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 [</, V, w] Displacements in rectangular coordinates , Uj\ Displacements in spherical coordinates 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 <x * a( 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 <p Cos** & ~| - L J °r rzio/vmj [u.VjwJ in dirzci/ons J Up - ^ S/*/z ? Co&2e*5/vZ<t>SnJa& + Cosz4>j ^ 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 <tj/A/Z0-2Cos d\ * Sfi Ef>3 L ^ + Cos3-<t> Cos*8~3S/aj*? Cosz0+A^ X4) 2 z 2 _ AO+V) \(4v-2) S/u 0-3 S/a/ <6 Cos & •*•/1 ~£fil 1 J 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*<p Cos $ +(4v-z) s>*/**j )<5) ^ ^TcjjjS 0>s**J pr^ s ^ f/+v) S//J ^ S/a/ 9 Cos <j^.