FEASIBILITY OF CONSTRUCTING LARGE UNDERGROUND CAVITIES THE STABILITY OF DEEP LARGE-SPAN UNDERGROUND OPENINGS
TECHNICAL REPORT NO. 3-648
Volume II
June 1964
Sponsored by Advanced Research Projects Agency ARPA Order No. 260-62 Amendment No. I
U. S. Army Engineer Waterways Experiment Station CORPS OF ENGINEERS Vicksburg, Mississippi FEASIBILITY OF CONSTRUCTING LARGE UNDERGROUND CAVITIES THE STABILITY OF DEEP LARGE-SPAN UNDERGROUND OPENINGS
TECHNICAL REPORT NO. 3-648
Volume II
June 1964
Sponsored by
Advanced Research Projects Agency ARPA Order No. 260-62 Amendment No. I
U. S. Army Engineer Waterways Experiment Station CORPS OFENGINEERS Vicksburg, Mississippi
ARMY-MRC VICKSBURG, MISS." COLORADO SCHOOL OF MINES RESEARCH FOUNDATION, INC.
Golden, Colorado
THE STABILITY OF DEEP LARGE SPAN UNDERGROUND OPENINGS
Prepared for
U. S. Army Engineer Waterways Experiment Station Corps of Engineers Vicksburg, Mississippi Contract No. DA-22-079-eng-334
Approved:
CoaU 0,. Free! JaVid Cardard, Jr. / Director of Research Mathematician
William H. Jurney Project Mathematician
Thomas I. Sharps Geologist
Project No. 320327
COLORADO SCHOOL OF MINES RESEARCH FOUNDATION TABLE OF CONTENTS
Page
Part I. Theoretical Considerations 1
The Spherical Cavity 1
The Elastic Case ...... 1
Introduction . 1 Derivation of Solutions for a Spherical Cavity in Uniform Stress Fields 2 Stresses and Displacements Due to Force Operative at a Point 3 Potential Stresses ...... 9 Field Stresses 11 Notes on Derivation of Solutions B and C . . 14
The Plastic Case 20
The Prolate Spheroidal Cavity...... 38
The Elastic Case 38
Introduction 38 Statement of the Problem 38 Three-Function Approach in Elasticity. ... 39
The Oblate Spheroidal Cavity . . 51
The Elastic Case 51
Introduction ^ 51
Openings in Layered Media . 59
The Multilayered Case ...... 69
Discussion of Theoretical Results and Optimum Cases .... 77
Spherical Cavity . 77 Prolate Spheroidal Cavity. .... 79 Oblate Spheroidal Cavity ... 81 Conclusions and Optimum Shapes ...... 82 Stresses in Vicinity of Cavity 82 References . 83
COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Table of Contents (continued) Page
Part II. Practical Considerations Relating to Cavity Stability 85
The Lithology 85 Local Structure 85 Regional Geology 88
Construction Consideration to Promote Stability. . . 90
Possible Linings and Support 90 Rock Bursts 91 References 94
Part III. Exploration Program for Selected Sites .... 95
Introduction 95 General Geological Reconnaissance. . 96 Detailed Surface Mapping in Area 97 Seismic Surveys 99 Drilling for Exploration 100
Well Logging 102
Initial Stress Measurements 110 Core Logging and Laboratory Tests 114 Estimated Cost and the Time for Exploration Program, Site 3 116
Reference 120
Appendix A, Tables 1 — 53 121
Table 54, Directional Surveying 104 55, Drill Stem Testing 105 56, Dipmeter. 107 57, Sonic Logging 108 58, Summary of Estimated Exploration Costs, Site 3. 119
Figure 1, Rectangular and Spherical Coordinate Systems. . 10 2, Reference System for Prolate or Oblate Spheroid , 40 3, 65 4, 71 5, Tentative Schedule Exploration Program, Site 3. 117
Map 1, Proposed Drilling Site 3, Inyo County, California 2, Preliminary Structural Geology Site 3 Area, Argus Peak, Inyo County, California
COLORADO SCHOOL OF MINES RESEARCH FOUNDATION ii
\ Nomenclature
Elastic Cases
fx, (/> z) Rectangular coordinates (see Figure 1)
Cx, y , z') Rectangular coordinates (see Figure 1)
Spherical coordinates (see Figure 1)
D'j j, O Unit vector triad for rectangular coordinate system
[ct^o , U. p , u Unit vector triad for spherical coordinate system
[
ect: Stresses in rectangular coordinates 3 , *c±._
) ec^ Stresses in spherical coordinates
faacfy,"?] "teurvilinlinear components of displacement and stress, respectively
>&, Harmonic stress functions (displacement potentials) K A , 9) Orthongonal curvilinear coordinates in general, and spheroidal coordinate in particular
// ; hz } /}$ Local scale coefficients
(j Cosh ac> f»5/'/iA ^Auxiliary position parameters for spheroidal p «Cos j coordinate system
?) 9a > Qo Values of <} , ^ ,/r , Q at a* c Cosh cc0 1 Polar and equatorial semiaxes, respectively, r of prolate spheroidal cavity (interchanged for b ' c 57/7/? oz0 J oblate cavity) a b Shape ratio COLORADO SCHOOL OF MINES RESEARCH FOUNDATION C Size constant for spheroidal cavities £*,?,£ Shear modulus, Poisson's ratio and Young's modulus Note: < „*/r dV. $*/* 2V. . . ^7 r ^-srr^ "5~x^^rrLaplaciana operator 3a y c* ^7 F • Grad Fm 7* J-4^ dx ^ <>y j&zi&e Uniform stress field constants in x, y, and z directions, respectively X/ Ratio of uniform horizontal stress field to s vertical stress field, i.e., crf * cr^ /Jcs * Plastic and Multilayered Cases ¥ Dissipative function 5 Incremental functional 3 Represents partial differential f Integral sign Uj, u/ Tensor components SI Angular velocity vector X; y Xk Denote coordinates juu Elastic constant 5 Second viscosity coefficient &/k > rfk Denote components of two stress tensors (fi A contour integral sign S Denotes small increment ^ Expression denoting analytic functions and 0 S constant X P Denotes analytic function It A constant 6/j Denotes components of strain tensor COLORADO SCHOOL OF MINES RESEARCH FOUNDATION iv (o Density q Acceleration of the gravity field u Displacement vector ux. ~l Components of displacement vector Ur * j'/j Components of stress tensor £ Modulus of elasticity (Young's modulus) V Poisson's ratio f Friction force Critical friction force expressed as a stress Critical friction force to maintain sliding motion G, Shear modulus r Position vector 6(k The Kronecker delta Ujh Time derivatives of strain components or, j- Friction factors C;jki General elastic constants (matrix) y Velocity vector p Pressure Viscosity for incompressible fluid 77^. Momentum flux density tensor cr'k Viscosity stress tensor COLORADO SCHOOL OF MINES RESEARCH FOUNDATION PART I. THEORETICAL CONSIDERATIONS THE SPHERICAL CAVITY The Elastic Case Introduction This portion of the report is concerned with theoretical stress distributions on spherical cavities in homogeneous, isotropic, elastic media. The medium in this solution is con sidered to be infinite but the results are applicable to a semi- infinite body, i.e., one having a free surface, provided the cavity is sufficiently far below the free surface. In the discussion of the results it will be shown that the depth to the top of the sphere may be as small as a diameter of the sphere. The basic solutions developed are for cavities in uniform stress fields in the coordinate axial directions. The stress fields being the stresses existing in the medium in the absence of the cavity. The stresses depend on only one of the elastic constants (Poisson's ratio v ). Basic solutions for uniform stress fields are given in Tables 1 — 25, at the surface of the sphere for v = 0.1, 0.2, 0.3 and 0.5. From these tables any desired type of solution (using the proper V ) for a spherical cavity in a uniform type stress field can be computed. These solutions were obtained using the same classic methods as Southwell (Reference 1) and Goodier (Reference 2). They are presented here in a more complete form than in the references cited. Stress concentration factors mentioned herein are of the form where <^3 is the vertical component or the uniform 1 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 1 stress field in which the cavity is located. A sample derivation follows. Derivation of Solutions for a Spherical Cavity in Uniform Stress Fields The method of obtaining the solution for a spherical cavity in a uniform stress field, is so well-known that only a brief outline of procedure is given in this report. First the "simple" solutions (so called by Boussinesq) of Lord Kelvin (Reference 3, Chap. VIII) connected with forces in the interior of an indefinitely extended elastic body are presented (i.e., those needed). Next potential type solutions are given. These consist of using a harmonic function to generate displacements and stresses which satisfy the equations of equilibrium and compatability for an elastic medium. The procedure is to choose a harmonic function F(x, y, z)( = 0) and assume displacements (u, v, w) = ^Vjr grad F where G = 1 2(1+v ) The stresses and displacements for the uniform stress field under consideration are, of course, needed. Having these a combination of stresses (from those mentioned! is obtained so as to nullify the normal and tangential stresses at any point of the cavity. The method described above has been considered as a special case of the three function procedure in elasticity by Neuber (Reference 7). COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Stresses and Displacements Due to Force Operative at a Point (a) Double compressive force in x direction at (0, 0, K): _.A(/*v) [?4v-/)_X_ -3 x31 " ~£— L ** ^5-J y •3 x*y (7) ; & /2 s" 2 w s~Mi±vl z -3X sZ E &3 f> Where A is an arbitrary constant or in spherical coordinates. U » AJ2ULL 3/*j Cos 9 "3 Cas3 &\ £/** L J v= /4 0+ v) T5/M 4> S/aj 9-3 S/u3 $ S/a/ & Cos 9~\ £p2 L J (Z) z w - AO* v) f Cos $ "3 5//j Cos Up - ^ S/*/z ? Co&2e*5/vZ lAj .ALLtJlL- \^(4v2) 5/m Cos 4> Cos2 & J ^ Ue » A(i + v) j~C2-4js) S/a/ s/m B Cos <^J COLORADO SCHOOL OF MINES RESEARCH FOUNDATION From (3) z 2 <= T3UP. Aa+V) [-8CV-0 Siu* P Coz*&-2 S/Ki v e - ePP +e„+eH [« -V + From (3), (4) and stress-strain relations - ~^r ,3/a/3 ?s c©**-,2 r/*soJ J ^ a pv-^; coS ^ 2 Cg _ jr/-jzv; * 3(zv./) s,aj* */**j )<5) ^ ^TcjjjS 0>s**J pr^ s ^ f/+v) S//J ^ S/a/ 9 Cos COLORADO SCHOOL OF MINES RESEARCH FOUNDATION (b) Double compressive force in y direction at (0, 0,. K): X -3 xy; u -= z V • B d+V) Mv-t) y 3 y (6) w, agi-v) - 3?* Z w £ R5 Where B is an arbitrary constant. From (6) in spherical coordinates 3 z w.g 8Cl+v) tp Cos 6-3 <5/A/ $ S/*i 6 Cos _ BC/+VJ \(4v-f) S/a/9-3 S/a/3<& S/a/3 & 1 L J > 17) v/« ; B (/*• is) $ - 3 S/iV* pf . £~os & S/A/2-&\ ffi* L J From (7) etc, . BO'V) \-(5-4v) 3/a/z ^ S/*JZ8 +t 1 * yf/O* L J z ,,. . 8O+v.) 5/a/ ^ s/a/ /, «. £C/-I-V) \(i v -2) S/sJ & 5/*/& Cos#1 ,£>*- L J COLORADO SCHOOL OF MINES RESEARCH FOUNDATION From (8) 6 z €fiP * zps™ \z(5-4v) S/a> ? S'M*9-Z] e/* S'»z ? S/m*P H4V-Z) S/A/*#*/] e*' * BCJp*> [~3 0 S'M* 9+C4V-2) Cos* e*/] e* B\ From (8), (9) and stress-strain relations x 2 2 x \f2 v- o 5/^2 S/A/2 Og w ~ ~(h~ |?f2V'/J S/H2 f S/A/* 0 +(4v2) Ces* &+(/-Zis)\ wo) TRT * ~PT \J2(/+TS) S/A/ £ CES ? S/*/Z $(*$* "fiT S/A/ yi. 5/*/ $ Ce>s d] ,•4pT J*'Iv-Z') Cos S/A/9 Ces COLORADO SCHOOL O W MINKS NK8KANCH F O U M DA TIO N 6 (c) Double compressive force in z direction at (0, 0, K) . Cd X -3xz' U= CZL [. m r ^ -3VZ*-1 V' I~RF rs J (if) Q(!tVi \C4p-/)'z -3 Z w~ T L &> or in spherical coordinates ca+Ts> U [j/A/ Cos tf] V = [S/^ ¥> 9--3 3/*/ pi Cos3- ft 5/fi 9 j )(/2) C(j*zO \C4V-!) Co* fi - 3Cas3/] w £/**• Where C is an arbitrary constant. From (18)-etc. «. • * / 4Ct*-0 Cos vj ) CC Uf - /J^ \2a-2.v) Sw r> Ces /J \(/3) Uff' o COLORADO SCHOOL OF MINES RESEARCH FOUNDATION A From (13) etc. CC/f v) Qf>p £"<2 S/a/ 2 ^ /• 3 C/ - v} Cos * ^'J t C f/ + v} eft \^.C/-2zr) C COS*? - S/w* ?)+S,»3 ji 4(~ts~/) Ccs^/J or e. . - C(/+v) z W Eei [(4v-/) $,u ?-ZCat*/\ - Ca+ is) $99 ^S/iiz ^ - 2 Coszft j C (!+ V) s e £/~- [ZCZV-D Sw* 4 f 40-2v)Cos- /} From (14) and stress-strain relations 0^> = pi \^-!L(H-v) Z/Nzfi + 4fZ-isJ Cos*? J °j» a ~jhr \fZv-0 •T'"2A t zrzv-o Ce,%^f*\ &$' ^C/-Zv) St fJ •*• j- 2. fZv-l) Co*-*-y!> J >r/5; Jjop ~ p 3 (/ 1" y) •S/M ^ C&S COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 8 Potential Stresses a Let ax F where XP'^ and p z\jxz *• X* *• z.4 From above v ! V F I ^ , S 4, 3 wj TT - ^ or ££ • 2,Q 3/M Cos & ^">5 J/zv-2^ v = Sg. S'» ? &"'' V*"^ Cos ft P'4 \_/S S,*,x J ^j4^] Where £k, V, vj has been expressed in spherical coordinates. The vector £u, Y} wj is now projected in the directions of ~ufi ,~Uf and ~u g (see Figure 1) by computing [u,Ywj , ~uP etc, to obtain displacements and u$ . The results are: 3 (S//J ^ f/£ Cos ^ (17) -3 UFI ' S/A/ / S/A/ Z (9) COLORADO SCHOOL OF MINKS RESEARCH FOUNDATION z-z' x =/> Sin^ Cos® - T Sin£ Cos6+ j Sin£ Sin® + K Cos Jjig - Sin ®+J Cos® RECTANGULAR 8 SPHERICAL C QOR DI NATE SYSTEMS Figure I By use of the stress-strain relations and (17), corresponding stress components are obtained as: Of 3 P~5 |'/ Go's ' 7 •S'H* j£ Cos *" Off -3fs Cos*0-S <3'**/* £ Cos*-#] )(/e) fpjt ~ 3fi'^ 4 5/*j2fi Cos 2& j Uffi ' 3 i°'s [-/ S/Af fi S/M 2- *] — & Derivation of Solution A i.e., uniforms stress field <5, , in x direction such that Ox*"®i » &z. > Ix* > 3** j 0 as * t y^-°° Field Stresses For this case £ U-, V, V J m which by the same transformation outlined for potential stresses above gives £^ fi Cos*-0- v 5/*/*- S/f/3- 9 - V Cos*- /] ^ * X VS;a/*0 t* v] S,*/2 ^4? s - X 0+ V) S/v / S,*/ Z 0 & , COLORADO SCHOOL OF MINES RESEARCH FOUNDATION (Jp " <5, S/*/2" Cos* & O^ m O, Cos*-fA Cos* & (Sfi • ff, S/a/2 (ZO) S//V 2 $6 Cos z & %e M ~ ^ ^~/A/ £ & & The conditions to be satisfied at the surface of the cavity are: Bp ' ft * 3p$ ~ O (ZD To satisfy (21) the following combination of stresses is used: 1. Double forces (equal) in y and z directions, i.e., combine (10) and (15) with C = B. 2. Double force in x direction (5). 3. Potential stresses as given above multiplied by an arbitrary constant - — . (The factor - — is used to 12 12 eliminate the coefficients in equations 17 and 18 and change sign to simplify ensuing algebra.) 4. Field stresses (20). COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 12 The application of the conditions (21) to the sum of stresses 1, 2, 3, and 4 lead to the simultaneous equations '£• (/0-2v> t £ (2V-/0) ~ + a, ~° \ZZ) ~4r ^ Otvi t-f- •<> The solution of (22) yields: i _ a*(9- /S v) a3(Sv-l) At <"aS 5v-7 Resultant displacements and stresses are now obtained by in serting A in equations (3) and (5), B in equations (8) and (10), replacing C by B in equations (13) and (15), multiplying equation (17) and (18) by-^ , leaving (19) and (20) as they are. After which corresponding components are added. The results are the combined displacements and stresses given in solution A in the subsequent sunpar^ of solutions. COLORADO SCHOOL OP MINES RESEARCH FOUNDATION Notes on Derivation of Solutions B and C Solution B This solution is obtained by using the following basic system: 1. Equal double forces in x and z directions 2. Double compressive force in y direction _ a 3. Potential stresses given by r- where ^« yp~3 le. grad 4. Field stresses obtained from equations (20) on replacing cos & by sin & and sin & by - cos & On the other hand stresses and displacements for solution B can be obtained directly from solution A by using the interchange of sin & and cos 0 given in 4. above. Solution C Basic stress systems used in this solution are: 1. Equal double compressive forces in x and y directions 2. Double force in z direction 3. Potential stresses resulting from F = d ^ —3 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 4. Field stresses and displacements s Gfi a3 Co* 2 » <5S 5//V y4 *-0"3 5/*/ / £bs ^ Lip - " 1/ .S/v-Z-^J a,. Uff •- o Solution A. Uniform stress field in x direction i.e., crx-»-cr, CTy , crx , Cfcy ect, — <9 as A and y-*c« c-,a° ~ z(5y-7 \j&-5v) SCS"1v) S/a/2 pi Cos*-0] -^r " ] -ff »" v) \s'A/z fb Cosze-v\ P COLORADO SCHOOL OF MINES RESEARCH FOUNDATION £U~+ _ _ „ , f C/C v-S) a*- _ 3 a4 2 T §^-S,»fi S,"20 f/>p> [ r*(s°.% " *- o> _ S(S'V) ^ _*£ . 6 a3 _ o-/ Sv7 & 5 /<''7 5"v-7 <°* s -/•ff Coz*-fi a + <* a* + ijfl r ^J_y pS * Sv-7 ~rr- S/t ft Ces*0 z 3 &£ m f/avs) Cos 0 a ^ f6-/Sv) a* ^ C 3 r t Z.f£V.7) P* ZCSf-7) fi + (ZO V - to) • Cos*#~T ^ CoS- 6,4 •/• f Ces2 ^ Cos*B Og ^(3 + ~^" * *i*V'7) &(5 Cos* 9-3+2 Cot*9) + Siuz 9 ~§f ~ v-7) 2/a/%4 Co^-'S -pf *" sv-7 S/aj Zft Cos3, & "pT" ^ X°g „ S(n-v) a? - ^ ST/A/ & S/aj £ 0 a. * G/ 2.(Sv-7) /A/ ^ S/M 20 ps SV-7 Ps - ^ 5>/ COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 16 C| 2-(5T> for /° » CL z a4> m /S Z/aj*- <£ Cos 6 f. jS Cv-/) Cosj. C$ ~J5 Sv-7 5v 7 &(5v-7) 0$. /5 1? S/j/^&Cas*# + /5 (/-1?) Cos*-& ^ /S - 2.7 ~W~ SV ~7 5"t/ - 7 2C5V-7J Solution B. Uniform stress field in y direction, i.e. Cy—•cs2 }aA &Z, U^a cf'xij Cfy*—-/0 <*S x and y— Stresses and displacements are obtained for those of Solution A z by replacing Cos* for P • a O* _ /5 S/«* S/u3- & /Sfv-OS/M*# _ f3-/Sv) a* £»-7 r 5-w-7 2.V-7 <59 m /5v ^ /sO-P) S,A/*0 + /Sir-*7 ~7=T" SV'77=3—_ —: ' Sv-7"3 = r *(5v-A > — 7)#-\ COLORADO SCHOOL OP MINES RESEARCH FOUNDATION Solution C. Uniform stress field in z direction i.e. ax-—; <3"a , cry , y cct. —o as x and tj •- *° Spf - [Ut-w-ste-ivy c.t*& *• + (9 Cos*f>-3$ ~pr\ (Co** f& - V S/a/ * £) -%&- ~Jv0b S/M z+ 3 -JL(/ + V-) 5/*J z_]t Ufi H o 3 <5p - r * J f . csv-t*) a 3 . /e a*] _z± a + Cos fi6 r (/ 5v-7 P* Sv-7 J £f~7 p* 6 a.* f 5V-7 P* o* - r * - F ' • Oov-s) a-* J. 2/ a*~] s of ~ Cos f [-/* z(sv.7) lor xcSv-V <° l ^ fSv-4) a3 - 9 A* M 5V-7) i°5 2-(Sv-7) cs COLORADO SCHOOL OP MINES RESEARCH FOUNDATION _g> C30-U-/S) as /S Cos ** ^ a? Oj .2-(5i"7j J-CSV-7) p* 3 ft-/Sv) a _ 3 aa 2 CSV- 7> P2, *•(•517-7) P* - S(' + v> *3 /2> a* ^ Cos / *v-7 of SV-7 -pir ^pif — — & for p • a cry _ /s Co* *• £ ^ C/sv-z7) 03 Sv-7 ZCSV-I) _ /S V Cos* ^ + 3-/St? 03 ' Sv-7 X(Sv-7) COLORADO SCHOOL Or MINES RESEARCH FOUNDATION The Plastic Case Time rates of change of deformations and stresses become important under certain circumstances. In this case, the primary concern is that, although the materials involved ordinarily behave like linear elastic solids, the stresses and times associ ated with the proposed opening might exceed "ordinary" conditions causing the materials to exhibit yield. Time rates of deformation suggest consideration from the viewpoint of the viscous flow of an incompressible fluid. Analyses based on such an analogy contribute heavily to the understanding of plasticity. It should be noted that these time rates need not be constant; a system under consideration may come to rest in a stress field which is constant for a long period of time. Westergaard (16) defines two important time effects; relaxation, which is the reduction of stress over time while strains remain constant, and creep, which is the increase in tota.. deformation over a period of time during which the stresses are constant. These phenomena are also temperature dependent, but in the analysis of structures wherein temperatures remain within a certain relatively small range, the effects are considered to be sensibly isothermal. It is also conceivable that systems exist in which the yield stress has been exceeded and in which creep is occurring, relaxation also occurs, the two phenomena being mutually dependent, thus; e • f ( COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Before completing the discussion of modelaL i>f solid behavior in a stress-field, it is desirable to state some observations about the physical nature of rock, which will serve as an orientation to the type of physical substance that any complete model will simulate. Rock must be considered to be an aeolotropic, heterogeneous granular substance, which is composed of several polycrystalline substances held together in a matrix or bonding material. This is a description which may be applied to the gross structural level of rock and lies somewhere between the macroscopic descrip tion of a homogeneous substance, and the microscopic description involving crystal structure, chemical bonding, etc. In the macroscopic ease, crystal orientations, minor flaws and inclusions, and other structural inconsistencies are ignored for simplification, and the stress-field is an idealization which can be, at best, a representation of averages in values and distributions over an area or volume larger than the area or volume in which the effects of actual nonuniformities are significant. The microscopic case involves the description of behavior of single crystals, with respect to an imposed stress field, as well as the behavior at boundaries between crystals. Lattice imperfections, slip, and plastic rupture are microscopic behavior patterns. Both types of analysis are useful; the macroscopic descrip tion affords information about expected gross distributions of COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 21 forces and energy; whereas, the microscopic viewpoint is of value in predicting the important mechanisms in deformation and failure. However, somewhere between these two idealized cases lies the region of interest to the engineer or the "rock mechanics" theorist, because departure from the idealized theory to the practical case occurs just due to the imperfections and structural features of the available material. It is expected that any change in load will require grains, the matrix, and even their molecular structures to change in order to accommodate the new load condition. Thus, the simplest law which may be stated relating behavior of a material to the stimulus - the force input - requires a change: * Cjj£i £ici where J/y is the stress tensor, is the strain tensor, and Cjjf much more complex; the important point is that the material is deformed from point to point relative to the reference system*. Added to this are the empirical observations that all the elements in the structure are involved in the deformation. Evidence indicates that not only are these several different materials in the structure of the rocks, but also that there are many discontinuities in the medium itself. These discontinuities or microcracks, also will affect the reaction of the material to COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 22 different loading systems. The loads may become sufficient to make some of these cracks unstable, resulting in their propaga tion and the formation of fractures. Relative rigid body displacements may occur across fractures. Such effects are not predicted by equations derived from continuum mechanics;: however, the discussion of these effects is delayed, since only the elastic and simple inelastic behavior can be described in the framework of continuum mechanics. In the practical case, only the ultimate effects of microscopic discontinuities is considered, i.e., flow and failure. A state of equilibrium will be attained by the medium, or the deformation will proceed to failure, which effectively changes the distribution of the forces, and therefore, the geometrical conditions of the medium in the field. If rigid displacements are ignored, it can be stated that any deformation of the medium must represent a deformation of the particles and matrices forming its structure. Having made initial assumptions which limited consideration to continuous media, the preceding statement becomes redundant, but it is well to clearly define the critical limitations of the theory; i.e., when considering the displacement of points in a geometric space, under field laws, only the space under consideration is available to deform the continuous medium. The models for inelastic behavior are investigated in order to determine the significance of inelastic effects for the practical cases which have been proposed. A simple fluid dynamic COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 23 model which superimposes the Hooke elastic relation with a fluid condition involving the viscosity rj is £*1-1$ (!) where & is the stress, e the strain, E is Young's modulus. This equation represents the total stress short of yielding, and is associated with the name of Kelvin. For strain rate, a relationship due to Maxwell exists: de «. a_ y / dor /£) ¥T n eft ' The equations do not account for dissipative effects due to thermal conduction. A solution to equation (1) is e • mxp * '[*• + 4 *ax 0*P If" '1 d*\ If e" f~\!~exp ^ These equations are much simplified and unidimensional in the form presented. Added to the initial elastic strain is a creeping strain which increases with the time after application of the load. The implication of this model is that the total strain is attained when time has extended infinitely; and conversely, complete restoration occurs only when an infinite time follows the removal of the load. It is noted, however, that the model is not complete; da/c/i\.& taken as zero, and thermodynamic effects were ruled out. A more complete model for elastic-plastic deformation, e , must' include . #.n where T is the temperature. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION A fluid dynamic model, stemming from physical theory, is not necessarily the only model available nor the best possible predictor of solid behavior. Some successful models are based on empirically derived stress-strain relationships. The model of Ramberg and Osgood predicts strain for ductile materials, using the formula £• cr / £ +• M (cr/£")h (5) where K and b are material constants. In A report by Patel, Venkatraman, and Bentson (17) it is shown how such a model can be combined with general stress-strain relationships, such as Prager's for isotropic, incompressible materials in the form £/j a C Jj" 5/ j in which C:and m are material constants, and the term J2 is the second invariant of the stress deviation tensor, j" 3/y 3',j, $,j ~ °/j ~ 3 °"kk. &ij The authors show, employing Hencky's hypothesis, that the combination of the empirical strain law and the stress-strain relationship is useful in the formulation of equations which apply to inelastic behavior of materials subjected to combined stresses. The constants C and m, which are difficult to evaluate experimentally, are related to the constants in the Ramberg - Osgood uniaxial equation in this method. Two cases were considered, theoretically and experimentally, in the above investigation: a hollow sphere under uniform pressures, and a thick circular cylinder under uniform pressures. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 25 In the case of the sphere, predictions were compared with data for an aluminum sphere subjected to various external pressures. Both theory and experiment corroborate that results vary from Lame's solution by a sensible amount only at high pressures. This variation occurs in the vicinity of maximum stresses, the area close to the inside surface, forming an inelastic "plastic zone". Both the variation from elastic behavior and its depth are small. The pressures exceed those expected in an under ground opening - about 30,000 psi. The investigation of the feasibility of opening large cavities, due to Deere, Langhaar, and Boresi (14), contains a discussion of plastic effects derived from the solution by Sokolovskii, and a discussion of inelastic effects in the "zone of decompression". Taking into account relevant physical param eters, the conclusion of this present report is that a phenomenor like the "zone of decompression" is the only important inelastic effect in hard rock at the depths suggested. It will be found that no calculations are performed for the plastic case, due to the fact that physical argument and the orders of magnitude of the parameters involved adequately demonstrate that plasticity is not a serious consideration at the depths and pressures presently contemplated. This conclusion is intended to include viscoelastic behavior, which is most likely to be insensible for practical purposes over any reason able length of time in hard materials. For general cases, Deere, et al., have shown graphical data for plastic regions in their report. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 26 Values of viscosity for "hard" rock are so great, even when compared with stresses and Young's modulus (the order of 77 = 1016 24 98 for weak types and 10 - 10 for "hard" types), that viscous flow could not be noticeable for long periods, even at the depths contemplated at present. Also, rocks such as granite exhibit fairly linear elastic behavior with no significant yielding to the point of failure; hence, plastic creep in continuous masses of granite would not be expected. In order to summarize the conclusions regarding viscous effects in continuous media, let us review some of the implica tions of the general theory regarding the viscosity of solids. In the cases of practical interest, deformation is thermo- dynamically reversible only if it occurs with infinitessimal speed; i.e., equilibrium is established at every point at every instant in time. Actual motion, however, is associated with finite velocities and, since the body is not in equilibrium at every instant, processes occur which tend to bring about an equilibrium state. The result of such processes is that the motion is irreversible and mechanical energy is dissipated. The dissipation of energy can be ascribed to two mechanisms: when the temperature varies from point to point, irreversible thermal conduction occurs; and, if internal motion takes place, there are irreversible processes arising from the finite velocity of that motion. The latter type of energy dissipation is referred to as internal friction, or viscosity. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION If a mechanical system is investigated whose motion involves the dissipation of energy, the motion can be described by the ordinary laws, with the forces acting on the system being augmented by the frictional forces. These forces can be written as the velocity derivatives of a certain quadratic function, , of the velocity, called the dissipative function. This frictiona], force, say fa. , corresponding to a generalized coordinate The above equation can be generalized to the case of motion with friction in a continuous medium. The state of the system is then defined by a continuum of generalized coordinates; these are the displacement vector _u at each point in the body. The above relation can be written in integral form, over the volume of the body, Sui dV (7) where they/ are the components of the dissipative vector f per unit volume. The total dissipative function is written as jf'dV , where J?" is the dissipative function per unit volume. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION The general form of the dissipative function 3f for deformed bodies can be determined. The function must be zero if there is no internal friction; hence, IP" * 0 when the body executes only a general translation or rotation. Thus,]?"" 0 if U = constant, or (i »A xr the angular velocity). The function therefore depends on the velocity gradient and contains only combinations of the derivatives which vanish when 6;*fi x_r ; these are the sums 3 u j 3 tlk dxk dxj which are the time derivatives U;k of the components of the strain tensor. The dissipative function is a quadratic function of the most general form of which is iktm U/k The tensor7^;^is the viscosity tensor, and for an isotropic body, it has only two independent components. An expression for can be written in a form analogous to the form for the elastic energy 5,> Uu ^ for an isotropic body. t-?(uik - j s,> ua)*' { s UU* (.9) where ^ and are the two viscosity coefficients. Since the expression (7 ) is analogous to that for elastic free energy, h f FdV = -J/6ujdV , where// - d&ik/&> fi = dcr;^ / (10) COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 29 where the dissipative stress tensor C/^'is defined as crj/ ' ct/d u.;k • u.,M (if) Thus, the viscosity can be taken into account in the equations of motion by adding the term for the dissipative stress: 9 <*ik 21(U;« - J 5ik Ujj) * Satt h;k f/2) It can be seen that substituting an expression er/k' into the ordinary equations for motion represents a negligible change when 77;klm » cr/^ and the time. The conclusion must be that viscous effects and plastic flow are not problems for concern in an ideal medium under the conditions proposed for the cavity and under the anticipated loads. However, since the medium is not ideal, and local discon tinuities are present, it cannot be said that elastic-plastic effects are nowhere present. What is stated is that motion of the contour of the cavity cannot be attributed to general viscous or plastic effects acting over the whole medium in the vicinity of the cavity; i.e., no significant viscous or plastic flow will occur due only to the opening of spheroidal cavity in the medium. It is known, however, that motion does occur in deep openings in rock. Having ruled out any contribution to such motion from inelastic behavior in continuous media, there remains the possibility of motion due to conditions which have not been considered because they are not associated with a continuous medium bounded by the single, simple contour, which was the initial assumption about the geometry. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 30 In order that a discussion of motion includes the effects of practical interest, the discontinuities in the vicinity of the boundary of the opening need to be considered. The simplest concept related to the effect of these discontinuities and irregularities is the "zone of decompression", mentioned by Deere, et al. Here, the primary purpose is the discussion of the physical behavior related to this effect. In general the reasoning can be limited to the facts that natural discontinuities, such as joints, fractures, etc., already exist in the undisturbed medium; added to these in the process of constructing an opening are many more fractures and cracks, the affected region extending some distance into the medium from the boundary. The result of disturbing the medium has been to require a different equilibrium state for the forces acting at a point in the neighborhood of the cavity (theoretically, a different field for the whole space except at infinity). This \ general statement applies to all space occupied by the medium given any configuration of openings. Therefore, the case has been considered in which the only disturbance of the initial equilibrium has been the introduction of a large cavity. Relaxing the assumption of uniformity to include structural defects already present changes some of the statements about the stress-field: the stresses are now average stresses rather than absolutely determinate stresses from point to point, along a con tour; so that, were it required to determine the forces acting on a specified volume bounded by a contour, in the ideal case, there COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 31 would be given f rf- dV * $cr;k (/3) where F; are the components of the force\F; , and the df; are the components of the surface element vectors'/ along the outward normal. However, in the nonideal case, /?/ d V * jf d fk (/ 4) where the F; * £ a;k/dx^ . As the volume investigated increases in space dimensions, and becomes large (statistically) with respect to the linear dimensions and distribution in space of the non- uniformities, then fFjdV —/ft dV , and in the limit J/f c/Y -J/>• d V For practical consideration,^/^dV-fffdV*- 6 , over an arbitrary volume element; and € is less than engineering factors of safety. The specialized assumption allows consideration of further "openings" (the discontinuities) in the space being investigated. The means available for detailed analysis of a practical case leads to great computational complexity very rapidly. The physical foundation for such an analysis is the structure of the ends of an equilibrium crack in a brittle solid, and the stresses associated with them. Once again, let us consider an idealized geometry - an isolated slit in the elastic medium. The stresses acting on the slit, g(x) (the direction of the slit is the positive x semiaxis) are equal to the difference between the stresses applied to the surface of the slit in the resultant field G(x) and the stresses which act on the slit, p(x), corresponding to the first state, which is the stress field due COLORADO SCHOOL OF MINES RESEARCH FOUNDATION to a force acting in a continuous body. The first state can be solved by the methods of the classical theory. The expressions for stresses in the second state are determined using complex variable techniques and appear in the form Oxuy + cr^ * 4 Re. $ C*) CTy(2-) - /O-xyM - + 0 fz) +• (z-2) §'(z) )f/s) tfJL (u(Z>+ /zs^) * %f(z)~ -U? (l)-(z-z) §(Z) X ' 2-4 v wherez^^/y , the bar denoted the complex conjugate \ - /y , and ^are stress components for the second state; dZ^and V^are the shear components along the x- and y-axis;/^ is the shear modulus,V is Poisson's ratio. The functions/ , tQ , j? , ® are given by the formulas 4>(z> • a (Z) •-f-rx) • f/6) fi*>. f%ro /- at (/y) (see Barenblatt(18)) In theory, it is possible to consider the stress distribu tion about a crack contour in any stress field. Complications arise when the a number of nonuniformly-oriented cracks are introduced whose spacing may be such that adjacent cracks also affect the stress field. The physical arguments may be presented COLORADO SCHOOL OF MINES RESEARCH FOUNDATION however, since the general conditions which must be satisfied in any case are known. In addition to the discontinuities already present in the structure, there are cracks due to mining methods which extend from the cavity surface to some point in the medium. Say that the linear dimensions of the crack in its plane are > the problem is to determine the tension required, normal to the crack, S0 , to extend the crack. Opposing extension are molecular cohesive forces (or the related surface tension), which are forces interior to the medium. Under these conditions, the change^/1", in the free energy of a brittle body containing a crack, as compared with the same body without the crack, is equal to the difference between surface energy, U , of the crack and the decrease, V/ , in the elastic energy caused by formation of the crack. Thus, in order to lengthen th* crack, it is necessary that an increase in linear surface dimension be associated with a decrease in the free energ; change <5 f. The minimum condition for the equilibrium state is 9 ' o //8) da. 1 The geometry of the cracks allows great concentration of stress at the ends. When introduced into a relatively high stress field which already contains locally concentrated stresses (aroumJ joints, blasting surface irregularities, etc.) the available elastic energy will be sufficient to exceed the critical extensio: stress, and cracks will extend until equilibrium is attained. COLORADO SCHOOL OF MINK. RC.KARCH FOUNDATION 34 The processes involved in crack extension and in relative rigid motion of segments along contact surfaces may locally involve plastic flow, but the motion of the medium surrounding the opening cannot be said to be due to plastic or inelastic deformation; it is almost entirely rigid motion involved in the opening of cracks and establishment of equilibrium. Such motion does not necessarily occur instantaneously, but may require hours or days to reach a sensibly stable state. The length of time involved results from the number of discontinuities involved, the locally plastic conditions in some cases, and the change of the stress field in any volume element associated with crack separa tion and motion. The zone of decompression can be seen to be a useful concept of the ultimate effects in the neighborhood of the opening. The maximum stresses in the field in general will be away from the surface: the attainment of equilibrium requires this. The final contour of maximum tensile stress, for example, will be determined by the average joint spacing, the depth and space-density of blasting cracks, the equilibrium crack parameters of the medium, and the gradient of the stress field from the cavity surface. That the motion can result in a displacement of the surface in the range of 8-10 inches appears quite reasonable for the depths and stress fields now contemplated. Nonhomogeneity and anisotropy of the physical parameters in the solid material may affect the distribution of stresses; however, these departures from ideal media are usually not large. COLORADO SCHOOL OF MINIS R1SKARCH FOUNDATION 35 It can be seen, therefore, that the most serious difference in behavior between the ideal theory and a practical case will be due to the violation of the assumption of continuity, which results in local distributions of stress (and therefore also strain energy) much different than the stress field predicted for a bounded continuous medium subject to the same conditions of loading. A macroscopic structural defect worth mention is an adverse orientation of joints in the immediate vicinity of a cavity. Such a precaution is immediately obvious to designers of openings. In a large opening, i.e., one whose linear dimensions are large in comparison to average joint spacings, it is conceiv able that blocks of material facing into the cavity may become free from resistance to rigid motion. This condition results from either an inverted wedge situation in which the block is entirely free to move in the gravity field, or a situation in which constraint is present (by frictional wedging) but the force associated with the block and the angles of the joints bounding it exceeds the static frictional force. Failure of the walls due to these mechanisms becomes more likely as the size of the opening increases compared with the average spacing of inter secting joint systems. Associated with this probability is the further physical fact that larger free-faces are permitted by larger openings, which represent much greater masses which, in turn, may be restricted from rigid motion by frictional forces which do not increase proportional to the increase of exposed COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 36 areas. The probability in this case, given the same distribution of joints, is disadvantageously weighted, inasmuch as the mass for a given distance back from the free surface increases as the third power of linear dimensions, whereas the area increases only as the second power and the friction increases only linearly, if at all. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION THE PROLATE SPHEROIDAL CAVITY The Elastic Case Introduction The general problem of ellipsoidal and spheroidal cavities in uniform stress fields has been considered by various writers of which the investigations of Sadowsky and Sternberg (References 12 and 6) and Edwards (Reference 10) seem to be the most signif icant. Reference 12 treats the general ellipsoidal cavity but unfortunately this case is not easily reducible to the special cases of the prolate and oblate spheroids. Reference 10 gives an excellent treatment of the prolate inclusion or cavity of which only the latter is considered here, since there was no interest in the former, i.e., where a spheroidal core is filled with an elastic material with different properties than those of the surrounding medium. The elimination Of the core in Edward's solutions requires that the quantity H be replaced by -1, The solutions of Reference 10 in which this investigation is concerned with checked and the results of their computation are given in Tables 26 — 37. An outline taken from this reference is given below. Statement of the Problem Consider an infinite body in a given state of uniform stress prior to the introduction of the cavity, which changes stresses in the vicinity of the cavity but these changes diminish rapidly Q@URAE|Q SCHOOL. OF M I N CH FOUNDATION 38 with distance from cavity so that the uniform stress field is approached. At the surface of the cavity the normal and tangential stresses must vanish. The problem therefore reduces to finding valid elastic solutions vanishing at infinity which combined with the uniform field stresses effect the conditions required at the surface of the cavity. Three-Function Approach in Elasticity Let the cartesian co-ordinate system Oxyz be introduced with 0 at the center of the spheroidal cavity and Oz coincident with the axis of symmetry of the configuration, Fig. 2. Since the state of stress at infinity is uniform, Oxy may be oriented so that fxy - O at infinity. The stress distribution throughout the body may then be obtained as a linear combination of the solutions to the following two loading conditions at infinity: Case 1: (plane hydrostatic state of stress) cr^cry-—0as X ard a 00 Case 3: (state of uniaxial tension) c?z, —cr^ 3 cr* > &y > j ® ^ 3rd 00 The classical three-function approach in the theory of elasticity will be usedi/ in the particular formulation employed by Sadowsky and Sternberg (6) whose notation will be adhered to. According to this formulation, the general solution of the 1/ The three-function approach was apparently originated by Boussinesq (8), and subsequently investigated by Papkovich (9) and Neuber (7). COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 39 z Note: a) For prolate spheroid a=cq0 b=cq0 b)For oblate spheroid b=cqft REFERENCE SYSTEM FOR PROLATE OR QBLATE SPHEROID FIGURE 2 displacement-equilibrium equations in the absence of body forces is taken as \ [u, V,w]* qrad / \u,V,w\* ^ CUr/ [0,0,0] ) m rad X = J [«. V5 w]' Tqc ? [°, <3, £^r *] where <7^ • <7* v o . It is readily shown that any solution of the displacement-equilibrium equations may be represented as a linear combination of these solutions provided the harmonic functions , £> , and X. are chosen properly. These solutions will be referred to as basic solutions 1, 2, and 3, respectively. Sadowsky and Sternberg (6) have determined the stress and displacement fields corresponding to Equations (1) and referred to prolate spheroidal co-ordinates. For the sake of completeness these results are summarized here. The prolate spheroidal co-ordinate system is defined by the equations of transformation x * C S/V7/1 a S//7 Cos If y * C Sinfi The surfaces cc = const, /& = const, and 2r = const form a triply orthogonal family of prolate spheroids, hyperboloids of two sheets, and meridional half-planes. COLORADO SCHOOL OF MINKS RESEARCH FOUNDATION For convenience, the following auxiliary variables are introduced q- cosh c ^ If the differential of arc length is written in the form the local scale coefficients become / / / ?' -/>* J if ' W (Q) V ' m "^x In this co-ordinate system., the two basic solutions used herein yield the following representations of the displacement fields a 9 /»! A'P+n Ut' ln (7) and 2C, fi? U* * ~2q p9 J-/D X./0 A xj P U*' m ZQTr pq x r COLORADO SCHOOL OF MINES RESE A RCH FOUNDATION The corresponding stress fields become First basic solution 2, z 4 aA' h p +• h ? * Cqtf>9 - p f hz ( ' ['A* fa + h4 fq^ -p^Jjpq = a 9 ~ 9&* Jc" " -2TZ P9 ~P * *" p y •Jar = h P*9 Third basic solution z a* ~ 6* CfK9f. -2 *.<, )p? -Z vh*(9p*' ~Pq* >~v) t z - f>\p*)p z z Op - h - 2'Xpj qp* * 2vh (q>p* X,.- pq* X?) +• 4 z +A -pXP} p a z * • t 0-Zv)/t pq c 'Tor ~ ~p~ [V X91 - (2-• 2v + ) X,] $0%' [v77^/** -t~ (Z~ Zv -pz ) ~Ky j COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Let the boundary of the inclusion be defined by the equation am In order to fulfill the boundary conditions at infinity, stresses will be represented in all cases as a sum of the form !?j * R0j + G.jj Qtj /?xj * £ sS (/& where Ro\ is the uniform field solution which would be obtained in the absence of a cavity. The S;j are solutions obtained from Equation (1) for which the corresponding stress fields vanish at infinity. The a;j are coefficients of superposition. Solution for Case 1: The boundary conditions for this case are <7 o> , o~jf j Oje, j j 0"xr j ^3z~ OS) or equivalently Z Z as q—©o a-q—•- q* -h COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Suitable harmonic functions to generate basic solutions for this loading case are found to be: Solution R, For Rll> g For R21, X - Q.p /.e. ^atv,vvj ^Cf0 qrad X - (7<^ For R31, ^ /; ' *C3 where Q a. / + ±. /0?g -LJ-. The stress and displacement fields corresponding to the solutions under consideration are as follows: Solution R01 "'-Mtv> [(>-»>} "» 'zs%v> Y'" ?l-0•v)] U-, = 0 z z z Cfi -• ~q + /> <7?? <*r " °7 Sap *• h <7 V r ~ ^ COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 45 Basic solution Rn h . Ut 2qq * ufi - ur - 0 Oct " -All - h * £ •^r ~~ ^ Basic solution R21 Zq" [^v-2; ^ p] /»*f a/3m IKf (2~ 4v) pp C/jr ~ O ^ * YZO) y- ^TS -Z) ot. -Zg-, C-3*4r>9*-2 z> Z + f \c-/-2v)9,(2-iv)9 f-/+z*)Q*'-Zr . 2*4, + /,^\f/-Xv) 9 *- ^ ?r = >**/>£ -/,* pp (? +T> COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 46 Basic solution R31 A - u.* 7 2q I* f39Ki)a 2q 2 Uf •• o + i*'[J$frL+J*9**] )(Zt) ar'"zr*'i£r3 'v 0"7 =, - xJ L - JL £_ , 99 .2 ^ (f+ 3fQ) Jay ~ ^07 3 ^ C O L OR A D O SCHOOL OF MIMES iEf EAiRCfM FF OU >NDA Tl O *1 47 Equations 11 will be satisfied if the constants a,j satisfy eight simultaneous linear equations. The equations are compatible and their solutions are bci n ' 2. Cja £ Q0 /- J &aj> ' 4 qo [/* jfr - C/-v.) \(tz) and &/i " ~<22/ ' Q 31 with */* '»{-$&•) (22) and Z--20.V) QS [ e *'ft'-")]* fa > C24J Combined solutions for Case 1 ( (X- 4 2 + 4 •f -«• qi + 4 he2, 9-*" (Q.O+ ) D *4* A.* 9* [ ® O^Ct p " m O 0 Computations of equations (25) are given in Tables (26), (27), and (28). COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Solution for Loading Case 3: The boundary conditions are f ! J / °"y > i 3jc.K O an< Rl3, R23, and R33 are taken as equal to R^, **21' * ®31» the stress and displacement components of which are defined by Equations 19, 20, and 21, respectively. The components of stress and displacement corresponding to R{)3 are given in Equation 27. Solution Rq3: \ Ur'0 cr* • -9* +h*-<)xci Op- qq* / COLORADO SCHOOL Or MINE* KEIEAKCM •FOU'NOATIO-N The coefficients a;j are determined through the continuity conditions defined by Equations 11 and are found to have the values; Q/3 m ~ " a3i / Where Qo and D are given by 23 and 24, respectively. Combined solutions for Case III ( c( * Oi0 ): &a m * Jaty - O . ^ Computations for equations (29) are given in Tables (29), (30), and (31). Combined solutions for 6# and of cases I and III are given in Tables 32 — 3.7. The combinations are in the form obtained by taking . COLORADO SCHOOL OF MINES RESEARCH FOUNDATION THE OBLATE SPHEROIDAL CAVITY The Elastic Case Introduction The stress concentration problem for an oblate spheroid in uniform stress fields has been considered by Richart in Reference 11. The approach apparently was to attempt a direct transformation of Edwardfs results for the prolate spheroid, leading to an incomplete and unwieldy investigation. It was therefore deemed necessary to derive the complete solution using the three (or two in this case) function procedure analogously to that followed by Edwards for the prolate case. The oblate spheroid is assumed to be one obtained by rotating an ellipse about its minor axis which is on the z axis (vertical). Since the method used is so similar to that of Reference 11 only a resume is given here with frequent reference to results presented in P* Equations of part P will be denoted Eq. P. ( ), etc. Two principal differences are present in the two spheroid solutions. First the equations of coordinate transformation, second the harmonic functions <^> and >. used to generate the necessary potential stresses and displacements. Identical nomenclature is adapted to the oblate solution as it will be evident which of the two is involved in any discussion related to either. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 51 The coordinate system is the same as in P. (See Figure 2 in P) and the coordinate transformation is given by X*• c Cosh of S/u /9 Cos y \ = y c Cos A a nSV/? /Q 3/ft n (» z. - c S/'nh The local scale coefficients are — V Af /re *qz\z + p*-o ** a,a nd 9P solution for Case I (at t e;j —*>*, { Ox&f , ft* ,($2—0 as X and y >; The basic solutions needed for this case are: [c^w] - -Jq qract / with (H) » ^ = A rcooi g • 7 C«.H' x • [°'o'J^3L x 1 with '(3) •x « Qp Arccot q). fy,Vj w] • -jq grod COLORADO SCHOOL OP MINES RESEARCH FOUNDATION The field stress solution Rqi is; Ucc - h cjq [f/-v)-0 + v) p*\ Ujg hpp -2 v\ ar*o - o, h",o* a& » • f/ &/S' a, The solutions following are obtained from Equations 2, 3, and 4 in conjunction with the stress-strain relations given in Reference 5, p. 357. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Solution RJX; «cc*i^ 5 ur*0 ^ -J**?' a/3 * 4 q Solution R21; VP*? ^4vZ)Q ~-^j- J PH* UfO z ar = ~~j{ ZvQ ~ Zv-Cl-Z-*) f ]+ f/-zv- dot# ' 6* f>p (-~f~[ O-ZJ?) -hlZ}z3 + 2(t'2v) ^cc-y ' ~ Q ' COLORADO SCHOOL OF MINES RESEARCH FOUNDATION Solution R31; »«••& 9?« •(> £')A (i'9«) "" " ~3$0 ChCl u?* o -Sf'4,4*1] (3) 3 , / * 29 * J#?= 5 ^ The general solution of the problem for any stress is nom formulated in the form = e ^• ^ a,j &,j + agj +a3j 3j where the a!j are coefficients to be determined and the Roi> Rll> R21, and R31 terms denote corresponding stresses in solutions Roi ^11» ®21> and ®31> respectively. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION The continuity conditions Pll lead to three simultaneous equations for the determination of *21 > an<* ^31 and their solution gives (_/_ _ [*«•"> *# &u * ®2.1 ~ & 3) Where z D• -2(t+v) q0 + \e f*- 4f/-v\-2 +zf/-T7) 7- L J 9o* >(/Z) and «./- pm Arc cot ^ Stresses and cr? were computed for Equations 6, 7, 8, 9, 10, and 11. The results are '*' •§- -v «• 4*-o] -fr ar'-f- +(*v3)*Q.(fp-3)+Z(3-2v)<)*') -}('?) * C4S-/J] -$z c 3 Values of ^ Solution for C^so III C^«Q« j j ^*1 j ^ COLORADO SCHOOL OF MINE* RESEARCH FOUNDATION The basic solutions required here are identical to those used in Case 1 and are given by Equations 2, 3, and 4, i..e.., ^^i' etc• The field stress solution Rq3 is; U «•=-§*• * 99 + -y] \ *-jp- hpp [- v-n+ V) ?*\ 01, -= CTQ - cTs/?*'£*•?'*' a fajs = s As in Case 1 the combined solution for any stress is expressed as G/3 ^ ^23 * Q 33 ^3£ 05) COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 57 The continuity condition Pll leads to three simultaneous equations in a 13, a.23, and a33. Their solutions are; a it - (/6) and n '^o &H3 ~ &33 where D and Q0 are given by 12. Stresses and o 7 for or = cx0 are: \('7) c, --a The computations of *5»/ Combined solutions of the form N(<7g of Case I) + C# of Case III, etc., were computed for various values of N and tabulated in Tables 46 — 53. COLORADO SCHOOL Or MIMES RESEARCH FOUNDATION OPENINGS IN LAYERED MEDIA Site 3, tentatively selected as a site suitable for cavity construction, is situated in a nonlayered homogeneous formation. Because of a lack of available information on the elastic or plastic behavior of layered media, bedded formations have been ruled out for initial selection. However, since future interest may make necessary the selection of other sites, a general discussion of layered cases is contained here. It is not the purpose of this report to include a logically complete mathemati- ical development of the theory of deformation for layered solids., but rather to present some of the more general physical arguments which will form a foundation for any further elaboration. Layered formation have been dealt with previously toy means of simplification. As an example, in the paper concerned with the design of openings by Obert, Duvail, and Merrill ((15), the layered case is considered only from the point of view of layers which form flat roofs of spans; cases in which the number of layers is small for any linear dimension of the opening., or for which the roof is curved or arched, are referred to the ideal theory of elasticity under the hypothesis that such a simplifica tion will present no serious design problems. However, for large openings, located at considerable depths below the surface, it may be argued that the simplification to homogeneous media may not give accurate results in general. The purpose of this section of the report is to outline the important interactions among the several materials involved, and their COLORADO SCHOOL OF MIMCC «CS£*KCM fOUtlDJlTilOtl 59 mutual effects upon deformation behavior which become important in the construction of a large opening. Except where otherwise stated, the initial assumptions are that the medium is made up of layers whose interfaces are plane surfaces; each layer.is of uniform thickness and the bounding surfaces are normal to the direction of a uniform gravitational field; the material composing each layer is homogeneous and isotropic; parameters determining behavior at the interfaces are uniformly distributed throughout. In order to simplify the analysis of deformation consider first a semi-infinite medium composed of two materials, separated by a plane boundary. The reference coordinates are cartesian, with the surface and interface lying in xy-planes. In equilibrium, the only forces acting are taken to be the body forces; thus, Sy » 0 5x' 0 SJJ « - /o, q z, for the upper layer z for the lower and 5ZZ * ~/°z3 z~ layer. Since the assumption is that there are no other forces acting, the nature of the contact at the interface is immaterial in the undisturbed medium. The vertical displacements, , can be calculated easily, knowing only the elastic parameters of the two materials. Once the medium is disturbed by the introduction of a superimposed stress field, the nature of the interface becomes important. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 60 Let a spherical cavity be introduced into the combined medi um, such that the interface lies at the equator, with the origin of the coordinates at the center of the sphere. A stress condi tion is superimposed which required that £V«2at r* a if a is the radius of the sphere (transforming to spherical polar coordinates X Assume that the cavity is sufficiently small with respect topqz that a uniform stress can be used as the first approximation. If the interface between the two materials transmits no traction outside the normal load (friction-free), the distribu tion of stresses can be calculated for each material for a geometry which includes a hemispherical pit at the free surface under the action of normal compressive load $z.*-ptgzt , for r>a . Even in this case, the computation is involved, but is available in principle. The ideal condition of a friction-free interface is not attained in practical cases ; some force acts at an interface between media whose elastic properties are different. The extreme condition of the practical case is an interface which allows no "slip", or relative motion between adjacent elements. In this case, any value for tractions is transmitted as a deformation at the interface. The important contributors tp the magnitude of deformation, given the same compressive load, will be the differ ences in the elastic properties; the stresses at the interface will be determined by some function of these properties T/j - Sz f (£, * ) V, * vz) , COLORADO SCHOOL OF MINES RESEARCH FOUNDATION which must satisfy the boundary conditions, as well as the equilibrium and compatibility conditions. The result is that a smooth surface remains on the deformed boundary of the sphere (in the theoretical case) - the tractions must have a zero resultant across the interface, and continuity demands that the points at the surface of the sphere do not undergo relative rigid displacements. The difference between the materials in elastic response must be "shared" in the neighborhood of the interface, which results in an oppositely directed deformation for the materials,, of equal magnitude, along any direction in the interface plane. It is implied in the preceding discussion that the reaction across an interface may be considered as the result of a resistive force, or friction. Since this mechanism appears to describe the relative motion of bodies in contact rather well, it is the only one which needs to be incorporated: i.e., small segments juxtaposed across the boundary between substances resist relative motion in the xy-directions with a force whose direction of application is in the xy-plane. The friction force may be considered to have a critical value fp , below which value an oppositely directed force component does not generate motion. The friction of the coupling between layers is in general also related to the motion, as in other areas of mechanics, and is to be taken as a dissipative force proportional to the speed of motion. Other mechanical and thermodynamic mechanisms could be invoked which would make the reaction at the interface more complex. COLORADO SCHOOL OP MINES RESEARCH FOUNDATION 62 Mathematical solutions relative to coupled elastic systems become very complex when the friction is included in the system; hence, physical arguments are presented which serve as a guide to an intuitive visualization of the problems associated with the opening of a cavity in a layered medium. It should also be noted that friction effects upon motion are only partly under stood for laboratory-produced conditions, and that further confounding by the introduction of varying contact surfaces and irregular interface shapes into the problem leads to extreme complexity. However, a brief argument which employs simple geometries as well as a simplified relationship for the friction forces acting at"the interface is believed to be valuable because of the possibility of selecting bedded site locations in the future. The reasoning should also lead to general conclusions about the ideal variations of parameters among the materials in the structure of any formation as well as the types of physical response which should be measured in order to determine the suitability of any prospective site location. Consider a spherical opening in a two-layered medium, the interface of which is a horizontal plane in the plane of the center of the sphere. In this case, assume that the relative motion at the interface is restricted by a friction force, f, between the materials. As in ordinary motion problems, it can also be assumed that there is some critical value of the force - (difference of forces between media), fp , which must be exceeded COLORADO SCHOOL OF MINES RESEARCH FOUNDATION ^efqre relative motion occurs. This friction force can be interpreted as a stress immediately, say , and it can be hypothesised that relative rigid motion does not occur as long as ^ik ^ "^3 * A further simplification is implied by the above statement: is uniform and the same in all directions. A typical geometrical configuration is given in Figure ( 3), for two adjacent materials, A and B.. Obviously the displacements, and therefore also the stresses will be dependent on the values of the elastic parameters and the variation between them. For the purposes of the initial arguments, the materials will be assumed to be ideally elastic. Incorporation of a horizontal bedding plane further simplifies the argument in that relative motion will occur across the interface due to the stress differences and not due to forces of the gravitational field alone. A form of solution for problems in elasticity may be applied to the present case: this form is restricted to spherical geometries: where u is the displacement vector, ^ is a function of the coordinates, i.e., of the position vector r_ , and G is the elastic modulus, ^m^/z . Displacements, therefore strains, are defined directly by jzS , which is called the strain function, or the strain potential. COLORADO SCHOOL OF MINKS RESEARCH FOUNDATION 64 Materia/ 3 ( zqui / ibrum positto/) C«9 ui/ibrum of c/emtnt ' FIGURE 3 (a)S(b) from this equation, a Poisson equation is immediately derived? 2 q cfiv _u. « 9z <£ It is immediately obvious that the value of the displacements With respect to the initial fixed reference frame depend upon the value of G, one of the set of physical parameters which is characteristic of the material. If G exhibits a discontinuity at some surface in the space, then a discontinuity in motion is implied. However, the determination of the motion, even in the unconstrained case (no friction at the interface), becomes complicated because of the equilibrium conditions and boundary conditions which must be satisfied. The nature of the dissipative force is such that for any opposing force > 5^ > relative rigid motion occurs between segments across the interface. There may also be some minimal force which is required to sustain rigid motion and related to the system* in general, by The displacements of the media can be determined front the gsnerail expressions for elasticity (formally); U-ik * &ik * (^ik " &ik tine- strain tensor in terms of the stress tensor; where K andjx are constants of the material. Thus, it can be seen that the strains and therefore the displacements will be different, under similar stress fields, if the material constants are different (the ease at the interface). Furthermore, in the vicinity of the CO l. OI» A O © • C M O O L. OP MINKI RESEARCH FOUNDATION 66 interface, the nonconservative dissipative force must ibe accounted for when motion occurs, and there will be a decrement from the elastic, conservative strain, ; the dissipative stress tensor,, of the general form: It can be seen that complexities arise immediately from the form of the general expressions. However, let us state the qualitative arguments which are compatible with the physical theory . For stresses acting at the plane of the interface,, the tangential tractions (i.e., the stress due to the differences In physical parameters) exist at this boundary and mo relative motion occurs until the condition 3^ is realized.. At {this point, relative motion occurs across the boundary,, resisted toy the dissipative force .. As long as this motion is steady,, the quasi-static stress (tangential traction)/transmitted across the boundary must be J/% ^ $#£ + It should be rem ember "fik . Hence , it is possible to have stresses along the I interface, most likely in the range This stress is shared by both media; since the theory of deformable media states COLORADO SCHOOL. OF MINES RESEARCH FOUNDATION 67 in general U-i/c s Gijkl J,'ik •> and the force jf* is opposed by an equal dissipative, regardless of which side of the interface is considered. In order to satisfy the usual boundary conditions, these stresses must decrease to zero as the distance from the inter face and from the cavity increases indefinitely, so that the stresses due to the different media in the field are local to the interface and the boundary of the opening. The empirical strength characteristics of the materials must now be incorporated into the arguments in order that they have meaning for establishing selection criteria. At this stage of development, there is no particular advantage to be gained by choosing one of the usual strength criteria. Instead, it can be stated, in general, that if some value of stress or stress- difference in the elastic substance attains or exceeds an empirically determined critical value, failure will occur. In the presence of friction, it is expected that certain stresses in the vicinity of the intersection of the cavity surface and the interface will be greater than for a homogeneous medium, or a "nonfriction" interface. Thus, if these stresses provide values in excess of the strength criterion, localized failure will be expected. The extent of failure will depend upon the magnitude of the differences among the physical parameters, since it is these differences which affect the relative displacements. COLORADO SCHOOL OF MINKS RESEARCH FOUNDATION 68 The Multilayered Case It is now possible to make some very general extensions and additions to the arguments presented above. Of greatest interest is extension of these ideas to a multilayered case. The geomet rical assumption will be that the surface of the opening intercepts many of the horizontally bedded layers in the medium.. Conversely, the thickness of each layer is small compared to the linear dimensions of the cavity. The terms many and small are purposely somewhat vague — it is intended that in cases where the deformation differences are great between layers, the curvature on the "edge" of the opening normal to the plane of the layer will be small enough that the opening in the layer can be considered to have cylindrical generators without loss in applicability. The first case will have the general assumption that all layers are ideally elastic. All bedding planes may be taken ;as horizontal, and it will not be necessary to assume that all layers have the same thickness. In rectangular cartesian coordinates, it can be assumed that, neglecting the additional traction at the interfaces, ^zz ~ "7°// g h ^ Pii % h where the z-direction is vertical, and fa is the depth of the layer from the surface. Since the medium is layered, p;; repre sents a summation of densities over depths for each of the overlying layers. The stress distribution for the layer is approximately that for a cylindrical opening in a heavy medium. COLORADO SCHOOL. OF MINES RESEAR CM FOUNDATION 69 Tq this must be added the forces due to constraint by the over lying and underlying layers. As an illustration (Fig. 4 ), assume I that both neighboring layers have the same properties, but are different from the layer being considered, such that they exhibit greater deformation per unit stress than the "sandwiched" layer. Also assume that the thickness of the sandwiched layer is small enough that the body forces can be considered to be uniform in the immediate vicinity of the layer and the material confining it on either side. The net result of the traction at the inter faces will be a radial tension in the layer being considered. This effect will be localized in the vicinity of the opening, because of the localized nature of the distribution due to the opening- If the tension exceeds the tensile strength of the material, cracking can be expected, in directions normal to the direction of the tensile stress. However, the material can be expected to retain its compressive strength in the vertical direction, although the value may be somewhat reduced due to the partial release of confinement by the opening of cracks. All the previous general arguments still hold in the multi- layered case, so that the value must be greater than the critical value of tensile stress, or rigid relative motion will occur, reducing the potential strain energy. Thus, the stress due to the "shared" displacement at the discontinuity must exceed critical stress values for strength criteria before failure will occur. COLORADO SCHOOL Or MINKS RESEARCH FOUNDATION MARSAJAU A 1 * Hi:. a < MArt*/ai. B I—" r( u* \ Ur* MAF£*/AL A FIGURE 4 No special attention will be devoted to roof and floor conditions. Because of the dome configuration resulting from spherical geometry, no large span of a single layer will be exposed. However, it should be noted that, as the angle of inclination from the horizontal zero increases to 90° at the vertical, the layers become less representative of a plate with a cylindrical hole. Each successive plate will have more over hang, and some breaking may result from bending in the overhang; however, it is believed that sufficient measures can be taken to prevent excessive breaking. In the elastic case, characteristic stress distributions in any layer considered will depend upon the differences in properties between that layer and each of the two layers bounding it. Because of the hypothesized interactions, the stresses will also, depend partly on the differences exhibited by layers which are further distant, but this contribution will be expected to be much less than that for immediately adjacent layers. If the materials composing layers are granular, and if localized stresses, about the interface exceed the critical stresses for the material, localized failure or crushing might be expected: at the interface. This variety of failure would create an interface with different friction characteristics, relieving the excessive stress concentrations. layers which lie above and below the cavity are not discussed here. Although motion may be expected in these layers, the stress concentration fields fall off rapidly away from the COLORADO SCHOOL OP MINES RESEARCH FOUNDATION 72 cavity and these layers are not expected to be important contrib utors to the behavior of the materials at the opening surface. The second case to be discussed is the layered medium containing plastic or viscoelastic layers; i.e.,, layers subject to flow. The inclusion of layers which exhibit time ratio of strain are expected to have significant effects on the stress field in the vicinity of the cavity. Supposing that the layer can be adequately described as a viscous fluid, it will be expected to flow into any opening, confined by its "channel" of underlying and overlying structure, in response to the imposed stress field. It will also be assumed that the material of the layer is incompressible, so that the equation of continuity takes the simple form cf/y V-0( v is the velocity vector). The equation of motion, or Stokes-Navier equation then has the form ff'(v -2) +• -iL ^ V where p is the density , ^ the viscosity, and p tthe pressure. iFor the stress tensor, as before., An expression for the force acting on an element of surface bounding the viscous medium can be written; the force is just the momentum flux through this element. If the surface element is df » the momentum flux is ITik « ( fViVk - & it) dfk , where TT]^ is the momentum flux density tensor, TT;k - p tf)lc i- pv; vk -cr!k *- COLORADO SCHOOL OF MINES RESEAflCH fF O U N D A TIOfN as above. If dfk is written in the form dfu'r>kdf, where n is the unit vector along the normal, and under the condition_V"0 at a solid surface, the force _P acting on unit surface area is s Pi ~ <*ik "v. ' pn-, - Oik nk . The tensor Formally, the problem can be solved for given boundary conditions, and the motion can be predicted. However, since the stress field is in general not uniform, the actual solution is very complicated. As mass flows out of the "channel", under the assumption that voids are not formed between layers, the volume of the viscous layer changes over time. The result must be further deformation of the adjacent layers, in the form of bending. Additional stresses due to bending may lead to failure in these confining layers. Whereas the stresses in the viscous layer may undergo stress relaxation as a result of displacement, the unfavorable condition of increased stress concentration in adjaceijt layers indicates that plastic or viscoelastic layers in the forma tion should cause rejection of the site, especially if the opening must be stable for a long period of time. COLORADO SCHOOL OF MINKS RESEARCH FOUNDATION 74 The conclusion which may be drawn from the above arguments is that layered formations can be included for consideration in the selection of sites for large openings. This statement must be qualified by specifying the conditions under which bedded materials would be suitable^ At present there are DO mathematical. solutions which are directly applicable to these cases, although solutions could likely be had. Consequently, no quantitative limits can be stated for important relationships relevant to the stability of a given opening. Certain qualitative conclusions can be suggested from physical reasoning. Bedded formations may appear initially to have definite advantages over homogeneous igneous structures. These properties include a minimum degree of jointing and fracturing, and a possibility of fewer residual tectonic stress distributions. Some types of rock, such as hard limestones, have very high critical stress values. A combination of these factors would increase the probability of opening a large, stable cavity; the joint spacing alone may make bedded rocks more favorable than highly jointed homogeneous granite. Coupled with these structural properties will be an evalua tion of the differences among layers in the vicinity of the cavity. If the physical properties (especially the elastic parameters) of the materials do not vary greatly from one layer to the next, then localized failure could not present any serious problems. The nature of the surface at interfaces would need to be evaluated in order to determine whether serious fracturing COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 75 or bursting might occur in any case. Numbers of layers per linear cavity dimension (i.e., relative layer thickness), does not appear to affect overall structural stability. Although the stress fields would be markedly different the extent of local failure due to the stress concentrations at the interface is expected to be similar under similar loading conditions. The considerations and conclusions made in both the plastic case and the layered case have been with reference to a spherical cavity. However, similar conclusions apply to other shapes which approach the smoothness and curvature of a sphere. Note: References for Chapter 3 and Chapter 4 follow Chapter 4. COLORADO SCHOOL OP MINKS RESEARCH FOUNDATION. DISCUSSION OF THEORETICAL RESULTS AND OPTIMUM CASES Spherical Cavity An inspection of Tables 1 — 17 shows that tensile stresses will exist at some points on a spherical cavity located in a uni axial uniform stress field (e.g. a*-*-CT,, Oy ,<5^ j as x and y-*©©). For this reason the cases studied were reduced to those of equal uniform stress fields art in the x and y directions together with a uniform stress field a, - It is noted that i.t&, * C O L O RA D O S, C H O O. L OF MINES RESEARCH* FOUNDATION In view of the above it is apparent that a spherical cavity would be unstable (or questionable stability) under the follow ing conditions: a) N = 4 at all depths considered b) N = 2 at depths greater than 3,000 feet c) N = \ at all depths considered due to the near vanishing of cr ^ and Og at the top (or bottom) of the cavity. Displacements were computed at the surface of the spherical cavity for the hydrostatic case N = 1 and the radial displacement is up -- o-y) Thus for the example cited and a cavity with center depth 4,000 feet; Up - 2.25Y/-z?)a /O'3 For a = 300 feet and & = 0.15 a decrease in the diameter of the cavity due to the field stress 15$ = 4520 psi in all directions would be 11-5 inches. Prolate Spheroidal Cavity Only one ratio a/b = 1.998 was considered in computing stress concentrations for the prolate spheroidal cavity. Results for other ratios could be computed very easily using LGP30 computer programs available. It was thought that extreme ratios would not be desired in constructing the cavity for which this report has been prepared. For example, for the case of ,a/b = :2p00 and a volume equivalent to a sphere of radius rc it is found that A COLORADO SCHOOL OF -MINE9 ;R :E 9 E A RC« .F O U *N D jA T I O N 79 a - - AZC, ,c \ 14 > fa* £•_ s 0e.&Jr* A.+ J and fox rQ = 300 feet a = 375 ft. b * 16$ ft. Again as for the sphere the hypothetical case of a granite is used to illustrate the manner in which the computations can be used. For V = 0,15 Tables 32, 33, 35, and 36 show that Gja is the maximum stress for CC = cc Q (at surface of cavity) at * O* or 180° (top or bottom of cavity), if N 2 1 and for = 90° if N < 1. These maximum values are shown in the following table for <5& = 1.13 H, where k is the depth of the center of the cavity Maximum <3* / in psi { 77 = 0.15) 4 ; 2 1 k\ : h j 2 y 000) 18,800 9,000 ; 4,100 2,900 3,000 |3,000 j 28,200 13,500 i 6,*00 4,300 4,500 14 y 000 37,600 18,000 8,300 5,700 j 6,000 The minimum stresses involved for the cases above are of the order of 0»26; k and therefore do not indicate trouble with tensile stresses in the case of the spherical cavity. CObaKADia S C H. O O L or HINEI RESEARCH foundation* It is thus seen that the prolate cavity is not feasible for N = 4 at any depth considered or for N = 2 at a depth greater than 2,000 feet. It is feasible for all other cases considered. Oblate Spheroidal Cavity The computation of stress concentrations was limited to the ratio a/b = 0.5005 for the same reasons regarding extreme ratios mentioned for the prolate spheroid. The ratio a/b = 0.5005 gives the same values of a and b (except that they are inter changed) for the spheroid having a volume equivalent to that of a sphere of radius rQ. The maximum stress (from Tables 46, 47, 50, and 51) concen trations are those of - 90° if N&l. The following table gives the maximum v.alues of ** - Maximum &/» in psi y = 0.15 4 2 1 1 2,000 13,600 6,200 5,700 4,900 •6,900 3,000 . 20,400 : 9,300 8,600 9,700 10.,.3 00 4,000 J 27,200 | 12,400 11,500 13,000 13 ,800j The oblate spheroid is seen not suitable for N - -4 at .all depths of the table, questionable for N — 2 at a (depth approaching 4,000 feet:; the latter remark including the situations for = .§ and J, In addition, Tables 46, 47, ;50, and .51 diow that N = J gives rise to tensile stresses at the top and bottom of the cavity.. Other*• wise stresses are safe in so far as tensile stresses are concerned." COLORADO SCHOOL Of Ml««« *«««*« C-H Stresses in Vicinity of Cavity All solutions for the spherical cavity involve terms containin )3and Thus the quantity £-p-J3is a measure of the rapidity wii;h which stresses approach the field stress in the medium surrounding the cavity- It is seen that if p* 3a (i.e. , one diameter from the cavity)^£|*< 0)..04 and therefore stresses will be within 4 percent of the field stress value. C Q UO *ADO ft C(+ OOL Or Mi I MB* RlftlARCH FOUNDATION 82 References 1. Southwell, R. V., and Gough, H. J., 1926, On the concentra tion of stress in the neighborhood of a small spherical flaw; Philos. Mag., v. 7, no. 1, p. 71. 2. Goodier, J. N., 1933, Concentration of stress around spherical and cylindrical inclusions and flaws; Am. Soc. Mech. Engineers Trans.,55, v. 7, p, 39, 3. Love, A. E. H., 1944, Mathematical theory of elasticity; Dover Publications, New York. 4. Goodier, J. N., and Timoshenko, S., 1951, Theory of elastlciit New York, McGraw-Hill Book Co. 5. Sokolnikoff, I. S., 1946, Mathematical theory of elasticity;; New York, McGraw-Hill Book Co. 6. Sadowsky, M. A., and Sternberg, E., 1947, Stress concentr.atio around an ellipsoidal cavity in an infinite body under arbitrary plane stress perpendicular to the axis of the cavity; Jour. Appl. Mechanics, Am. Soc. Mech. Engineers Trans., v. 69, p. A-191. 7. Neuber, H., and Edwards, J. W., 1944, Kerbspannungslehre: Ann Arbor, Michigan. 8. Boussinesq, J.., 1885, Application des potentielsj Gauthier- Villars. 9. Papcovich, P. F., 1932, Solution generale des equations differentielles fondamentales de elasticite, exprim^e par trois fonctions harmoniques: Acad. Scl. ((Paris) Comptes Rendus, v. 195, p. 513-515. 10. Edwards, R. H., 1951, Stress concentrations .around spheroidaj inclusions and cavities: Jour. Appl. Mechanics.. 11. Richart, R. E., Jr., and Tetfzaghi,, Karl, 1952, Stresses in rock about cavities: Geotechnique, v.. 111.. 12. Sadowsky, M. A., and Sternberg, E.., 1949, Stress concentra tion around a triaxial ellipsoidal cavity:: Jour.. Appl.. Mechanics, v. 16, no. 2, p. 149. 13. Balmer, G. G., 1953, Physical properties of some typical foundation rocks; !U. :S. Bur. Reclamation, (Concrete Lab. Kept. No. SP-39. COLORADO SCHOOL Of WINES FF vO ill rN dD A 1T il (O rN 83 14. Deere, Don V., Langhaar, H. L., and Boresi, A. P., 1959, An evaluation of the factors influencing the stability of a large underground cavity: Rept. to AEC, AECU- 4654. 15. Obert, L., Duvall, W. I., and Merrill, R. H., , Design of underground openings in competent rock: U. S. Bur. Mines Rept. Inv. 16. Westergaard, H. M., 1952, Theory of elasticity and plasticity: Harvard, Cambridge, Mass. 17. Patel, S. A., Venkatraman, B., and Bentson, J., 1962, A stress-strain relation for inelastic material behavior: Contract No. NONR 839(23), Proj. No. NR 064-433. July. 18. Barenblatt, G. I., 1961, The mathematical theory of equilibrium cracks formed in brittle fracture: Transl. Service Branch, Foreign Technology Div., Wright-Patterson A.F.B., Ohio, FTD-TT-62-132/1. Source - Zhurnal Prikladnoy Mekhaniki i tekhnicheskoy fiziki, no. 4, 1961. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION PART II. PRACTICAL CONSIDERATIONS RELATING TO CAVITY STABILITY The Lithology The composition and physical characteristics of the rocks, irrespective of local structures within the rocks themselves, are important because they govern the load-withstanding capabilities of the structure. The ultimate tensile strength, compression strength, and other mechanical and elastic properties developed in the last analysis depend upon the molecular structure of the mineral grains. However, before this point is reached the nature of the discrete grains in the rock, their size, the nature of the boundary between grains and the degree of their coherence governs the "macro" strength of the rock. Crystals, in general, show different properties along different crystallographic axes. If the discrete minerals are randomly oriented and dispersed the rock strength reaches an average that is fairly consistent in all directions. However, it sometimes happens in nature that mineral grains are lined-up in certain directions. In this case the rock properties will be different in different directions. As far as can be seen from surface observations and without laboratory work, the site 3 granite does not show any particular lineation in its petrofabric structure and it is, therefore, reasonable to assume that the properties are more or less uniform in all directions. This assumption will, of course, be checked by actual measurements on drill core. Local Structure Local structure, in the vicinity of the proposed cavity, is the most important aspect of planning for the work. Theory may COLORADO SCHOOL OF MINES RESEARCH FOUNDATION say that stability is independent of the span of the opening and that loading in various directions can be either measured or calculated, but because local variations and structure are found in a random manner and because there is no way to determine where the local structure is at depths outside of drilling, there is always a good deal of uncertainty about opening up a cavity. It is usually customary in calling for bids on various under ground structure to describe the rock in more or less general terms. One useful classification is given by the California Department of Water Resources (2 ). Rocks are divided on this basis as follows: "Intact Rock—Intact rock contains neither joints nor hairline cracks. Consequently, when breaking, it breaks across sound rock, and breakage is not influenced by joint and fracture patterns. Stratified or Schistose Rock—Stratified or schistose rock consists of individual strata with little or no resistance to parting along boundaries between strata. Strata may or may not be weakened by transverse joints. However, if transverse joints and fractures are spaced so closely as to destroy bridging action of the strata, rock is classified as very blocky and seamy, or moderately blocky and seamy. Distance between stratifications is generally less than five feet. Where distance between bedding planes is greater than five feet, the rock is better classified as moderately jointed, moderately blocky and seamy, or very blocky and seamy, COLORADO SCHOOL OP MINES RESEARCH FOUNDATION 86 depending on spacing of joints and fractures. Massive, Moderately Jointed Rock—Massive, moderately jointed rock contains joints and hairline cracks, but the blocks between the joints are locally grown together or so intimately inter locked that vertical walls do not require lateral support. Moderately Blocky and Seamy Rock—Moderately blocky and seamy rock consists of chemically intact or almost intact rock fragments that are entirely separated from one another and imperfectly interlocked. In such rock vertical walls may require support. In moderately blocky and seamy rock, the joints and fractures are so spaced that individual blocks are larger than two feet in diameter. This classification applies to both sedimentary and crystalline rocks. Very Blocky and Seamy Rock—Very blocky and seamy rock consists chemically intact or almost intact rock fragments which are entirely separated from each other and are imperfectly inter locked. In such rock vertical walls may require some support- Very blocky and seamy rock differs from moderately blocky and seamy rock in that the joints and fractures are so spaced that the intervening blocks are less than two feet in diameter. Completely Crushed or Unconsolidated Rock—Crushed or unconsolidated rock consists of sand to pebble sized particl.es that are chemically intact and are very loosely consolidated or unconsolidated. Fault gouge is sometimes present. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 87 Wet Competent Rock—Wet competent rock includes those rock types ranging from intact through very blocky and seamy under a saturated condition. Water inflows into the tunnel come from joints and fractures separating the individual blocks. Estimated inflows of 100 gpm or more from the heading must be anticipated before the ground is classified as wet competent. Wet Crushed or Unconsolidated Rock—The term "wet" is applied to this classification when the material is saturated. Inflows into the tunnel come from interstices between the individual particles. Estimated inflows of 100 gpm or more must be anticipated before the ground is classified as wet crushed or unconsolidated." Site 3 is probably classified as moderately blocky and seamy rock; conditions here, however, may change in depth and improve somewhat. Regional Geology The distinction between regional and local geology is not easily drawn. For present purposes we consider everything within three diameters from the periphery of the cavity as being local structure, but certainly any structure near the three diameter limit would be of more importance than structure farther out. The far out structure, however, may be of importance because it acts at a distance to cause stress on the periphery of the cavity. These stresses may be caused by such things as faulting, isostatic adjustments of mountain ranges, thermal COLORADO SCHOOL OF MINES RESEARCH FOUNDATION stresses caused by heat sources or sinks, and probably other geological processes. Very careful consideration should be given to these facts before making a definite decision about site 3. As stated previously, there is probably a fault through Wilson Canyon within one mile of the site. If the fault has acted to relieve under ground stresses, this is a favorable feature. However, it may act to create underground stresses. It may also project downward into the cavity area and may be accompanied by en echelon faults that go through the cavity area but are not apparent on the surface. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION CONSTRUCTION CONSIDERATION TO PROMOTE STABILITY The sphere or ellipsoid contemplated in connection with the ultimate use of the cavity are favorable geometries to promote stability. Excavation problems may be a little more difficult but in our opinion are not necessarily enough more difficult to consider changing shape of the cavity. From a theoretical stand point a sphere or ellipsoid is amenable to analytic solutions of equations from the theory of elasticity and the stress conditions can in principle be calculated. From a practical standpoint they avoid "stress risers" where the local stresses may exceed the ultimate strength of the rock. Rectangular or other openings where sharp corners are formed are particularly undesirable. In the particular case being considered there may be stress risers at the points where the access tunnel joins the sphere or where temporary construction features such as raises or subdrifts are constructed. In general, all openings should have arched backs, be more or less circular in cross section, and should / intersect other openings at approximately right angles. These geometries will minimize stress risers and arched backs will tend to be in equilibrium. The excavation plans given in Volume III of this report appear to meet the above criteria. It may however be necessary to modify construction plans to meet local conditions. Possible Linings and Support Customer criteria at present do not allow for any internal support in the cavity, and would therefore preclude permanent use COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 90 of struts, posts, or girders across the opening. It is conceivabl that the ultimate use would permit supports that were tight across the periphery of the opening or would permit use of a concrete or steel lining. The cost of such arrangements would, however, be prohibitive. Therefore, the design must be limited to rock bolts, cyclone fencing, grouting, roof sewing and perhaps temporary local supports. Rock bolts will to some extent take the place of lining and can be perhaps of the order of 50 feet long and of sufficient diameter to take shear and be torqued to develop an arch surrounding the opening. They can be used to pin fairly large rocks together but if the local jointing at depth is such as to dissect the rock into large pieces, bolting will probably not be effective. Grouting or cementing with the epoxys or other suitable material in connection with bolting will help but there can be no guarantee that the ground can be held. Gunite and other thin layers of sprayed on materials can be helpful in preventing air slacking and in holding light slabs or pieces of broken rock. Guniting will also improve reflectance of the rock and the underground lighting. It will not support heavy loads. Rock Bursts A rock burst is a sudden, unanticipated violent breakage of rock in an underground opening. Rocks are spalled from the sides of the opening with considerable force. The size spalled may vary from small grains to very large blocks of rock. The exact COLORADO SCHOOL OF MINES RESEARCH FOUNDATION mechanism that causes rock bursts is not well understood but in general is related to the amount of energy that can be stored per unit volume of rock. In general, hard rocks are more likely to be subject to rock bursts than soft rocks. Granites, quartzites, limestones, and gneiss are more susceptible than shale, sandstone and siltstones although there are exceptions. Bursts are more likely to occur in deep mines than in shallow but this does not mean that rock bursts have not taken place in shallow mines. Tectonically active regions or regions that exhibit many structural features such as anticlines and faults are more likely to have rock bursts than areas that do not contain these features Rock bursts may occur immediately after blasting or may "lie in wait" and take place a considerable length of time after the opening has been made. A geophone or seismophone may give an indication of abnormal activity inside the rock but interpreting the noises heard in such devices is a subjective matter and depends upon experience in the particular mine. It is not likely that there will be any experience available at the cavity site. After a rock burst occurs, the ground is in a state of equilibrium for that particular stress field. Subsequent mining operations, however, may change the stress field and rock bursts can recur. The shape of the opening, a proper mining sequence and the use of backfill in mined-out areas help to a limited extent. Blasting to deliberately trigger a rockburst is also helpful in some cases. However, there is really no effective way to handle rock bursts and they cannot be prevented because COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 92 the forces involved are too great. Effect of Construction Sequence on Cavity Stability If the in situ stress measurements show that the stress field is uniform, the mining sequence should be planned to maintain a reasonably uniform load over the volume to be opened. In general a uniform stress field implies that mining should be more or less symmetrical around the centroid of the cavity. If, however, the in situ stresses are not uniform and change during the course of construction it may be desirable to sequence the mining operations in an unsymmetrical pattern. The pattern to be adopted will depend on the nature of the stress field and, of course, must be considered in the light of construction requirements. Present knowledge of site 3 is insufficient to see whether or not this should be done. If the ultimate customer lays down criteria of smoothness similar to those specified in the Tatum Dome project, construction costs will be high. A smooth surface, however, has the advantage of not presenting stress risers. This consideration probably resolves into a trade-off between cost and suitability for end use. It will be important to prevent overbreak in the mining operations and to avoid shattering of the walls. Smooth blasting techniques near the periphery are advisable. It will also be necessary to load very lightly and perhaps use a low velocity powder near the edges of the cavity, since it is important to prevent cracking of the cavity walls. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 93 References 1. Hopper, Richard H., 1947, Geologic section from the Sierra Nevada to Death Valley, California: Geol. Soc. America Bull., v. 58, p. 393-432. May. 2. State of California, Department of Water Resources, 1959, Investigation of alternative aqueduct systems to serve southern California; Appendix C. Procedure for estima ting costs of tunnel construction: Bull. no. 78. September. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 94 PART III. EXPLORATION PROGRAM FOR SELECTED SITES INTRODUCTION The exploration program discussed in this chapter points toward obtaining information on those factors outlined in the previous chapter. It is not in any sense a planned exploration program either for areas in general or for site 3 in particular. If a different site were to be selected it is quite possible a much different approach to the well logging and drilling programs would be advisable. The costs and the times of execution for the various items making up the program are rule-of-thumb estimates only and are based upon information published by well logging companies and by drilling companies as of the latter part of May 1963. No adjustment has been made for future upward or downward trends of exploration costs. Furthermore, a. potential contractor would of necessity examine the particular site '.before making a bid. It also has been assumed that the job would be organized an much the same way as other jobs by the Corps have been and that some of the services such as the bore hole camera work, keeping track of costs and progress, safety during exploration and tthe like would be furnished by an area engineer's office.. We lhave however included a cost estimate for outside work and for engineering specifically required for interpretation of the specialized data that must be gathered. It does not provide for construction surveying or for engineering outside of the actual COLORADO SCHOOL OF MINES RESEARCH FOUNDATION exploration work. The estimated times for completion of the various phases are optimistic. We have included a down time factor for drilling but we have assumed that there will be no lost time on account of weather, fish jobs or for other unforeseeable reasons. GENERAL GEOLOGICAL RECONNAISSANCE The following discussion is based on the assumption that the cavity will be located at site 3. The general recon naissance should cover an area perhaps for a radius of 3 miles surrounding the actual sphere. The area is not critical, but should be governed by the number and nature of geologic features, particularly faults that are found in the area. It need not be in great detail but should provide information on rock types, major structures, the geologic history of the region and the prevalence of ground water and mineraliza tion. It is necessary in our opinion to make such a survey in order to get an idea of what stresses and other factors may be active in the immediate region of the cavity. Special attention should be paid to faulting, major folding and to the thermal history of the area. Much of this could be done from examination of aerial photos and published reports. COLORADO SCHOOL OP MINIS RESEARCH FOUNDATION 96 It will be helpful to prepare maps showing possible projection of faults or the axes of major folds into the immediate con struction area. The throw along faults should be determined as it may be indicative of the magnitude of the stresses causing the faulting. The presence or absence of associated en echelon faulting and fracture will be indicative of possible stress relief. The lithology of the region and the knowledge of when and how the rocks were formed will be helpful in determining whether or not one can expect stressing due either to static loading or to thermal stresses within the rock mass. If it is possible to get access to any mining workings in depth in the granite, a study should be made of the closing of joints with depth. It will also be helpful to have an idea of prevalence of sheeting planes and other lineation within the rock, first, so that we have an idea of how homogeneous and isotropic the rock mass may be and secondly and more important, to know what kinds of planes or weakness and possible troublesome discontinuities occur in the cavity region and may be encountered in the construction. DETAILED SURFACE MAPPING IN AREA A detailed study should be made of the area in the immediate vicinity of the access tunnel and cavity. The area covered should be, say 1500 feet on either side of the access tunnel and 1500 feel beyond the center of the spherical cavity itself. The scale of this mapping should be perhaps of the order of 100 feet per inch. \ COLORADO SCHOOL OF MINES RESEARCH FOUNDATION The purpose of this detailed study is to prepare plan and sections of major features such as joints, contacts, fault planes and the like and to see where they may go in depth. If it is possible to orient underground workings to avoid major structures, speed and cost will be improved and greater safety will result. Enough data should be taken during the course of this mapping program so that a three-dimensional picture of a region can be inferred. It will be necessary to have facts about stream run-off and water flow conditions during various periods of the year. It is unlikely that any very wet conditions will be found with depth but information concerning the water table should be compiled so that grouting during construction can be anticipated. The methods of making both the regional and geologic maps and sections are standard and need not be discussed in this report. We have assumed that large-scale aerial maps of the area immediate ly surrounding the construction are already available and that preliminary office work with these large-scale maps can point to the area where most field attention should be devoted. There would seem little purpose in an extensive drilling program in the immediate area other than along the center of the proposed access and in the region of the cavity. The end use of the cavity itself may dictate that a hole from surface to cavity be provided and it will be worthwhile to investigate the requirements of the ultimate customer at this point in order to avoid duplication of drill holes. COLORADO SCHOOL OF MINKS RESEARCH FOUNDATION 98 The geologic work should precede actual construction and should be scheduled so that detailed projections of structures intc the cavity area are available before final location of the cavity and its associated access drifts is made. i , SEISMIC SURVEYS At this time it is not clear whether jor not seismic work should be done. Seismic work would be helpful in locating possible faults that project into the area and are not evident from conventional surface mapping. It might also be useful in determining whether or not the granite is consistent over ,a large area at depths considerably below the 4000 foot depth contemplated for the cavity. At the present time we do not recommend that .any seismic work be undertaken for these purposes because any nearby faults will likely .show up during surface mapping, and because of the nature of the granite itself it is mest unlikely that ;any other rocks would lie below the center of the cavity. It may be that the end use of the cavity will require that some seismic work ibe undertaken but this need not be completed before construction begins. Reflection shooting would be best for the locations of structures and also for possible changes of rocks with depth.. The work would have to be contracted.. Depending upon the detail desired and the area to be covered, costs might .approach an order of magnitude of $6000 per acre shooting a limited number (of profiles per acre.. No cost for seismic work has, .however., ibeen COLORADO SCHOOL. OF DRILLING FOR EXPLORATION Two kinds of drilling are advised. The first is drilling to obtain 4-inch diameter cores from the immediate cavity area and from the rock surrounding the cavity to a distance of 1 radius out, This drilling will be done with a heavy oil well type rig and will not start taking core until perhaps within 3800 feet below surface. It is also advisable we believe to start deflected holes at perhaps 3:800 feet depth and drill several through the immediate cavity area. The techniques for doing this are well known and depend upon whipstocks for deflecting the drill bit in the required direction. Bore hole surveys are made to follow the course of the deflected drill holes. The second type of drilling will be with lighter rigs and will get NX drill cores from various points near the sphere. Some of the NX holes will be collared in off-set alcoves from the main access drift and will be taken during the latter stages of tunnel driving. They will be scheduled so as to not interfer with the tunnel contractor. It will be necessary to provide access roads and site preparation for the rigs. It is very difficult to estimate the cost of doing this until actual holes have been spotted and surveys made. It may be well worthwhile to consider placing the rigs by COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 100 helicopter. A study of the topography maps and aerial photos of site 3 indicates that something like 6 miles of access road will be involved. A figure of $10,000 per mile for the roads has been estimated and an additional $5000 for the drill sites themselves has been assumed. It will be possible to spot the collars of the drill holes after about 1 week of field mapping. Drilling mud can be used for the entire distance although it is probable that water alone will be sufficient. There may be losses of water during the drilling because of the joints, fissures and other openings in the granite. If this is the case it will be necessary to do some cementing and if there is an in flow of water through such openings it will be advisable to make drill stem tests to determine its nature and quantity. Consideration should be given to drilling with air in the immediate vicinity of the cavity. Whether or not this is feasible will depend upon the contractor, his past experience and the type of equipment he has. The reason for preferring air drilling to a wet drilling in the cavity area is that the elastic and mechanical properties of the core may be very sensitive to moisture. It is very likely that the granite itself will be impermeable but there may be some weakening of the core by virtue of the fluid entering pre-existing microcracks. We have assumed that the drilling will be conducted on a 21 tour (shifts) per week basis and that 2 rigs will be available. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION In arriving at the time estimates we have assumed that a good penetration rate will be 3 feet per hour. Depending upon the type of rig it will be possible to pull core in 60-foot lengths and we have assumed that a trip from the bottom can be made in 1 hour and 45 minutes. We have assumed 6 percent down time on the rig. Well Logging Well logging will serve two purposes; first it gives an indication of construction difficulties to be encountered and will enable a potential contractor to assess possible difficulties and thus make a more realistic bid. The second function it serves is to furnish data which teilfls much about the character and nature of the rock between surface and the points where actual openings will be made. We recommend that the following types of well logs and surveys be made; All bore holes into the cavity should be surveyed. There are two general types of bore hole surveys. The first depends upon magnetic compass readings to determine azimuth. The second determines azimuth by gyroscopic means. In either case, the dif> or inclination is measured by displacement of a plumb bob. Eastman Oil Well Survey Company deals primarily with the first method and Sperry-Sun sells the second type of service. It is possible to obtain readings that are accurate to plus or minus 15 minutes both in azimuth and dip with either method. The methods can be both continuous or so-called single shot. It is customary COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 102 for the survey contractor to furnish reports showing the coordinates of the hole at various depths. If it becomes neces sary to resort to directional drilling surveys must be made to orient the whipstock properly. Cost of bore hole surveys are based upon a move-in charge that depends upon the location of the contractor's nearest office and operating charge is based upon time in the hole. Table 54 shows the general nature of these charges but of course it may be out dated at the time the Corps actually wi&hes to make the survey. A second type measurement that it may be necessary to make is a so-called drill stem test. It is probable that it will not be necessary to make these tests since their main objective is to determine fluid flow and to catch samples of the fluid within the bore hole. However, if water is encountered in depth it is well worthwhile to provide this information. A drill stem test gives an idea of the formation pressures, the quantity of water that will flow into the well or shaft itself, and gives a sample of the actual fluid flowing at a given horizon. This may be useful if the chemical composition of the water is such as to present difficulty in grouting. Costs of drill stem tests are likewise based upon move-in charges plus cost of making the tests. Table 55 shows the.general nature of these charges. A further device that will be very useful in connection with the exploration program is the bore hole camera tied COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 103 1 June 1963 320327 TABLE 54 Directional Surveying (over sphere) Transportation 7 trips from Bakersfield, California at $100.00 per trip equals $700.00 Surveying One complete survey over entire distance of 5,500 feet of vertical hole plus 3,400 feet of deviated hole at $0.13 per ft equalling $1,157.00 Plus 25,000 ft traversed but not surveyed at $0.05 per ft equalling $1,250.00 Plus 100 ft surveying charge for each of first surveys or 600 ft at $0.13 equals $78.00 Total for over and around sphere - $3,185.00 (for incline) Transportation Nil, conducted in conjunction with surveying over sphere. Surveying One complete survey of each hole totaling 12,000 feet at $0.13 equals $1,586.00 plus $1,000.00 for footage traversed but not surveyed making, a total cost for surveying holes collared over the incline of $2,586.00. (underground drilling) Transportation - $100.00 One complete survey of 6,400 ft at $0.13 making $832.00 Total underground - $932.00 Total Directional Survey Costs Over sphere $3,185.00 Over incline 2,586.00 Underground 932.00 $6,703.00 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 104 1 June 1963 320327 TABLE 55 Drill Stem Testing Estimates 1 and 2 Mileage Same as for sonic logging,. $150.00. Service Nil if performed in sequence with sonic log. Depth Instrument positioned by FT-Gamma Ray. 0-20,000 ft at $0.07 per ft with minimum of $145.00. 5,500 ft at $0.07 equals $385.00 Operation Depth Per Test 0-4000 $ 265.00 4001-5000 295.00 5001-6000 325,00 Auxiliary bottom hole pressure recorder run in conjunction with D.S.T. at $50.00 per test. Probably 3 tests at $265.00, 6 at $295.00, 3 at $325.00 and 3 bottom hole pressure test will be adequate with a cost of $3,690.00 Mileage $ 150.00 Service Depth 385.00 Testing 3,690.00 $4,225.00 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 105 into a closed circuit TV set. The Corps of Engineers has used the bore hole camera and has the equipment on hand. The camera takes pictures as it travels through the hole and by proper manipulation enables one to identify discontinuities. In some cases three coordinates at three points can be obtained and computation of its dip and strike can be made to correlate it with projections of discontinuities downward from the surface. It is, our recommendation that the bore hole camera be used. We have not tried to determine the cost of this work since it presumably will be a Corps of Engineers service. There are many other types of well logging that can be applied. Most of them have been developed for oil well purposes and are particularly applicable to the sedimentary rocks usually encountered in petroleum exploration. There would be little apparent use for logs such as gamma ray neutron or SP logs. We do> recommend that some sonic logging be done in order to determine where possible changes in the longitudinal velocity of sound occur at depth. Since the velocity of sound is a function of density and the elastic modulus of the rock these logs will give an indication of consistencies of elastic proper ties within the volume of rock to be considered. Table 57 shows an approximate cost of making sonic logs. We also recommend that a continuous dipmeter be run in each of the holes. The dipmeter finds the magnitude of the formation tests across the bore hole. It works by a resistivity principle recording a difference in the resistivity as the electrodes COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 106 X June 1963 320327 TABLE 56 Dipmeter Estimates 1 and 2 Continuous Dip Meter digital Mileage Same as for sonic, should not exceed $150.00. Service Nil if performed in sequence with sonic log. Depth 0-20,000 ft, $0.07 per ft with minimum of $140.00. At 5,500 ft by $0.07 equals $385.00 Operation $625.00 plus $0.13 per ft logged. Logged over .entire depth of hole 5,500 ft by $0.13 equals $715.00. $625..00 plus $715.00 equals $1,340.00 Computation $100.00 Cost Mileage $ 150.00 Depth 385.00 Operation 1,340.00 Computation 100.00 $1,975.00 Say $2,000.00 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 1 June 1963 320327 TABLE 57 Sonic Logging Estimates 1 and 2 Mileage $0.50 per mile for all mileage in excess of 150 miles round trip from nearest Schlumberger location at Bakersfield, California. Should not exceed $150.00. Service Only one service charge made for a series of operations performed in sequence, without interruption for additional drilling or running casing, during one trip to a well, $150.0(i. Depth 0 to 20,000 ft to $0.08 per ft with minimum charge of $160.00. As maximum depth would be 5,500 ft, the charge would be $440.00. Operation $0.08 per ft with a minimum charge of $80.00. 5,500 ft by $0.08 equals $440.00. Transit Time $0.02 per ft, minimum of $60.00. 5,500 ft by $0.02 equals $110.00 Stand-by Time $15.00 per hour. An allowance of 10 free hours of stand by time will be made prior to completion of first operation at the well. An additional allowance of 5 free hours will be made for each different type of service performed under the same,service charge. Therefore, probably no charges should be made for stand-by time. (continued) COLORADO SCHOOL OP MINES RESEARCH FOUNDATION Table 57 (continued) Sonic Logging Total Costs for Sonic Logging Mileage $150.00 Service 150.00 Depth 140.00 Operation 440.00 Transit 110.00 $1290.00 Say $1,300.00 Allowed for fishing and lost instruments 700.00 $2,000.00 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION crosses a boundary between two dipping formations. It is probable that drift angles can be measured plus or minus 15 minutes and bearing within plus or minus 5 degrees. It is a continuous recording detector and can be run on wire line tools. Sonic logging records the time required for a sound wave to travel definite length of formation since the sonic travel times are inversely proportioned to the speed in various formation and since the speed of sound depends upon the elastic properties of the rock matrix and the porosity of the formations and their fluid content and pressure which in turn can be related to stabili ty of a cavity we feel this type of logging is very necessary. Sound velocities in granite will be perhaps of the order of 12,000 feet per second. It is unlikely that the granite will con tain any interstitial liquids. INITIAL STRESS MEASUREMENTS The theoretical work has shown that the initial state of stress of the rock in three dimensions is of utmost importance and that it will be necessary to have a clear picture of the stress field before the size or shape of the cavity can be determined. If high stresses are encountered it will be necessary to orient the axes in a manner that it is dependent upon the existing stress field. The initial stresses existing in rock mass will also to a large extent determine whether or not rock bursts may be a problem. Assuming that the Argus Peak site is finally selected we recommend that some preliminary in situ stress measurements be made in the Ruth mine if it is at all possible COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 110 to get into it. The state of the art of in situ stress measuring is unfortunately such that it is not possible to make the measure ments at distances from a free surface of over perhaps 100 feet. This means that it will be necessary to determine stresses from the access tunnel during the course of construction and that the number of measurements to be made will increase as the tunnel approaches the final cavity site. The stresses may also change. As a result of mining operations; the time required to re establish equilibrium cannot be predicted a priori but can be approximated by periodic stress measurements. A change of stress rather than stress itself is measured in place in the rock. A suitable sensing device is placed in a drill hole preloaded to a given stress dependent upon the individu al situation and the anticipated stresses and then cut away from the rock mass, usually by over coring. When the stress is relieved the initial cavity tends to deform and the deformation is measured by the sensing device. Panek lists three different types of devices, first those that respond to deformation only, second those that have an elastic modulus matched to the modulus of the rock and third those devices that respond to deformation and stress in an arbitrary manner. The projecting stud type developed by USBM in which the stud acts on a cantilever and strain gages measure bending in the cantilever is an example of the first kind. Changes in hole diameter can be converted to stress when the elastic modulus of the rock is known or a calibration curve can be made. The gage COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 111 readings are not easily interpreted in rocks that undergo plastic flow. Fractures in the rock may also make use of this gage difficult. The second type in which the gage modulus matches the rock modulus operates by measuring a strain or fluid pressure within the gage. It is satisfactory as long as the two moduli match and if the design is such as to give the necessary sensitivity. Strain gages mounted on various materials can be used and will operate in the plastic regime. They are being tried in potash mining and are operating reasonably well. Calibration of the individual gages is required. The third type of gage is not necessarily matched to the surrounding rock and responds in an arbitrary manner. It is necessary to calibrate these in the actual range of application and to avoid situations where the rock exhibits an hysteresis effe|ct. These gages are under development. Since the current problem is concerned only with the static stability of the cavity it is not necessary to be concerned with the dynamic response of the gages. Linearity is desirable as is a gage that is stable over long time periods and fairly independen of temperature change although temperature effects can be balanced out by dummy gages and suitable circuits. Because it is not likely that plastic flow, at least in granite will be a problem, either a type one or type two g^ge would appear to be suitable. It will be possible to estimate the stress range over which the gages must act by assuming the lithostatic load in the vertical COLORADO SCHOOL OF MINES RESEARCH FOUNDATION direction and by assuming a Poisson's ratio of J to make the lateral loads equal to the vertical load. The real situation of course may be much different because of geological, thermal, or other factors. Enough measurements must be made at a given point to specify the stress ellipsoid at that point and to allow for lost readings or for averaging. Consideration should be given to making 12 measurements at each point. The in situ stress measurements should also be made continually in the region of the cavity during con struction as an aid to monitoring the stability of the cavity, to aid in planning the excavation sequence and to help interpret the geophone results. An appropriate array can be designed as con struction proceeds. Since it will be desirable to follow the stress situation frequently over many gage points, a computer at the field office would be desirable. The computer would also be useful for other engineering work. It is estimated that $325,000 will be required for making the in situ measurements. This allows for the special drilling and for some development work. Gages are not stock items and must be specially made. Work on gages can begin in the second or third month after the go ahead on the site. Consideration should be given to trying to measure stresses in the large bore holes at depth but it is likely that special tooling will have to be developed. Dial gages should be installed across discontinuities in the tunnel or sphere to watch possible slipping, and to take corrective measures if necessary. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 113 CORE LOGGING AND LABORATORY TESTS Cores from the drilling should be accurately located on the basis of the bore hole surveys and should be oriented. Orientatior can be done by the methods used by Hanna Mining Company or by measurement of the residual magnetism of the rock and the core as is done in oil well operations. The physical rock tests should be made on cores from; selected intervals. The intervals should be ehoseni oa the basis of the visual logging and apparent changes in the. lithology.. Tests, should be concentrated in the area within one diameter from, the periphery of the cavity on all sides including the volume below the bottom. Physical rock tests can be made on either a static or dynamic basis;, the present problem is concerned only with the static situation, although the ultimate use of the cavity may require that som& dynamic tests be made. This can be coordinated with the ultimate' customer- The' following work should be done and there should be enough replication to give reasonably good averages—say 65 percent confidence limits. I- Determine the density, the mineral composition, the porosity, and if indicated, the permeability. 2. Compile stress-strain curves on uniaxially loaded cylinders. This will give such quantities as the Poisson's ratio, the elastic modulus, the shear modulus and the bulk modulus, and will determine the point at which plastic flow may begin. The specimens COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 114 will also be tested to failure in uniaxial com pression and the ultimate compressive strength found. 3. Determine elastic modulus and Poisson's ratio by sonic methods since in some cases they are more reliable. These should be compared with the results from number 2. 4. Determine tension strength of the rock. 5. Make modulus of rupture tests on suitable bars of rocks in accordance with standard procedures. 6. Determine the shear strength when placed in double shear. 7. Make a series of triaxial strength tests at con fining loads close to that anticipated at the cavity location and over such a range that data can toe interpolated when in situ stress measurements have been made. Compile Mohr circles and compute octahedral shear- 8. Determine the creep properties of the rock when subjected to long term loads* This will tend to prove or disprove the mathematica1 prediction (con cerning the relative lack of creep expected in granite under the environmental (conditions specified- 9. Consider making a model cavity of a suitable material and on a basis of dimensional analysis predict behavior of full-scale opening. 10. Get samples of clay seams from any jointing and determine their swelling properties. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION The measurements should of course be made with properly calibrated equipment. Depending upon the results of the petro- fabric structure, it may be necessary to determine the properties in various directions and to take additional cores from holes collared underground in the access drifts. It is likely that there will be ltttle difference in the site 3 granite case between the apparent and the true stress- strain curves and that moduli computed from the apparent curves will be sufficiently accurate. It is reasonable to use the secant value of Young's modulus at the estimated load in making elastic theory computations. ESTIMATED COST AND THE TIME FOR EXPLORATION PROGRAM, SITE 3 Table 58 summarizes the costs for the exploration program. The bar chart in Fig. 5 estimates the time required for each of the various steps starting with the time a definite go-ahead for the project is given. The time required for driving the tunnel is shown so that the relation of the tests to the construction program is evident. The required driving time is taken from Volume III "Cost and Constructability Studies". It may be desirable to change the direction of the tunnel after some of the data is available from drilling. But this can not be foreseen at this time and it is not believed that the change in direction would be such as to gre&t)y affect construc tion costs. If site 3 is eliminated, a new estimate would have to be drawn up based on the specific area. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 116 FIGURE 5 Tentative Schedule Exploration Program Site 3 (Times are Optimistic) Number Activity Months 0 12 3 4 5 6 7 8 9 , 10 II 12 13 14 15 16 Geol. mapping Mill l-m i i i I > i i I i i i I i I i 1 i ii I i i i 1 i i i 1 I i i I i i i I i i i I i i i I i i t 1 i i i I i i i I Access roads Drill sites 0 I 2 3 4 5 6 7 8 9 10 If 12 13 14 15 16 1 i fnfi i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i 1 i i i I i i i I i i i I i i i I i i i I i i i I Exploration 0 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 drlljipK (I) I i r t ... i ... i ... I i l . i . I i .. I i » i I ... I i . i I i ». I Incline driving 0 2 $ 4 5 6 7 8 9 10 II 12 13 l4 15 16 U-JL. 0 _2 ^ 4 5 6 7 8 9 10 II 12 13 l4 15 16 logRips (3? U-i- l n i~ i ITzo-Jl • • 1 • » » 1 » • • 1 » » • 1 • •' » 1 « » « 1 • » « l • * • 1 1 1 1 1 » « 1 I 2 3 T.... 4 15 6 J 8 9 10 II 12 13 14 15 16 In situ stresses (3) ? , , , • . ^. i . . . . i i t . I-e .<«» I +4 m *+ 2 3 4 5 6 7 8 9 10 II 12 \t 14 15 16 Physical rock tests (4) >. , ; , ,• r I -r fMr i >*». jFi-r • . I \ i I . • • M * * r* ,*», frf »• 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 Possible seismic *--1 • 1 » t . . .Iii» I i » « 1 • t i < i"i » 1 • i i I r l i I 1 i i I I 1 1 I I > I I i 1 i I l I 1 I 9 10 II 12 13 14 15 16 Engineering (5) crf- 6 2 i 4 6 7 I 9 to 11 1* 13 14 ib 16 I * T I i r I i i i 1 ttil 1 i 1 I V I J 1 i -Y H--V I , . 1 I . -ii 1 I . r . I » 1 .1 I t * ^ 1 . < . I See notes next page. Figure 5 Tentative Schedule Exploration Program Argus Peak Granite Site Notes — 1* Two rigs on 21 hour basis assumed. Six percent down time for logging and minor repair. No casing assumed. 2L D.^S^T* done intermittently. Other logs as holes become deeper* 3;.. Time for in situ stress determinations in Ruth mine not included.. More in situ measurements made as access approaches cavity location* 4* More physical rock tests made at job. Start when core available and near end of access construction in vicinity of cavity than at middle of job during tunnel driving. 5. Engineering continuous throughout job and includes keeping maps — particularly structure predictions up to date. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 1 June 1963 320327 TABLE 58 Summary of Estimated Exploration Costs Site 3 1. Access roads and drill site preparation $ 65,000 2. Rig moving charge 2,000 3. Exploration drilling 3800 ft 5|" hole at $56/ft $213,000 24,000 ft NX hole at $36/ft 864,000 Total Exploration drilling 1,077,000 4. Well logging and surveys Directional surveys $ 6,700 Drill stem tests 4,200 Dipmeter logging 2,000 Sonic logging 2,000 Total logging and surveying jL4,90Q 5. In situ stress measurement and physical rock tests 200,000 6. Engineering 190,(000 7. Contingency 100,000 TOTAL $1,€48,000 SAY $1,650,000 COLORADO SCHOOL OF MINES RESEARCH FOUNDATION 119 Reference Panek, Louis A., 1961, Methods for determining rock pressure: Fourth Symposium, Rock Mechanics. Pennsylvania State Univ. COLORADO SCHOOL OF MINES RESEARCH FOUNDATION APPENDIX A Tables 1 — 53 COLORADO SCHOOL Or MINES f(C*CARCM FOUUDtTION 121 TABLE 1 f°r Solution A, 17 = 0,1, /o = & H 0° 15° 30° 45° 1 60° 7.5° 90° 0° 1 .9615 1 .8224 1 4423 0 .9231 0, 4038 0,0237 -0.. 11.54 15° 1 .8069 1 .6782 1.3264 0 .8458 .0, 3652 0,0134 -0,1154 30° 1 .3846 1.2842 1 .0096 0 .6346 0, 2596 -0,0151 -0,1154 45° 0 .8077 0 .7459 0 .5769 0 .3462 0. 1154 -0,0535 -0,11.54 6 0O 0 .2308 0 .2076 0 .1442 0 .0577 -0. 0288 -0,0922 -0,1154 75° -0 .1916 -0 .1864 -0 .1725 -0.1535 -0. 1344 -0.1205 -0,11.54 90° -0 .3462 -0 .3307 -0 .2885 -0 .2308 -0. 1731 -0.1308 -0,1154 TABLE 2 for Solution A, v = 0.2, 7" = a 0° I 15° 30° 45° 60° 75° 90° 0° 2 .0000 1 .8667 1 .5000 1 .0000 0 .5000 0. 1340 0. 0000 15° 1 .8325 1 .7097 1 .3743 0 .9163 0 .4581 0. 1228 0. 0000 0 0 CO 1 .3750 1 .2829 1 .0312 0 .6875 0 .3438 0. 0921 0. 0000 45° 0 .7500 0 .6997 0 .5625 0 .3750 0 .1875 0. 0502 0. 0000 60° 0 .1250 0 .1166 0 .0937 0 .0625 0 .0313 0. 0084 0. 0000 75° -0 .3325 -0 .3102 -0 .2494 -0 .1663 -0 .0831 -0. 0223 0. 0000 90° -0 .5000 -0 .4665 -0 .3750 -0 .2500 -0 .1250 -0. 0335 0. 0000 TABLE 3 **+/$ for Solution A, v - 0.3, p = a 0 O 0° 15° 30° 45° < O 75° 90° 0O 2 .0454 1.9176 1.5680 1. 0909 0. 6136 0. 2643 0.1364 15° 1.8627 1.7472 1.4311 0. 9995 0. 5680 0. 2520 0. 1364 30° 1.3636 1.2815 1.0568 0. 7500 0. 4432 0. 2186 0. 1364 j 45° 0 .6818 0 .6453 0 .5454 0. 4091 0. 2727 0. 1729 0.1364 60° o .0000 0 .0092 0 .0341 0. 0682 0. 1023 0. 1272 0.1364 I 75° -0.4991 -0 .4565 -0 .3402 -0. 1814 -0. 0225 0. 0938 0. 1364 90° -0.6818 -0 .6270 -0 .4773 -0. 2727 -0. 0682 0. 0816 0.1364 TABLE 4 for Solution A, v = 0.5, ^o = a O o o CO o CO o o 0° 15° 45° 75° CO 0O 2 .1667 2 .0550 1.7500 1.3333 0 .9167 0,.6116 0 .5000 15° 1.9433 1.8467 1.5825 1.2217 0 .8608 0 .5967 0 .5000 30° 1.3333 1.2775 1.1250 0 .9167 0 .7083 0 .5558 0 .5000 45° 0 .5000 0 .5000 0 .5000 0 .5000 0 .5000 0 .5000 0 .5000 60° -0 .3333 -0 .2775 -0 .1250 0 .0833 0 .2917 0 .4442 0 . 5000 75° -0 .9434 -0 .8467 -0 .5825 -0 .2217 0 .1392 0 .4033 0 .5000 90° -1.1667 -1.0550 -0 .7500 -0 .3333 0 .0833 0 .3883 0 . 5000 TABLE 5 for Solution A, v = 0.1, /° = a 0° 15° 30° 45° 60° 75° 90° OO -0 .1154 0.0237 0.4038 0 .9231 1.4423 1.8224 1.9615 15° -0 .1308 0.0093 0.3924 0 .9153 1.4384 1.8214 1.9615 30° -0 .1733 -0.0301 0.3606 0 .8942 1.4279 1.8185 1.9615 45° -0 .2308 -0.0840 0.3173 0 .8654 1.4135 1.8147 1.9615 60° -0 .2885 -0.1378 0.2740 0 .8365 1.3990 1.8108 1.9615 75° -0 .3307 -0.1772 0.2424 0 .8154 1.3885 1.8080 1.9615 90° -0 .3462 -0.1916 0.2308 0 .8077 1.3846 1.8069 1.9615 TABLE 6 ^9/c} *-or Solution A, v = 0.2, /O = a 0 0 < O 0 O 0° 15° CO 45° 75° 90° OO 0. 0000 0.1340 0 .5000 1.0000 1.5000 1.8660 2 .0000 15° -0. 0335 0.1027 0 .4749 0 .9833 1.4916 1.8636 2 .0000 30° -0.1250 0.0173 0 .4063 0 .9375 1.4687 1.8576 2 .0000 45° -0. 2500 -0. 0993 0 .3125 0 .8750 1.4375 1.8493 2 .0000 60° -0.3750 -0.2159 0 .2187 0 .8125 1.4062 1.8409 2 .0000 750 -0.4665 -0.3013 0 .1501 0 .7667 1.3834 1.8347 2 .0000 9 OO -0. 5000 -0.3325 0 .1250 0 .7500 1.3750 1.8325 2 .0000 TABLE 7 ^ for Solution A, v =0.3, p = a 0 O 0° 15° CO 45° 60° 75° 90° 0° 0.1364 0 .2642 0 .6136 I .0909 1 .5682 1 .9176 2 .0454 I 15° 0. 0815 0 .2131 0 .5725 1 .0635 1 .5545 1 .9139 2 .0454 30° -0. 0680 0 .0734 0 .4602 0 .9886 1 .5172 1 .9039 2 .0454 45° |-0. 2727 -0.1174 0 .3068 0 .8864 1 .4667 1 .8902 2 .0454 60° -0. 4773 -0 .3083 0 .1534 0 .7841 1 .4147 1 .8765 2 .0454 75« i-0.6270 -0 .4480 0 .0411 0 .7092 1 .3773 1 .8665 2 .0454 90O -0. 6818 -0 .4991 0 .0000 0 .6818 1 .3636 1 .8628 2 .0454 TABLE 8 for Solution A, v =0.5, /o = a O O CO 0 0° O X 15° CO 45° 75° 90° 0° 0 .5000 0. 6116 0 .9166 1 .3333 1 .7500 2 .0550 2 .1667 15° 0 .3884 0. 5074 0 .8329 1 .2775 1 .7220 2 .0475 2 .1667 30° 0 .0833 0. 2228 0 .6042 1 .1250 1 .6460 2 .0270 2 .1667 45° -0 .3333 -0. 1659 0 .2917 0 .9166 1 .5420 1 .9992 2 .1667 60° -0 .7500 -0. 5546 -0 .0208 0 .7083 1 .4375 1 .9710 2 .1667 75° -1 .0550 -0. 8392 -0 .2496 0 .5558 1 .3612 1 .9508 2 .1667 90° -1 .1667 -0. 9434 -0 .3333 0 .5000 1 .3333 1 .9433 2 .1667 TABLE 9