1 1 17p- Si = JL'a + Sj-Si (3.17)so JL' is determinedas theinverseof the slopeof thelineobtainedby (7p - S;)-l asordinateagainst (a)-I asabscissa (Figure3.20b). Table3.4 hsts typical values of Lundborg's parameters. Strengths determined by ring(3.16)JL' a7p= Si + ----!...--:---'a1 +--!-JL_-=-Sr- SiT T(a3, al,p _ 1)qu quAnotherapproachfits datafromtheringshear test (Figure 3.3e) tofindanempiricalequation fora Mohr envelope of intact rock(Lundborg, 1966). Thepeak shear strength (7p )(Equation 3.4) is plotted against ato define the strengthenvelope, Figure3.20a. Lundborg foundthatsuch data definea curved enve-lopewith intercept Si and asymptote Sj, fit bystresses are compressive. In practice, empirical curve fitting is the best proce-dure for producing a criterion offailure tailored to any given rock type. See, forexample, Herget and Unrug (1976). Asatisfactory formula formany purposeswillbeafforded by theunion of atension cutoff, a3= - To, andapower law(Bieniawski, 1974):al,p =1 +N(a3)M (3.15)qu quTheconstantsNandMwill bedetermined by fittingacurvetothe familyofpoints8S Rock Strength and Failure CriteriaFigure 3.19 An empirical criterion of failure defined by theenvelope to a series ofMohr's circles: A, direct tension; B, Brazilian; C, unconfined compression; D, tri-axial compression. __....L.. --'- ---l ---JL....- a3/x Coai,Bieniawski (1968)Cedar City quartz diorite(altered)Pratt, Black,Brown, andBrace(1972)o:;--------->--------'0Calcareous ironore, Jahns (1966)15010070500.":2 30

-501C

1;;Q)> Q)a.Eou3.10 The E.ffect ofSize on Strength 91Specimen edgelength,mFigure 3.21 Effect of specimen size on unconfined compressive strength. (AfterBieniawski and Van Heerden, 1975.)2L-. ---J'-- ---'- -:-":- -'- --:'':- -::'O strength. Figure 3.21 demonstrates this pattern ofbehavior in a summary ofthetestsoncoaI andironore, aswellastestson analteredandfissuredquartzdiorite by Pratt et aI. (1972). This clever series of tests included specimens ofequilateral triangular cross section 6 ft (1.83 m) on edge, and 9 ft (2.74 m) long,loadedvia stainlesssteel flat jacks in averticalslot atone end. Figure3.22ashows a specimen being freed by drilling a slot inclined at 60 and Figure 3.22bshows the surface of the specimen, with completed slots, jacks in place on oneend, and extensometers positioned for strain measurements on the surface. Thequartzdioritetesteddisplayedalargesizeeffect becauseit contains highlyfractured plagioclase andamphibole phenocrysts in a finer-grained groundrnass with disseminated clay;lhe porosity of thisrock is 8-10%.The influence of size on shear and tension tests is less well documented butundoubtedlyas severefor rocksthat containdiscontinuities. Thesubject ofscaleeffect will beconsideredfurther inChapter7 inthecontext of under-ground openings.Table 3.4Some Values ofConstants for Lundborg'sStrengthEquationSi Sf/L' (MPa) (MPa)Granite 2.0 60 970Pegmatite 2.5 50 1170Quartzite 2.0 60 610Slate 1.8 30 570Limestone 1.2 30 870Rocks are composed of crystals and grains in a fabricincludes fissures; understandably, rather largesamplesarereqmredtoobtamstatlstl-cally complete collections of all the components that influence strength.":'henthe size of a specimen is so small that relatively few cracks are present, fal1ureis forced to involve new crack growth, whereas a rock mass loaded through alarger volume in the fieldmay present preexistingcriticaIlocations.Thus rockstrength issize dependent. CoaI, altered gramtlcrocks, andother rocks with networks of fissures exhibit the greatest degree of Slze depen-dency, the ratio offield to laboratory strengths sometimes attaining values of 10or more. .Afewdefinitive studies havebeenmadeofsizeeffect incompresslvestrength over a broad spectrum of specimen sizes. Bieniawski (1968) reportedtests on prismatic in situ coaI specimensup to1.6x 1.6x. 1 m, prepared bycutting coaI from a pillar; the specimens were then capped wlthconcreteand loaded by hydraulic jacks. Jahns (1966) reported results of sImIlar tests oncubical specimens of calcareous ironore; thespecimens werepreparedmeans of slot cutting with overlapping drill holes. Jahns a specI-men size such that 10 discontinuities intersect any edge.Larger speClmens aremore expensive without bringing additional size reduction, while smaller speci-mens yield unnaturally high strengths.Available data are too sparse to Jahn's recommendationfor all rocktypes but it does appear that there .ISgeneraIly a size such that larger specimens suffer no further decrease msheartendtobeslightlyhigherthancorrespondingstrengthsdeterminedbytriaxial tests.3.10The Effect ofSize on Strength90 Rock Strength and FailureCriteria3.11 Anisotropic Rocks 93(3.19)(3.18)(3.19a)(3.18a)Si = SI - Sz[cos2(t/J- t/Jmin,sWtan~ = 0.600- 0.280 cos2(t/J- 50)Si = 65.0- 38.6[cos2(t/J- 30)]3 (MPa)andandVariationof compressivestrength accordingtothedirection of theprincipalstresses is termed "strength anisotropy." Strong anisotropy is characteristic ofrockscomposed of paralIel arrangementsof fiat mineraIslikemica, chlorite,and clay, or long mineraIs like hornblende. Thus the metamorphic rocks, espe-cialIyschist andslate, areoftenmarkedlydirectional intheirbehavior. Forexample, Donath(1964) foundtheratio of minimumtomaximumunconfinedcompressive strength of Martinsburg slate to be equal to 0.17.Anisotropy alsooccurs in regularly interlayered mixtures of different components, as in bandedgneisses, sandstone/shale alternations, or chert/shale alternations. In alI suchrocks, strength varies continuously with direction and demonstrates pro-nounced minima when the planes of symmetry of the rock structure are obliqueto the major principalstress.Rock masses cut by sets of joints also display strength anisotropy, exceptwhere the joint planes lie within about 30 of being normal to the major principalstress direction.The theory of strength for jointed rocks is discussed in Chap-ter 5.Strength anisotropy can be evaluated best by systematic laboratory testingofspecimens drilledindifferent directions fromanorientedblocksample.Triaxial compression tests at a set of confining pressures for each given orienta-tionthendeterminetheparametersSi and~ asfunctionsof orientation. Ex-panding on a theoryintroduced by Jaeger (1960), McLamore (1966)proposedthat bothSi and ~ couldbedescribedas continuousfunctions of directionaccording towhere SI. Sz, TI. Tz, m,' andn are constantst/J is theangle betweenthedirectionof thecleavage(or schistocity,bedding or symmetry plane) and the direction(TIt/Jmin,s andt/Jmin,1> are the values of t/J corresponding to minima in Si and~ ,respectivelyFor aslate, McLamore determined that frictionandshear strength inter-cept minima occur at different values of t/J, respectively 50and 30. Thestrength parameters for theslate are3.11Anisotropic RocksFigure 3.22 Large uniaxial compression tests conducted in-situ by TerraTekon Cedar City Quartz Diorite. (a)Drilling a line of 1-l/2-inch diameter holesplunging 60 to create an inclined slot formingone side of the triangular prism"specimen."(b)A view of thetest site showing ftat jacks at one end and ex-tensometers for relative displacement measurement duriDg loading. (Courtesyof H. Pratt.)92 Rock Strength and Failure CriteriaReferencesBieniawski, Z. T.(1967a) Stability concept of brittle fracture propagation in rock, Eng.Geol. 2: 149-162.Bieniawski, Z. T. (1967b) Mechanism ofbrittle fracture ofrock, Int. J. Rock Mech. Min.Sei. 4: 395-430.Bieniawski,Z. T. (1968) The effect of specimen size on compressive strength of coai,Int. J. Rock Mech. Min. Sei. 5: 325-335.Bieniawski, Z. T. (1972) Propagationofbrittlefracture inrock, Proceedings, 10thSymposiumon Rock Mechanics(AIME), pp. 409-427.Bieniawski, Z. T. (1974) Estimatingthestrengthof rockmateriaIs, J. SouthAfricanInst. Min. Metal/. 74:312-320.Bieniawski, Z. T. andBernede, M. J. (1979)Suggestedmethodsfordeterminingtheuniaxial compressive,strength and deformability ofrock materiaIs, for ISRM Com-mission on Standardization of Laboratory and Field Tests, Int. J. Rock Mech. Min.Sei. 16 (2).Bieniawski, Z. T. andHawkes, I. (1978) Suggestedmethodsfordeterminingtensilestrength of rock materiaIs, forISRMCommission on Standardization of LabandField Tests,Int. J. Rock Mech. Min. Sei. 15: 99-104.Bieniawski, Z. T. andVan Heerden, W. L. (1975) Thesignificance of in-situ tests onlarge rock specimens, Int. J. Rock Mech. Min. Sei. 12: 101-113.Broch, E. (1974) The influence ufwater on some rock properties, Proc. 3rd Cong. ISRM(Denver), Vol. 11 A, pp. 33-38.Brown, E. T., Richards, L. W., and Barr, M. V. (1977) Shear strength characteristics ofDelabole slates, Proceedings, Conference on Rock Engineering (British Geotechni-cal Society,Vol. 1, pp. 33-51.Cook, N. G. W. and Hodgson, K. (1965) Some detailed stress-strain curves for Rock, J.Geophys. Res. 70: 2883-2888.References 95In general, the entire range of l/J from Oto 90 cannot be well fit with one set ofconstants since the theory (Equations 3.18 and 3.19) would then predictstrength at l/J = 0 to be less than the strength at l/J = 90; in fact, the strengthwhenloadingisparallel toslatyc1eavage, schistosity, orbeddingisusuallyhigherthanthestrengthwhentheloadingis perpendiculartotheplanesofweaknesswithintherock. (CompareFigures 3.23aandb.) For oHshale, arepetitive layering of marlstone and kerogen,McLamore used one set of con-stants for the region 0 ~ l/J 2 - O"h,maxor1 6 40"h,max-> -l( KO"vNow,O"h 20"h,max 2 = --=-:....;:.,;=--=N+1K O"h,min +O"h,maxO"VSubstituting (8) in (7) gives(5)(7)(8)IIIlIII(w.(-1)+ m. (3) Wo+(i) o(-1)) = (1 \m (-1) + Wo (3) m(3) + m (-1) O t)Equation 4.15The derivation of this relation, for the case withTxz =O, is given by Jaeger andCook(opocit. Chapter 1), Section10.4. Theresultsfor fi, fi, and 13canbefound in their Equation 26, which is therefore equivalent to (4.15) withTxz=O.Equation4.13. d3 I I t d fromtheKirsch solution ThestressconcentratlOns- 1 an were ca cu a e .(Equation 7.16) in the derivation to Equation 4.7. Substltute: or O" . andO"h in for O" vert O" for 0"0A 0"0 RlorO"oB O"h max 11 honz ,m O,w , , , ,Equation4.14(1 ) (-1 3)i i is the inverse of 3 - 1k==----r-....I..--+- ... x/

To findthe displacement inducedbyan additionalstresscomponent Txz ' pro-ceed as follows(see diagram). Consider a set ofprincipal stressesO"x'= -Txz 'O"y' = O,andO"z' = +Txz where Ox' is 45 clockwise fromOx. With respect to thex' axis, the displacement lid along a line inclined 8 with Ox is found from (4.15)bysubstituting-Txz for O"x, Ofor O"y, Txzfor O"z, for Txz , and 8 +45for (),436 Derivations 01EquationsIntroducing thesesubstitutions in Equation 4.15gives(1 - v2 dv2)t!.d = -Txzd[1+2 cos(90 +28)] -E- + E(1 - v2 dv2)+Txzd[l - 2 cos(90+28)] -E- + Eord(1 - v2) .t!.d= TxzE 4 sm 28= Txzf4Equation 4.17Thestress-strainrelationsfor isotropic, linearlyelsticbodiesarestatedinEquation 6.1. Theholebeing parallel toy, onthebottomsurface of thedrillholeCTy =Txy =TyZ = O. Then, the independent variables in (6.1) reduce to 3 andthestressstrain relations becomeEquation 4.23 437Equation4.21We needtoderiveaformulafor theradial displacement of apoint near acircularcreatingthat holeinthesurfaceof aninitiallystressed lsotroplc, elashcrock mass. If therocksurface is normal to y, CTy==TyZ = TyZ = O. The state of stress in the rock near the surface is, therefore, one ofplane stress. Equation 7.2a gives the radial displacement of any point around acircular hole in an initially stressed rock mass corresponding to plane strain. Asdiscussed with the derivation to Equation 7.2 later on in this appendix, there isasimple connection between planestress and plane strain. To convert a for-mula derived for plane strain to one correct for plane stress, substitute(1 - v2)Einplaceof E, andv/(1 +v) inplaceofv. G = E/[2(1 +v)] isunaffected. Making thissubstitution in (7.2a) givesa22(1+v) { [ ( v) a2] }Ur = -;: 4E (PI + P2)+ (p, - P2) 41 - 1 + v - y2 cos28 (1)Simplifying andsubstitutingCTx in place of P" andCTzin place of P2 yields(2)whereEquations 3, 4, and5 correspond to thethree rows of(4.17) ifthestrains arezero before deepening the drill hole.1 vex = ECTx - ECTz-v 1e z= ]fCTx + ECTz2(1+v)'rzx = E TzxMultiplying (2) byv, and adding (1) and (2) gives1 - V2-E- CTx= ex +vezMultiplying (1) byv, and adding (1)and(2)gives1 - v2-E- CTz= vex +ez(1)(2)(3)(4)(5)I IIIIIa2H== 4- (1+v)-y2Then using thesame procedure as in the derivation to Equation 4.15,we findthe influence of ashear stress:1 a2Ur== E-;: HTxz sin 28 (3)Arranging (2) and (3) in the form of Equation 4.21determines fi, fi, and hasgiven.Equation 4.23Leeman (1971, Chapter 4) presents complete formulas for the stresses around alongcircularholeof radius aboredinanisotropic, elasticmediumwithaninitial state of stress. The initial stress components will be represented here by438 Derivations of Equationsx', y', Z' subscripts with y' parallel to the axis of the bore asshown in Figure4.16. For apoint onthewall ofthebore (r =a) andlocatedbyangle ()counterclockwise from x' asshown, Leeman's equations reduce to

z ='Y- S (Tv (3)Since(Tv(Y = O) = qA= 'Y_4k tancf>qsFinally, Pb= (Tv(y = t),givings ( (4ktancf )Pb= 'Y- 'Y- q e-4ktaoq,yls4ktancf> sSimplifying yields(b)k = cot2(45 +cf>/2)= 0.406tis = 0.67, q= 21kPaThen (6) givesPb= 21e-O.50?= 12.65 kPa (=1.83 psi)If s = 1.5 m, the forceper support isT= S2 (12.65) = 28.5 kN (=6400 lb).11. (a) With self-weight, the free-bodyequilibrium givesS2d(Tv+41'sdy = 'Ys 2dy (1)Substituting as in the answer to ProblemlOCa) givesd - ( 4k tancf> )d(Tv - 'Y- s (Tv Y (2)S2d(Tv +4T sdy = O (1)Substituting in(1)T= (Thtancf> (2)and(Th = k(Tv (3)yieldsd(Tv 4(4) -=--ktancf>dy(Tv sSolving gives(Tv = A e-4ktanq,yls(5)when y = O, (Tv = q, givingA = q.Socketa I VolumePendT(Tmax,concrete(m) (m) (m3)PendiPtotal(MPa) (MPa) (MPa)1.0 8.97 28.2 0.31 2.00 0.24 6.370.9 9.54 24.3 0.255 2.00 0.28 7.860.8 9.94 20.0 0.201 2.00 0.32 9.950.7 10.15 15.6 0.154 2.00 0.38 12.990.6 10.13 11.5 0.113 2.00 0.46 17.680.5 9.85 7.74 0.079 2.00 0.60 25.460.4 9.27 4.66 0.050 2.00 0.82 39.790.3 8.63 2.95 0.024 2.00 1.20 70.74If the concrete has compressive strength equal to 20 MPa, and it is desired toachieve a factor of safety of 2, thesolution with minimum volumesocket is a pierwith radius 0.8 m and length 10 m. These results depend markedly on the choice forIJ- and Ecl E, and, to a lesser extent,on v, andVc'Another solution is to use a pier seated without a socket on the surface of therock, or, if that surface is weathered or inclined,seated inside a socket of enlargeddiameter. The required pier radius is 1.78 m. The most economical choice betweenthe altematives depends on thevolume of the pier passing through thesoil.8. Consider the sandstone roof as a continuous clamped beam. The most criticaI condi-tion is tensile stress at the ends on the upper surface. Using (7.5) with(Th = O, 'Y=150 Ib/ft3, andTo = 2MPa givesL=334 ft =100m. Thisisincreased if(Th # O.However, a beam 200 ft thick with L = 334 ft is too thick for thin-beam theory. (Afiniteelement analysis would be useful in a particular case.)2h9. H= B- 110. (a)Summing forcesinthey(vertical)directionactingonthedifferentialelementgives546 Answers to ProblemsAnswers to Problems 547T..-=--7C---t-L---fL-_-L.__-I..__....L..+--.........._1Ta =22.5, 112.5"U'I = 108.3U'2 = 51.7a =-31.7,58.3U'I =62.4, U'2 =17.6

For ProblemIb:3. For Problemla:(b) T2. (a)(4)"""----............. x lo