STABILITY ANALYSIS OF WEDGE TYPE Ly - Open...

97
Stability analysis of wedge type rock slope failures Item Type text; Thesis-Reproduction (electronic) Authors Sublette, William Robert, 1944- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 21/06/2018 14:00:31 Link to Item http://hdl.handle.net/10150/347880

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Stability analysis of wedge type rock slope failures

Item Type text; Thesis-Reproduction (electronic)

Authors Sublette, William Robert, 1944-

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.

Download date 21/06/2018 14:00:31

Link to Item http://hdl.handle.net/10150/347880

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STABILITY ANALYSIS OF WEDGE TYPE ROCK SLOPE FAILURES

LyWilliam Robert Sublette

A Thesis Submitted to the Faculty of theDEPARTMENT OF MINING AND GEOLOGICAL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree ofMASTER OF SCIENCE

WITH A MAJOR IN GEOLOGICAL ENGINEERINGIn the Graduate College

. - THE UNIVERSITY OF ARIZONA

1 9 7 6

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STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfill­ment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below:

RICHARD D. CALL DateLecturer in Mining and Geological Engineering

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ACKNOWLEDGMENTS

The author expresses his sincere gratitude to Dr. Richard D. Call, the thesis director, for guidance during the course of study. Review of the manuscript , by members of the thesis committee, Dr. Charles E.. Glass, Dr. William C. Peters, and Dr. Young C. Kim, provided meaningful constructive criticism.

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TABLE OF CONTENTSPage

LIST OF ILLUSTRATIONS . .VABSTRACT....... ./....... vi

1. INTRODUCTION ........ 1Review of Literature................... 3Objectives of Thesis.......................... 7

2. STABILITY ANALYSIS ....... 93. DETERMINATION OF TETRAHEDRAL WEDGE GEOMETRY..... 194. STATISTICAL APPROACH. .... .' ........... 295. SEQUENTIAL DEVELOPMENT OF PROGRAM. ....._____ 346. EXAMPLE PROBLEM ........ 397. DESIGN APPLICATIONS AND CONCLUSIONS. ..... 43

Suggestions for Future Research. ............. 44 •'APPENDIX A: PROGRAM DOCUMENT I ON. . . 46

. APPENDIX B: DESCRIPTION OF INPUT. .......... 52APPENDIX C: PROGRAM LISTING . . . ........ 57APPENDIX D: . EXAMPLE PROBLEM INPUT DATA. ......... 72APPENDIX E: RESULTS OF EACH ITERATION IN THE

EXAMPLE PROBLEM................ 74APPENDIX F: SUMMARIZATION OF THE RESULTS IN

THE EXAMPLE PROBLEM. ........____ ■ 78APPENDIX G: PROGRAM FLOW CHART ....... 80REFERENCES ..... . . . ...... 88

iv

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LIST OF ILLUSTRATIONSFigure Page

1. Typical Wedge Type Failure.......... 22. Tetrahedral Wedge ..... 103. Cross Section of the Tetrahedral Wedge

Parallel to the Line of Intersection. .... 114. Cross Section of. the Tetrahedral Wedge

Perpendicular to the Line of Intersection 125- Surface Roughness....... .......... 156. Surface Roughness Effect on the Resisting.

Shear Stress along the Potential -Failure Plane ....... 16

7. Effect of Surface Roughness (i) on theMohr's Envelope. .................... . . ......... 17

8. Rotation of Coordinate System....... . . . ...... 209. Spherical Coordinate System Used to Locate

the Exterior Surfaces and Fractures..... 2210. Determine Spherical Location Coordinates

for the Exterior Surfaces and Fractures...... 2311. Description of the Planes and Points Which

Form the Tetrahedral Wedge........... 2612. Cross Section of the Rock Slope in the

: Example Problem. ..... 4013- Schmidt Plot of Fractures. . ................... 4l

: v. . • ;

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ABSTRACT

An analytical method and its corresponding computer program is developed to analyze the probability of failure for a wedge-type failure in a rock slope. When analyzing the stability of a fractured or faulted rock slope, this method provides the capability of considering many possible wedge configurations that may exist in the slope and significantly influence its stability.

A probability approach is necessitated when many wedge configurations are considered in the determination of a slope's stability. The analytical method presented in this Thesis first determines the probability that failure is kinematically possible for randomly selected wedge configu­rations. Next it determines the probability of failure for those wedge configurations which have a kinematic possibil­ity of failure. The total probability of failure is then determined by multiplying the two previously mentioned probabilities together.

A vector analysis is used to determine the factor of safety for each wedge analyzed. The resisting shearing stresses developed along each fracture plane is determined from the Mohr-Coulomb strength equation.

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CHAPTER 1

INTRODUCTION

The stability of fractured rock slopes is a major concern in open pit mining and engineering projects. This is especially true in open pit mining where mine depths are increasing and the pit slope inclination becomes a significant factor in the economics of the open pit operation. As the pit slope inclination is increased the cost of stripping the overburden is decreased, however this increases the probability of a slope failure. Since a slope failure can be very costly, both the stripping costs plus the slope failure costs should be considered in determining the optimum pit slope inclination.

The subject of this thesis is the development of a analytical method which uses a three dimensional vectoral stability analysis to determine the probability of failure for a wedge-type block failure in a fractured rock slope. The wedge-type failure is a common type of failure in rock Slopes (Fig. 1). With this method and its corresponding computer program the design engineer can determine the probability of a wedge failure for various pit slope inclinations.

1

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2

Line o f In tersec t ionT o p

Surface

[etrahedral Wedge

Slope Face

Botto m Surface

Figure 10 Typical Wedge Type Failure

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■ ' . ..... 3;Review of'Literature >

Two distinct approaches may be taken in the analysis of a rock slope. The rock mass in question may be consider­ed as a continuum and the stresses and strains throughout the region of influence within this continuum may be Calcu­lated , or the rock mass may be described as a discontinuum • whose stability calculation involved the statics and kine­matics of a rigid block or an accumulation of rigid blocks. This paper will deal only with the mechanics of a discon- tinuum.

The analysis of discontinuous rock slopes is a product of a mostly European workers; most prominently Pierre Londe, Walter Wittke, J.E. Jennings, Klaus John, and Evert Hoek. Significant American contributions are from Richard Goodman and Robert Taylor.

Wittke (1965) developed a comprehensive vectoral stability analysis for a rock slope containing one or two fracture sets. Wittke discussed the stability Of blocks having cohesive as well as frictional shear strength on their boundaries, and considered rotational as well as translational failure mechanisms. Goodman and Taylor (1967) reviewed the development of the concepts and appli­cations of the two contrasting methods of approach — stability of a discontinuum and mechanics of a continuum.

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. 4In their paper they discussed both the finite element and vector analysis approaches. In the vector approach they analyzed possible rotational and translation wedge failures.

Klaus John (1968) used equal area hemispheric pro­jections to evaluate the stability of a. slope with two planes of weakness. With this method he was able to determine the direction of movement and its factor of safety. Since forces can not be considered in his approach, it has the following limitations: shear is only frictional, andall forces such as hydraulic thrust, and retaining forces (rock anchors, retaining walls, buttresses, etc.) must be expressed in proportion to the weight of the rock mass.It was also assumed that no deformation takes place in the failing rock mass. Klaus John (1970) considered a modi­fied wedge bounded by two mean failure planes. The analy-- sis was a graphical approach based on a reference hernia sphere. Cohesion, friction, and other forces,. such as hydrostatic forces, were considered in this approach.

Londe, Vigier, and Vormeringer (1969) used spheri­cal coordinates to develop a three dimensional approach to analyze the stability of a slope. This method allowed for an estimation of the relative influence of the vari­ous strength parameters and internal water pressure on

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;stability. In their "approach they assumed the following: no deformation of the failing rock mass; negligible.co­hesion and tensile strength; and no moments produced by the forces. In 1970 Londe, Vigier, and Vormeringer approached the stability problem by representing the three dimensional wedge on a plane and presenting the results on simple graphs. The major advantage of graphic representation of limit equilibrium conditions is the easy appreciation of the relative■"weight" of the parameters acting on stability In most cases a glance at the diagram reveals which para­meters are vital and should consequently be checked by measurement (either in situ or in the laboratory) or con­trolled by special arrangements of the design (either drain age or grouting).

J. E. Jennings (1970) developed a coefficient of continuity for joints in rock. He explained that quite often a potential failure plane will have joints separated by intact rock. Since there is greater strength in the • intact rock, it is important to know the percentage of jointed rock relative to intact rock along the potential failure plane. Jennings also discussed the development of a mean plane of potential failure. In this case there are two sets' of pre-existing joints with dips other than the dip of the mean failure plane. The initial

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. ' 6 movement and driving forde is in a direction paralleling - one. of the pre-existing dips, even though the overall movement of the rock mass is parallel to the mean failure plane.

Heuze and Goodman (1971) described the various approaches and analyses which can be used in three dimen- . sional slope stability.problems. In their.paper they explained that there are two basic approaches to stability analysis; stress analysis; and limit equilibrium analysis.

Photoelastic models and Finite Element methods are techniques used in the stress analysis approach. Their use is particularly warranted when large rock blocks are deformable to a significant degree so that the behavior of. the rock mass is no longer clearly governed by discontinui­ties. Except in weak rocks the blocks can be considered to behave in a linear elastic manner, whereas severe non- linearities exist at the level of the fractures.

Inert physical models, analytical vector operations, and stereographic projections are techniques used in the limiting equilibrium approach. Rigid body movement with no allowable deformation is assumed in these approaches.

McMahon (1971) introduced graphic procedures for determining the probability of failure of a rock slope by statistical analysis of fracture orientations in conjunc-

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' • ■ • ' ■■■ ' : 7tion with graphic kinematic and kinetic analyses of stabil­ity over a range of possible slope angles.

One of the most comprehensive books written on slope stability was written by Evert Hoek and John Bray (19740, entitled "Rock Slope Engineering." It attempts to cover all facets of slope design, such as: economic con­siderations, data collection, strength parameters, ground­water, mechanics of failure and the various modes of failure. Hoek and Bray discussed in detail three methods which can be used when attempting a wedge failure analysis; engineering graphics solution, spherical projection solution, and vector analytical solution.

Objectives of Thesis Previously design engineers used stereographic

techniques or manual analytical solutions which were very time consuming, error prone, and analyzed only one wedge configuration at a time. Using the method presented in this paper many different wedge configurations and various rock properties can be analyzed rapidly and their correspond­ing factors of safety calculated. From this a resultant maximum and minimum range, for the probability of failure is determined at a 95% confidence level for a given slope inclination.

The computer use is necessitated when taking a -— probability approach to slope stability due to the many

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8iterations which are involved. With the probability approach one must deal with the variations of wedge con­figurations , and the variability of rock properties.

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CHAPTER. 2

STABILITY ANALYSIS

A limit equilibrium, approach is taken in the follow­ing rock slope stability analysis. The potential failing rock mass consists of a tetrahedral wedge formed from two fracture planes, the top surface, and the. slope face, as illustrated in Figs. 1,2,3,4. Rigid body motion is assumed in a limit equilibrium analysis, therefore there is no. in­ternal deformation within the tetrahedral wedge during failure. The direction of motion during failure is in the direction the line of intersection plunges.

The driving and resisting force acting on the tet­rahedral wedge can be seen in Fig. 3* The driving force is developed due to the weight component of the tetrahedral wedge in a direction parallel to the intersection of the two fractures. The equation for the driving force is: DRIVEF=(WT) (sin(DIPINT))DRIVEF= the driving forceWT= the weight of the tetrahedral wedge (calculated by

multiplying the unit weight of the rock times the volume of the tetrahedral wedge - the determination of the volume will be explained in Chapter 3)

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10

2 , . « » « « « ! •pigur® 2 ‘

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11

XT

Tetrahedral Wedge

XTM K

XTML

X M

DRIVEF = (WT)(sin(DIPINT))

Top Surface

- D IP INTXT

W TSlope Face

L------------ Line of Intersection

Vertical Section ABX M

Figure 3. Cross Section of the Tetrahedral Wedge Parallel to the Line of Intersection

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12XT

Tetrahedral Wedge

XTMK

XTML

X M

Top Surface

W T

F rac tu re

Frac tu re

Section CDLine of Intersection

Figure 4. Cross Section of the Tetrahedral Wedge Perpendicular to the Line of Intersection

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. 13DIPINT = the plunge of”the.line of intersection of the

two fracturesThe resisting force or sheafing resistance

developed along the fracture planes is due to the in-situ cohesive and frictional properties along these planes. Using the Mohr-Coulomb strength equation,'7'= c + crfan <f> , the shearing resistance developed along the failing surfaces can be determined. " is the shear strength or resisting force developed along the potential failure planes, "c" is the cohesion value, and "0" is the friction value. "cr1' is the normal, force on the potential failure planes developed from the weight of the overburden rock above the potential failure plane. Since there are two fracture surfaces on the tetrahedral wedge, there are two normal weight reaction components. These two normal forces are NORMK and NORML as seen in Fig. 4. To determine NORMK and NORML, it is necessary to set the tetrahedral wedge up as a free body with all of its body forces acting on it and then resolve these forces in the direction of NORMK and then in the direction of NORML. Section CD in Fig. 4 depicts the free body. It should be noticed that the forces due to shear on each fracture are not included since they are both normal to both NORMK and NORML, and this therefore negates any shear force component in the direction of either normal reaction.

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- . 1 14

After resolving the forces in the two normal directions, the following, equations are developed for JNfORMK. and NORML:ANBN = sin(DK)sin(DL)cos((AK+90)-(AL+90))+cos(DK)cos(DL). NORMK = ( ( A N B N ( - c o s ( D L ) ) + c o s ( D K ) ) / ( 1 . - A N B N 2 ) ) W T

NORML = ( ( A N B N ( - c o s ( D K ) ) + c o s ( D L ) ) / ( 1 . - A N B N 2 ) ) W T

DL = the dip of the L fracture on the tetrahedronDK = the dip of the K fracture on the tetrahedronAK = the azimuth of the K fracture on the tetrahedronAL = the azimuth of the L fracture on the tetrahedronWT .= the weight of the tetrahedron

The surface areas of both fractures in the tetrahedral wedge must also be calculated and used in the determination of "'7-", "c", and " c r " . A description of the method used to determine the fracture surface areas is given in Chapter 3*

If failure surface roughness is considered, then the Mohr-Coulomb strength equation is altered to the following form: 'T = c + crtan(<̂ + i). The parameter"i" is the surface roughness inclination to the potential failure plane. Figs. 5>6 show the development of the strength equation when surface roughness is considered, and Fig. 7 shows the effect of surface roughness on strength. Fig. 7 also shows that after the surface asperities have sheared, the Mohr's envelope will return back to a slope "p". It can be seen from Fig. 7 that,

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I 0 I IT

cr =

T = L =the normal stress on the potential failure plane the shear stress along the potential failure plane the angle the surface roughness face makes with

the potential failure plane

<r

H -i

Figure 5• Surface Roughness

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W e ig h t of rock mass above the

fa i l u r e plane

Potential

Failure Plane

7' = C 4- CT tan (<t>+ l)

Figure 6. Surface Roughness Effect on the Resisting Shear Stress along the Potential Failure Plane

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T

c

(T

Figure 7• Effect of Surface Roughness (i) on the Mohr's Envelope

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18

for low normal stresses on the fracture plane, the shear strength along the fracture could he over estimated when using the Mohr-GOulomb strength equation T = c f c r tan(<£). It would therefore be best to develop two separate strength equations: one for lownormal stresses on the fracture plane t = o- tan (<#> +i), and the other for higher normal stresses -r = c + <r tan(^). In the later case the surface asperities would be sheared.

The tetrahedral rock mass is stable when the resisting shearing force developed along the two fractures is greater than the driving force developed from the weight of the tetrahedral wedge. If the driving force is greater than the resisting force failure occurs.It can be said in this case that the factor of safety is less than one.FS = RESIST/DRIVEF FS - factor of safetyRESIST = the resisting force developed along the fracture

surfaces of the tetrahedral wedge DRIVEF = the driving force developed in the tetrahedral

wedge

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CHAPTER 3

DETERMINATION OF TETRAHEDRAL WEDGE GEOMETRY

Since the tetrahedral wedge is composed of two fracture planes, it is necessary to segregate the fracture planes into distinct groups. This is accomplished "by rotating the reference coordinate system until the slope face has a azimuth of 90° (Fig. 8). Then depending on each fractures azimuth, they are segregated into two groups. One group has fractures with azimuths between 90° -270°and the other fracture group has azimuths between 27.0° -90°.In this text the words■"azimuth" and "strike" will be synonymous and refer to the direction of the fractures pro­jection on a horizontal such that the fractures dip direc­tion will be 90° clockwise with respect to the direction of the fractures azimuth or strike.

The segregation of fracture planes is necessary so that each tetrahedral wedge analyzed,, consisting of a fracture from each group, will have a kinematic possibility of failure and the ability to slide simultaneously along both fracture planes in a direction parallel to the line of intersection.If two fractures from the same group are used, sliding

• , 19

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PLAN VIEWAzimuth and dip of exi st i ng

or proposed slope f ac e

Rotat ion of coordi nate system, resulting in a slope f ace

azimuth of 9 0

SLOPE FACE

O rig in or Bench M ark

Figure 8. Rotation of Coordinate System

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21

would occur only along one fracture since the other fracture would be overhanging and carrying no normal weight component from the tetrahedral wedge. A kinematic possibility of failure is developed when the line of intersection formed by the two fractures in the. tetrahedral wedge intersects the slope face above the bottom surface plane and plunges in a manner such that the tetrahedral wedge will slide along the line of intersection away from the rock slope. If the line of intersection plunges into the slope face, no movement or sliding can occur.

To assist in the understanding of the tetrahedral wedge development, an example will follow each of the succeeding equations that will be developed.

Before the tetrahedral wedge can be defined (Fig. 2), its exterior surfaces (top surface, slope face, bottom surface) and fracture planes must be located and their corresponding planer equations developed. The strike and dip for each fracture and exterior surface plane must be measured in the field.A spherical coordinate system (Fig. 9) is used to locate the exterior surfaces and fracture planes relative to a origin or bench mark established in the field (Fig. 10). The spherical coordinates (r,©,^) refer to the location distance (r), the vertical angle from the horizontal (©), and the location azimuth ($) (Figs. 9 and 10).

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22

z

Figure 9• Spherical Coordinate System Used to Locate the Exterior Surfaces and Fractures

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23L o c a t i o n poi nt at which the a z i muth and dip of the

f racture is taken

Existing or proposed slope face

Locat i on d is tance = r

L ocat i on az i muth = (f)The locat i on ver t ic al a ng le ( ©)

must be measuredOrigin or

Bench M a rk

Location A z im uth

Figure 10• Determine Spherical Location Coordinates - for the Exterior Surfaces and Fractures

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These spherical location coordinates are then transformed into cartesian coordinates (Pig. 9)• x = r cos (©) sin(^) y = r cos (©) cos(^) z =' r sin (©)Knowing the cartesian location coordinates and the strike and dips for the exterior.surfaces and fractures, the unit direction numbers (direction cosines) can be calculated for each plane and their corresponding planer equations developed in the form of the following equation Ax+By+Cz-D A»B, and G are the direction cosines of the plane.A = 1 sin(@D)sin(0S+9O)B = 1 sin(eD)cos (0S+9O)0 = 1 cos(eD)D = Ax + By + CzAx + By + Cz + D (equation of plane)

The tetrahedral wedge ’can now be formed from the top surface, slope face, and two fracture planes. Their corresponding planer equations are used to determine the four corner points of the tetrahedral wedge by solving three equations simultaneously.

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+ B1y + °lz = D1

•2X + V ■+ C2Z = D2.̂ x + V + C^z-+

The solution of the three equations will be a corner point of the tetrahedral wedge. This is done for all four corner points in the tetrahedral wedge. Fig. 11 shows the four corner points of the tetrahedral wedge (XT, XM, XTMK, and XTML). A three dimensional diagram of the tetrahedral wedge and its failing displacement is shown in Fig. 1.

Once the tetrahedral wedge has been defined, it will then be possible to determine whether or not the tetrahedral wedge has a kinematic possibility of failure. This is accomplished by two tests. The first test determines whether the fracture intersection plunges into the slope face or out of the slope face. If it plunges out of the slope face it will- make failure kinematically possible. The second test determines whether or not the fracture intersection daylights on the slope face. A fracture intersection which daylights on the slope face will make failure kinematically possible.

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26

top exterior surface

XTPoint at which line o f intersection daylights on top surface

XTM K/

K% Fracture Plane \

slope face

Line of intersection

^XTMJ_

V X T M L

LFracture Plane

XMPoint at which line of intersection

daylights on slope face

XX M Bl Toe of slope face

bottom exterior suface

Figure 11. Description of the Planes and Points Which Form the Tetrahedral Wedge

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After the tetrahedral wedge has been defined, the surface area of the two fractures can be determined and also the volume of the tetrahedral wedge can be calculated. The determination of the surface area of the two fractures in the wedge is necessary to calculate! the resisting force developed from cohesion on the fracture surfaces in a direction paralleling the line of intersection. The following vector cross product is used to determine the surface area:

| a x b|= \ f (a-a)(b-b)-(a-b)2"a" and "b" are vectors on the fracture plane and also form two of the edges in the tetrahedral wedge. Vectors "a" and "b" are composed of vector components a ^ a ^ a ^ and b, bQ,bQ respectively. The vectors "a" and "b" are easily developed by subtracting the cartesian coordinates of one corner on the tetrahedron from the cartesian coordinates of another corner on the tetrahedron. The resulting direction numbers, or vector components, represent the vector between these two corners of the tetrahedron. The following procedure demonstrates the development of a vector and its corresponding vector components between two corners of the tetrahedron.

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The example presented here will "be the vector initiated at point XM and terminated at point XT. b = (XM1 -XM1, XT2 -XM2 , XT3 -XM3)Two other vectors, both initiated at XM and terminated at XTME or XTML, can also be developed in the same manner.

The determination of the tetrahedral wedge volume is necessary to calculated the weight and 'the resulting driving force along the line of intersection and also the normal force on each fracture plane which is utilized in the determination of the frictional resisting force along each fracture. The following scalar triple product is used to determine the volume of the tetrahedral wedge:Volume = l/6(a*(b x c))"a", "b", and "c" are three different vectors along the edges of the tetrahedral wedge.

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CHAPTER 4

; STATISTICAL APPROACH

The statistical analysis is initiated by generating random numbers for both groups of fractures (K group and L group). Then a K fracture and L fracture is sampled randomly from each respective group. These two fractures are the particular fractures in the tetrahedral wedge being analyzed at that moment.

The cohesion and friction values for each of the two fractures are determined by applying the mean and standard deviation values for cohesion and friction to a normally distributed random number generator. It is possible that negative cohesion and friction values may be generated, but these must be rejected and new valuesgenerated. It would be impossible to develop negative

( ' cohesion or friction.A count is kept of all the fracture combinations

being analyzed which have a kinematic possibility of failure. If a particular pair of fractures form a tetrahedron which does not have a kinematic possibility of failure, then the analysis is initiated again by randomly choosing a new pair of fractures. If the

29

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30tetrahedron being analyzed does have a kinematic possibility of failure, then the factor of safety is determined.A count is also kept of all those factors of safety which are less than one and for which failure would result, (i.e., driving force exceeds resisting force).

After one hundred fifty iterations involving the determination of the factors of safety has been completed, an estimate of the probability that failure is kinematically, possible (POSFAL) is determined. This is accomplished by dividing the number of iterations that have pairs of fractures which form tetrahedrons that have kinematic possibilities of failure (150), by the total number of iterations attempted (KKKK).POSFAL = 150/KKKKNext an estimate of the probability of failure for those pairs of fractures in which failure is kinematically possible (PCFL) is calculated by dividing the number of iterations with factors of safety less than one (IFAIL), by the number of iterations which have a kinematic possibility of failure (150). PCFL = IFAIL/150

The standard deviation is determined for both the probability that failure is kinematically possible and the probability of failure for those pairs of fractures

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which have a kinematic possibility of.failure. The following equation is used in this determinations .

er= V"- P(l-P)/n<r= standard deviation (SIG1 and SIG2)P= estimated probability (POSFAL and PCFL) n= number of iterations (KKKK and IPS)SIG1 = V POSFAL(1.-POSFAL)/KKKK

SIG2 = V PCFL(1o-PCFL)/lFSSIG1 = standard deviation for the probability that

failure is kinematically possible.SIG2 = standard deviation for the probability of failure

for those pairs of fractures which have a kinematic possibility of failure.Next a true maximum - minimum probability range

at a 95fo confidence level is determined using the following equationspt = pe - (*1.96) (<r)P̂ . = true maximum - minimum range of probability (P1MAX,

P1MIN, P2MAX, and P2MIN)Pe = estimated probability (POSFAL and PCFL) tp ^ = 1.96 (value for a confidence level of 95%)<r = standard deviation (SIG1 and SIG2)Using the above equation the maximum and minimum limits

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of probability (considering a 95f° confidence level) are determined for the probability that failure is kinematically possible and the probability of failure for those pairs of fractures which have a kinematic possibility of failure.P1MAX = POSFAL + 1.96(8101)PIMIN = POSFAL - 1.96(8101) .P2MAX = PCFL + 1.96(8102)P2MIN = PCFL - 1.96(8102)P1MAX = Maximum true range of probability that failure

is kinematically possible.PIMIN = Minimum true range of probability that failure

is kinematically possible.P2MAX = Maximum true range of the probability of failure

for those pairs of fractures which have a . kinematic possibility of failure.

P2MIN = Minimum true range of the probability of failure for those pairs of fractures which have.a kinematic possibility of failure.The maximum and minimum total probability of

failure (considering a 95?° confidence level) is determined using a range arithmetic operation. This is accomplished by multiplying the minimum probability that failure is kinematically possible times the minimum probability of failure for those fracture pairs which have a kinematic

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possibility of failure, resulting in a minimum total probability of failure (T0TPF2). The maximum total probability of failure (T0TPF1) is determined by multiplying the maximum probability that failure is kinematically possible times the maximum probability of failure for those fracture pairs which have a kinematic possibility of failure.T0TPF1 = ABS((P1MAX)(P2MAX))T0TPF2 = ABS((F1MIN)(P2MIN))ABS = the FORTRAN abbreviation for absolute value.

The mean and standard deviation of the distribution of the factors of safety, which were calculated during the analysis, was also determined using the following equation: XMEAN =■ SUM/IFSSTDEV = f (SUMSQ - (S.UM2/IFS) )/(lFS - 1)

IFSSUM = FSi

SUMSQ = 4 = (FS± )2

XMEAN = the mean factor of safetyFS = 150 - the total number of iterations in which.a

factor of safety was determined.STDEV = the standard deviation for the factor of safety

distribution.

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CHAPTER 5

' : SEQUENTIAL DEVELOPMENT OF PROGRAM

The rock slope stability program consists of one main program and four subroutines. The main program is called SLOPE and the four, subroutines are called SIMULT, RANDCF, RANDKL, and MNSTD.

The main program can basically be divided into two segmentso The first segment calculates the factor of safety for various fracture combinations while the second segment determines the maximum and minimum range of the total probability of failure in the rock slope considering a 95^ confidence level. An analytical three dimensional vectoral approach is used to determine the factor of safety for the tetrahedral wedge of the rock slope as shown in Fig, 1.

Subroutine SIMULT is a simultaneous equation solver. Its purpose in this program is to determine the intersection point of three planes. By using this subroutine the four corners of the tetrahedral wedge can be found.

Subroutine RANDCF is used to generate normally distributed random values for cohesion and friction.

34

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35Subroutine RANDKL is., used to generate random numbers for two predesignated groups of fractures«

Subroutine MNSTD calculates the mean and standard deviation for the distribution of factors of safety calculated during the analysis.

After the data is read in, the coordinate system is rotated such that the exterior slope face has an azimuth of 90° (Fig. 8). The purpose of this rotation is to make it possible to segregate the fractures into two separate groups.

The next sequence of events will consist of changing the location coordinates from polar to cartesian for the fracture planes and the exterior surface planes. Unit direction numbers (direction cosines) are formulated for the above mentioned planes. Following this the fractures are segregated into two groups, which are determined by their azimuths. One group consists of those fractures with azimuths between 90° -270°(L group) and the other group with azimuths between 270° -90° (K group). .

A fracture is randomly selected from each group.This is accomplished by using the subroutine RANDKL.A normally distributed random number generating subroutine called RANDCF, is used to generate values for cohesion

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and friction. Near the end of the program a factor of safety is calculated for each set of randomly selected variables, and these iterations are continued for each set Of randomly selected variables until one hundred and fifty factors of safety have been determined,.

Following the selection of two fractures and the cohesion and friction values, it is necessary to determine the location of the corner points of the tetrahedral wedge of which the two fractures are part of. This is accomplished through the use of the subroutine SIMULT, which is a simultaneous equation solver. All four corner points of the tetrhedral wedge are determined in this manner. Two other intersection points are determined by SIMULT, These points labeled XTMI(K,L) and XMBI(K,L) as shown in Fig, Jl. They are used to determine whether the line of intersection is dipping in a manner which makes failure kinematically feasible. If failure is not kinematically feasible, a new set of variables (cohesion, friction, and ”K" and "L" fractures) are randomly selected and the iteration initiated from the beginning.

The surface areas of the:i "K" and "L" fractures in the tetrahedral wedge are determined using vector cross products. The surface areas of the fractures are . used to calculate the shearing resistance developed by the cohesion. A scalar triple product is used to determine

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37'the volume of the tetrahedral wedge. Knowing the volume, the weight of the tetrahedral wedge can he determined, from which the tetrahedral wedge’s driving force can he calculated. The weight of the tetrahedral wedge is also used in determining the normal forces on the "K” and "1" fractures which in turn is used to determine the shearing resistance developed hy the friction angle. Once the total resisting and driving forces have been determined, the factor of safety for the rock wedge can be calculated.

The azimuth and dip of the line of intersection for the two fractures is determined once the location cartesian coordinates are known for two points on the tetrahedral wedge. The two points used in the determination of the line of intersection are XT(K,L), and XM(K,L) as shown in Pig, H , and as determined from subroutine SIMU1T, Next the coordinate axis must be rotated back to its original position so that the azimuth for the line of intersection will have its actual in-situ orientation.

To allow for the development of a factor of safety frequency curve, the factors of safety for various ranges are accumulated as they are calculated. The frequency of occurence for various ranges of factors of safety can be easily observed in the output. Following this it is determined if the factor of safety for the tetrahedral

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' ' 38wedge "being analyzed,.is less than one. If it is less than one, failure.occurs for that particular wedge configuration "being analyzed.

After one hundred and fifty iterations have occured in which the factbr of safety has been calculated, an estimate of the probability that failure is kinematically possible is calculated plus its corresponding standard deviation is determined, An estimate of the probability of failure for those pairs of fractures in which failure is kinematically possible is also calculated and its corresponding standard deviation determined.

Using the estimated probabilities of failures and. their corresponding standard deviations, the true maximum and minimum range of the total probability of failure can be determined considering a 95$ confidence level. Follow­ing this the mean and standard deviation for the factor of safety is determined by using the subroutine MNSTD.The final results are then printed in the output.

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CHAPTER 6

EXAMPLE. PROBLEM

A rock slope as shown in Fig. 12 is used as an example to demonstrate how its probability of failure is determined. The example problem being studied is a hypothetical case. The cohesion and friction values used in the example problem are typical of a clay gouge, Evert Hoek and John Bray (1974). The unit weight of the rock is typical of a porphyry or granite, Evert Hoek and John Bray (1974). A Schmidt plot (Fig. 13) shows the distribution of fractures. The "K" fracture set has a mean strike of 19.590 and a dip of 44.93°• Its standard deviation for the strike and dip is 7-51 and 7-53 respectively. The "L" fracture set has a mean strike of 145-58°. and a dip of 67'9°• Its standard deviation for the strike and dip is 6.97 and 7-22 respectively.

The input data is presented in Appendix C.This information consists of the following: numberof fractures, unit weight of the rock, cohesion and friction information, and azimuths, dips, and location information for the exterior surfaces and fractures.

39

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Bench mark or origin from which location azimuths, vertical angles, and distances are measured from

100 ft

Figure 12. Cross Section of the Rock Slope in the Example Problem

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41

N

6 030W-K — f r a c t u r e set

TSFigure 13 . Schmidt Plot of Fractures

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The results of all the ‘iterations is shown in Appendix D, while a summarization of the results, including the total probability of failure, is presented in Appendix E.

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CHAPTER 7

DESIGN APPLICATIONS AND CONCLUSIONS

Probably the most obvious design application for the method, and corresponding computer program, presented in this thesis would be in an open pit mining operation. Other possible uses would be in road cuts in rook, dam embankment excavations, and any other rock excavation in which stability is required. .

This program could be used as a tool in the design of an open pit mine or as a method of analyzing problem areas in existing open pits. If this program is used in the initial design of an open pit, it can be utilized in one of two ways $ 1) for each section of the open pit allthe structural and strength information available is fed into the computer as input and a stability analysis for various slope possibilities is done. 2) the second approach utilizes steroplots in initially developing the pit configuration and safe pit slopes, and at the same time the use of these steroplots can determine possible problem areas, which can then be studied in further detail with this program. This last approach is probably more practical since a steroplot is usually always developed

: 43 _

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. by the pit designers for the purpose of delineating fracture patterns = If the problem areas are determined by the use of the steroplot, then more fracture and strength parameter information from these areas might be needed to properly determine their stability.

This program can also be useful after excavation has been initiated in the open pit. Many times previously undetected discontinuities will be exposed as the pit is excavated,.or other unforeseen problems will arise which might have an effect on the future stability of certain slopes.

. Suggestions for Future ResearchThe analytical method developed in this thesis

is limited to only certain situations encountered in the field. This method does not consider water pressure, tension cracks, or external loads such as rock bolts and surcharges. The usefulness of this method could be increased considerably by attempting to incorporate the above mentioned considerations. How these forces or loads should be considered would be dependent upon their effect on the tetrahedral wedge. Those forces that would increase the driving force would be included in the driving forces calculations, while those forces ■ effecting the resisting forces would be included in their determination = Another improvement on this

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program could be realized by analyzing a slope in which its slope face consisted of two or. more predominant surface features. This is especially applicable to potentially large failures in benched slopes or in situations where the slope face azimuth changes directions at a corner or curve in an excavation.

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■ APPENDIX A

PROGRAM•DOCUMENTATI ON

46.

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' 1. Program"Identification1.1 Program Title: Rock Slope Stability Analysis1.2 Program. Code Name s SLOPE, TET1.3 Writers W.R. Sublette1.4 Organizations Department of Mining and Geological

Engineering, University of Arizona, Tucson, Arizona.1.5 Dates 9/1/751.6 Updatess This is the original.1.7 Source Languages CDG 6400 FORTRAN Extended. Non­

standard features of FORTRAN Extended used in the program includes1. Nonstandard Hollerith field delimiters:..2. Mixed-Mode Arithmetic.

1.8 Availabilitys Available only from the writer.1.9 Abstracts See the Abstract in the thesis.

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. • 2.' Engineering Documentationi

2o1 Narrative Description: Refer to Chapter 1 of thesis.2i2 Method of Solution: Refer to Chapters 2,3, and 4=2=3 Program Capabilities: The program in its present

form is limited to inputing a maximum of one- hundred fractureso

2.4 Data Inputs: Refer to Appendix B.2=5 Program Options: There are essentially no program

options o2=6 Printed Output: Refer to Appendixes 0,D, and E.2=7 Other Outputs: No other outputs are produced,2.8 Flow chart: Refer to Appendix F.2=9 Sample Runs: Refer to Appendixes C,D» and E.

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' 493. System Documentation

i.3-1 Computer Equipment: The SLOPE, TET program was runon a GDC 6400 computer with a 200K octal core central memory- Approximate minimum central memory access time is 1.0^ second.

3.2 Peripheral Equipments The following peripheral equipment were used in the execution of the.SLOPE, TET program: CDC 405 card reader, GDC 844disk drive, CDC 512 line printer.

3-3 Source Program: A source program listing and sourcecard deck must be obtained from W. R. Sublette.

3.4 Variables and Subroutines: A descriptive list ofthe variables is included with the source program listing and must be procured from W. R. Sublette.The following consists of a list of the subroutines used in the SLOPE, TET program.(1) SLOPE: Main control program.(2) SIMULTs Subroutine to solve three equations

simultaneously.(3) RANDCF: Subroutine to generate normally distributed

random numbers for cohesion and friction.(4) RANDKL: Subroutine to generate random numbers

for K and L fractures.(5) MNSTD: Subroutine to calculate mean and

standard deviation for the factor of safety distribution.

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3°5 Data Structuress No files,are created.3 = 6 Storage Requirements s As presently structured, the

SLOPE, TET program requires approximately 50 K storage.

3=7 Maintenance and Updates: There have been no updatesin the original program.

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...... Operating .Documentation4.1 Operator Instructions! The SLOPE, TET program is

operated under the GDC 6400, SCOPE 3.4 operating system.

4.2 Operating Messages: Error messages produced "bythe program are self-explanatory.

4.3 Control Cardss The program is executed with standard SCOPE 3.4 control cards. As run on the University of Arizona CDC 6400, the makeup of the input, deckis as follows:

Job CardFTN. 'LGO.7/8/9 . ■(SOURCE PROGRAM)7/8/9(INPUT)6/7/S/9

4.4 Error Recovery: Program must be restarted on error.4.5 Run Time: The example problem illustrated in

Appendixes C,D, and E was completed in approximately 15 seconds.

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. . APPENDIX B

DESCRIPTION OF INPUT

52

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53The input data consists of the following

parameters;' \ . .

lo number of fracturesZo unit weight of the rock3o mean and standard deviation for cohesion and

friction4= azimuths and dips for the exterior surface

planes, plus their location azimuths, vertical angles, and distances (Fig. 10)

5° azimuths and dips for the fractures, plus their location azimuths, vertical angles, and distances (Fig. 10.)The number of fractures read into the computer

is the number of fractures in a representative sample population of the area being studied. Azimi2ths and dips for each of these fracture surfaces are also included in the input data, plus their location azimuths, vertical angles, and location distances. The fracture planes are located with respect to an origin or bench mark established nearby. By using surveying equipment and the origin as a reference, the location azimuths, vertical angles, and location distances are determined for the fracture planes (Fig. .10). The exterior surface planes are located in the same manner.

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The mean and standard deviation values for the fracture’s cohesion and friction are determined from a series of strength tests on the fracture surface material.

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55• Input Format

Card # Columns Format Variable Description of Variable .1 1-5 15 M1 5-10 F10-2 UNITWT

2 1-10 F10.0 XMNC

2 11-20 FlOoO SDC

2 21-30 F10.0 XMNF

2 31-40 F10.0 SDF

3 1-5 F5.1 AZS(l)

3 6-10 F5.1 DIPS(l)

3 11-16 F6o2 STSA(l)

3 17-23 F6.2 STSD(l)

3 24-33 F10.2 DlSS(l)

4 1-5 F5ol AZS(2)4 6-10 F5.1 DIPS(2)4 11-16 F6o2 STSA(2)

4 17-23 F6.2 STSD(2)

Number of fractures readUnit weight of the rock (Ib/ft3)Mean cohesion value (lb/in2)Standard deviation for cohesion (lb/in2)Mean friction angle p. <%.. > value (degrees-)Standard deviation for friction angle (degrees)Azimuth of top exterior surfaceDip of top exterior surfaceLocation azimuth of top exterior surfaceVertical angle of location azimuth for top exterior surfaceLocation distance for top exterior surfaceAzimuth of slope faceDip of slope faceLocation azimuth of slope faceVertical angle of location azimuth for slope face

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564 24-33 F10-, 2 DISS(2)

5 1-5 F5-1 .AZS(3)

5 6-10 F5„l DIPS(3)

5 11-16 F6 .2 . STSA(3)

5 17-23 F6o2 STSD(3)

5 24-33 F10,2 DISS(3)

6 1-5 F5.1 AZF(J)6 6—10 F5.1 DIPF(J)6 11-16 F6o2 STFA(J)

6 17-23 F6,2 STFD(J)

6 24-33 F10.2 DISF(J)

Location distance for slope faceAzimuth of bottom exterior surfaceDip of bottom exterior surfaceLocation azimuth of bottom exterior surface .Vertical angle of location azimuth for bottom exterior surface ,Location distance for bottom exterior surfaceAzimuth of fracturesDip of fracturesLocation azimuth of fracturesVertical angle of fracture location azimuthLocation distance for fractures

* Card numbers 6 through M+5 contain fracture information, M refers to the number of fractures,

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• APPENDIX C

PROGRAM LISTING

57

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581 PROGRAM S L U P E d N P U T , O U T P U T ) A 10

C N-NUMBER OF EX T E R I OR SURFACES OF THE ROCK SLOPE ( 3 ) ' A 2 0C M » NUMBER OF FRACTURE PLANES I N THE ROCK A 30C U N I T V T - U N I T WEI GHT OF THE ROCK ( L 9 / F T 3 ) A 40

5 C A Z S ( I ) - A Z I M U T H OF TOPOGRAPHIC SURFACES - A Z S ( l ) I S TOP P L A NE , A 50C A Z S ( 2 ) I S MIDDLE PLANE (SLOPE F A C E ) , A Z S < 3 ) IS THE LOWER OR A 60C BOTTOM PLANE . A 70C D I P S ( ! ) • D I P OF THE ABOVE MENTIONED SURFACES A 80C STS A ( I ) - LOCAT I ON AZ I MUTH FOR THE POI NTS AT WHICH THE AZI MUTH AND A 90

1 0 - C D I P WERE TAKEN FOR THE TOPOGRAPHIC SURFACES ( I T WOULD RE A ICOC POSSI BLE TO USE S T A D I A I N LOCATI NG THESE P OI NT S FROM A PRESET A 1 10C O R I G I N ) A 1 20C S T S D ( I ) * V E R TI C A L ANGLE OF LOCATI ON AZI MUTH FROM THE PRESET O R I G I N A 1 30C TO THE LOCATI ON POI NT AT WHICH THE AZI MUTH AND D I P OF THE A 1 40

15 C TOPOGRAPHIC SURFACE WAS MEASURED <♦ UP, - DOWN) A 1 50C D I S S ( I ) » D I STANCE FROM PRESET O R I G I N TO P OI NT AT WHICH THE AZ I MUT H A 1 60C ' AND D I P OF THE TOPOGRAPHIC SURFACE WAS MEASURED ( I N FE E T ) A 1 70C A Z F ( J ) * AZ I MUTH OF FRACTURES A 1 80C O I P F ( J ) = DIP OF FRACTURES A 1 90

20 C S T F A ( J ) = L OCATI ON AZ I MUTH OF FRACTURES MEASURED FROM PRESET A 2 00C O R I G I N A 2 10C S T F D ( J ) * V E RT I C A L ANGLE OF FRACTURE LOCATION A Z I MUT H MEASURED A 2 20C FROM THE PRESET O R I G I N ( ♦ U P , - DOWN) A 2 3 0C D I S F ( J ) * D I STANCE FROM PRESET O R I G I N * TO THE POI NT ON THE FRACTURE A 2 4 0

25 C WHERE ITS AZI MUTH AND D I P WERE MEASURED ( I N FE E T ) A 2 5 0C ROTA- FUNCTION USED I N C A L C U L A T I ON OF A X I S ROTAT I ON A 2 6 0C Z S D ( I ) - ( Z ) D I R E C T I O N COMPONENT OF D I S S ( I ) A 2 7 0C X S D ( I ) - ( X ) D I R E C T I O N COMPONENT OF D I S S ( I ) A 2 8 0C Y S D ( I ) * ( Y ) D I R E C T I O N COMPONENT OF D I S S ( I ) A 2 9 0

30 C ZSU ( I >=• ( Z ) U N I T D I R E C T I O N NUMBER FOR TOPOGRAPHIC SURFACE A 3 0 0C X S U ( D * ( X ) U N I T D I R E C T I O N NUMBER f c r TOPOGRAPHIC SURFACE A 3 1 0C Y S U ( I ) - ( Y) U N I T D I R E CT I O N NUMBER FOR TOPOGRAPHIC SURFACE A 3 20C. THE FOLLOWING COMMENT STATEMENTS APPLY FOR a Z F ( J ) BETWEEN 0 - 9 0 AND A 3 30C 2 7 0 - 3 6 0 A 3 4 0

35 C AZF3 ( K ) * A Z F ( J ) A 3 50C D I P F l ( K ) - D I P F ( J ) A 3 60C COHE S 1 ( K ) = C O H E S ( J ) . A 3 70C F R I A N K K ) - F R I A N G ( J ) A 3 8 0C . Z F U l ( K ) - ( Z) U N I T D I R E C T I O N NUMBER FOR FRACTURE SURFACE A 3 90

40 C Z F C K K ) - ( Z ) D I R E C T I O N COMPONENT OF D I S F K K ) A 4 0 0C X F U 1 ( K ) - ( X ) U NI T D I R E C T I O N NUMBER FOR FRACTURE SURFACE A 4 1 0C Y F L ' K K J - ( Y ) U N I T D I R E C T I O N NUMBER FOR FRACTURE SURFACE A 4 20C X F D K K J - ( X ) D I R E C T I O N COMPONENT OF D I S F K K ) A 4 3 0C Y F t l ( K ) « < Y) D I R E C T I O N COMPONENT OF D I S F K K ) A 4 4 0

45 C THE FOLLOWING COMMENT STATEMENTS APPLY FOR A Z F ( J ) BETWEEN 0 0 - 2 7 0 A 4 5 0C A Z F 4 ( L ) - A Z F ( J ) A 4 6 0C D I P F 2 ( L ) - D I P F ( J ) A 4 7 0C C 0 H E S 2 ( L ) =COHES( J ) • A 4 8 0C F R I A N 2 ( L ) - F R I A N G ( J ) • A 4 90

50 C Z F U 2 ( L ) • ( Z ) U N I T D I R E C T I O N NUMBER FOR FRACTURE SURFACE A 5 00C Z F D 2 ( L ) « ( Z> D I R E C T I O N COMPONENT OF D I S F 2 ( L ) A 5 10C X F U 2 ( L ) - ( X ) U NI T D I R E C T I O N NUMBER FOR FRACTURE SURFACE A 5 2 0C YF02 < L ) * ( Y) U N I T D I R E C T I O N NUMBER FOR FRACTURE SURFACE A 5 30C X F D 2 ( L ) * ( X ) D I R E C T IO N COMPONENT OF D I S F 2 ( L ) A 5 40

55 C Y F D 2 ( L ) * ( Y ) D I R E C T I O N COMPONENT OF D I S F 2 ( L ) A 5 5 0C XT - THE X COORDINATE FOR THE I N T E R S E C T I O N CF THE TWO FRACTURES A 5 6 0C AND THE TOP SURFACE A 5 7 0

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YT ■ THE Y COORDINATE FOR THE AND THE TOP SURFACE

I N T E R SE C TI O N OF THE TWO FRACTURES

ZT ■ THE Z COORDINATE FOR THE AND THE TOP SURFACE

I N T E R SE C TI ON OF THE TWO f r a c t u r e s

XM ■ THE X COORDINATE FOP THE AND THE M I DD L E SURFACE

I N T E R SE C T I O N OF THE TWO f r a c t u r e s

YM ■ THE Y COORDINATE FOR THE AND THE MI DDLE SURFACE

I N T ER SE C T I O N OF THE TWO f r a c t u r e s

ZM ■ THE Z COORDINATE FOR THE AND T'HF MI DDLE SURFACE

I N TE RS E C T I O N OF THE TWO f r a c t u r e s -

K3 « A VALUE FROM 1 - 3 WHICH I N D I C A T E S WHICH TOPOGRAPHIC SURFACE D I R E C T I ON NUMBERS THAT W I L L BE READ

KD » IS USED AS A METHOD OF I N D I C A T I N G WHEN A S P E C I F I C PART OF THE SUBROUTINE SHOULD BE USED

KC - A I N I T I A L VALUE FOR KKM « THE MAXIMUM NUMBER OF K FRACTURES 0 - 9 0 AND 2 7 0 - 3 6 0 LC- « A I N I T I A L VALUE FOR LLM * THE MAXIMUM NUMBER OF L FRACTURES 9 0 - 2 7 0 VK » I N D I C A T E S WHICH K FRACTURES SHOULD BE DISREGARDEDVL * I N D I C A T E S WHICH L FRACTURES SHOULD BE DISREGARDEDKE « I N D I C A T E S WHICH TOPOGRAPHIC SURFACE THAT IS BEING CONSIDEREDW » I N D I C A T E S WHICH COMB I NA T I ON OF FRACTURES SHOULD BE CONSIDERED

FOR S T A E I L I T T A N ALY SI S X I U - ( X ) D I RE C T I O N NUMBER FOR V E P T K AL I N T E R SE C TI O N PLANEY I U • ( Y) D I R E CT I ON NUMPEP FOR V E R T I CA L I N T E R SE C TI O N PLANEZ I U ■ ( Z ) D I R E CT I ON NUMBER FOR V E R T I C A L I N T E R SE C TI O N PLANEX • THE RESUL TI NG X COORDINATE FOR THE I N TE R SE C TI ON OF THREE

PLANESY » THE . R E S U i T I N G Y COORDINATE F q R THE I N T E R S E C T I ON OF THREE

PLANESZ ■ THE RESUL TI NG Z COORDINATE FOR THE I N T E R SE C T I O N OF THREE

PLANESKM1 » THE MAXIMUM D I ME N S I ON A L STATEMENT FOR K FRACTURES L M l s THE MAXIMUM D I ME N S I ON A L STATEMENT FOR L FRACTURES K CK- I N D I C A T E S WHETHER OR NOT A S P E C I F I C ROW HAS BEEN USED YET AS

A P I VO T ROWLOC - I N D I C A T E S WHAT ROW WAS USED AS A P I VOT P OI NT FOR A GI VEN

COLUMNAMAX = MAXIMUM VALUE I N EACH COLUMN F * M U L T I P L Y I N G FACTOR FOR THE P I V O T ROWX I « RESULTI NG UNKNOWNS FOR THE THREE SIMULTANEOUS EQUATIONS I F S - NUMBFF OF FRACTURE SETS FOR WHICH THE FACTOR OF SAFETY IS

CALCULATED XMEAN * MEAN F . S .STDEV - STANDARD D E V I A T I O N OF F . S . D I S T R I B U T I O NKJ J K - INTEGER TO I N D I C A T E I F THE VARIANCE OF THE PAST AN D MOST

RECENT M E A N AND STANDARD D E V I A T I O N I S LESS T HA N . 0 5 KKKK > COUNTER TO DETERMINE THE TOTAL NUMBER OF FRACTURE SETS

ANALYZEDXMBI * X COORDINATE FOR THE I N TE RS E C TI O N OF THE SLOPE FACE, BOTTOM

SURFACE, AND VE R T I C A L I N T E R SE C TI O N PLANE YMRI * Y COORDINATE FOR THE I N T E R S E C T I O N OF THE SLOPE FACE, BOTTOM

SURFACE# AND V E R T I CA L I N T E R SE C TI O N PLANE ZMBI « Z COORDINATE FOR THE I N T E R SE C T I O N OF THE SLOPE FACE, BOTTOM

SURFACE, AND V E R T I CA L I N T E R S E C T I O N PLANE XTMI - X COORDINATE FOR THE I N T E R S E C T I O N OF THE TOP SURFACE, SLOPE

FACE, AND V E R TI C A L I N TE R SE C TI ON PLANE

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115 C YTMI » r COORDINATE f o r THE I N T E R S E C T I O N of THE TOP SURF ACE, SLOPE A 1 1 5 0C F A C E , AND V E RT I C A L I N T E R S E C T I ON PLANE A 1 1 60C ZTMI « Z COORDINATE FOR THE I N T E R SE C T I O N OF THE TOP SURFACE, SLOPE A 1 1 7 0C FACE, AND V E R T I C A L I N T E R S E C T I ON PLANE A 1 10 0C XTMK « X COORDINATE FOR THE I N T E R S E C T I O N OF THE TOP SURF ACE, SLOPE A 1 1 9 0

120 C FACE, AND K FRACTURE A 1 2 0 0C YTMK - Y COORDINATE f o p THE I N T E RS E C T I O N OF THE TOP s u r f a c e . SLOPE A 1 2 1 0C FACE, AND K FRACTURE A 1 2 2 0c ZTMK » Z COORDINATE FOR THE I N T E R S E C T I O N OF THE TOP SURFACE, SLOPE A 1 2 30c FACE, -AND K FRACTURE A 12 40

125' c XTM.L * X COCROINATE FOR THE I N T E RS E C T I O N OF THE TOP SURFACE# SLOPE A 1 2 5 0c FACE, AND L FRACTURE A 12 60c YTHL « Y COORDINATE FOR THE I N T E R S E C T I O N OF THE TOP SURFACE, SLOPE A127Cc FACE, AND L FRACTURE A 12 8 0c ZTMI « Z COORDINATE FOR THE I N T E RS E C T I O N OF THE TOP SURF ACE, SLOPE A 1 2 9 0

130 c FACE, AND L FRACTURE A 13 00c D ( , ) ■ VECTOR COORDINATES FOR THREE CORNERS OF THE TETRAHEDRON A 1 3 1 0c OKS * SURFACE AREA OF K FRACTURE A 1 3 2 0c ELS * SURFACE AREA OF L FRACTURE A1 3 3 0c VOL ■ VOLUME OF THE TETRAHEDRON A 13 40

135 c WT * WEIGHT OF TETRAHEDRON A 13 50c ANBN * PART OF EQUATI ON USED I N CA LC U LA T I N G THE NORMAL LOADS ON A 1 3 6 0c THE FRACTURE PLANES A 1 37 0c NORMK - NORMAL WEIGHT COMPONENT ON THE K FRACTURE A 1 3 8 0c NORML - NORMAL WEIGHT COMPONENT ON THE L FRACTURE A 1 3 9 0

140 c CCKDKS * COHESIVE SHEAR STRENGTH FOP K FRACTURE A 1 4 0 0c COHELS = COHESIVE SHEAR STRENGTH F I R L FRACTURE A 1 41 0c R E S I S T * TOTAL SHEAR STRENGTH ( R E S I S T I N G FORCE ) A 1 4 2 0c D I P I N T = L I N E OF I N T E RS E C T I O N PLUNGE FOR THE TWO FRACTURES A 1 4 3 0c A Z I N T « AZI MUTH FOR L I N E OF I N T E RS E C TI O N A 1 4 4 0

145 c D RI V E F « TOTAL D R I V I N G FORCE A 1 4 5 0c F S ( I F S ) « FACTOR OF SAFETY A 14 8 0c POSFAL ■ E S T I MA T E OF THE P R O B A B I L I T Y THAT I N D I C A T E S THAT A CERTAI N A14 7 0c . PERCENTAGE OF FRACTURE COMBI NATI ONS W I LL HAVE A I N T E R S E C T I ON A1 4 P0c L I N E ORIENTED I N SUCH A MANNER THAT F A I L U R E IS P OS S I BL E A 1 4 90

150 c PCFL - ESTI MATE 0 c THP P R O B A B I L I T Y OF F A I L U R E FOR THOSE I T E R A T I O N S A 1 5 00c OR FRACTURE C OMBI NAT I ONS I N WHICH F A I L U R E I S K I N E M A T I C A L L Y A 1 5 1 0c P OS S I BL E A 1 5 20c XMNC = MEAN COHESION VALUE ( I N ROUNDS/SQUARE I N C H) A 1 5 3 0c S DC » STANDARD D E V I A T I O N FOR COHESION ( I N POUNDS/SQUARE I N C H) A 15 40155 c XMNF » MEAN F R I C T I O N ANGLE VALUE A 1 5 50c SDF = STANDARD D E V I A T I O N FOR F R I C T I O N ANGLE A 1 5 6 0c COHESK = COHESION FOR K FRACTURE A 1 5 7 0c COHESL ■ COHESION FOR L FRACTURE A 1 5 8 0c FR1ANK r F R I C T I O N ANGLE FOR K FRACTURE A 1 5 9 0160 c F P I A N L * F R I C T I O N ANGLE FOR L FRACTURE A 1 6 0 0c I F R C F S ( I ) - ACCUMULATIVE FACTOR OF SAFETY V A R I A B L E A 1 6 1 0c S I G 1 * STANDARD D E V I A T I O N FOR THE P R O B A B I L I T Y THAT F A I L U R E IS A 1 6 2 0c K I N E M A T I C A L L Y POSSI BLE A 1 6 3 0c S I G2 * STANDARD D E V I A T I O N FOR THE P R O B A B I L I T Y - O F F A I LU R E FOR THOSE A 1 6 4 0165 c PAI RS OF FRACTURES IN WHICH F A I LU R E I S K I N E M A T I C A L L Y P OSSI BL E A 1 6 5 0c T 0 T F F 1 - MAXIMUM TOTAL P R O B A B I L I T Y OF F A I L U R E C ON S I DE RI NG A A 1 6 6 0c 9 5 PERCENT CONFIDENCE LEVEL A 1 8 7 0c T 0 T P F 2 - MINIMUM TOTAL P R O B A B I L I T Y OF FA I LU R E C ON S I DE RI NG A A 1 6 8 0c 9 5 PERCENT CONFIDENCE LEVEL A 1 6 9 0170 DI ME N S I ON A Z S ( 3 ) , D I P S ( 3 ) , S T S A ( 3 > , S T S D ( 3 ) , D I S S ( 3 ) , A Z F ( 1 0 0 ) , D I P F ( 1 0 0

1 ) , S T F A ( 1 0 0 ) , S T F D ( 1 0 0 ) , C I S F ( 1 0 0 ) , F S ( 6 0 0 ) ,A 1 7 0 0A 1 7 1 0

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2 Z S 0 ( 3 ) , X S D ( 3 ) > Y S D ( 3 ) > Z S U ( 3 ) , X S U ( 3 ) > Y S U ( 3 ) , * 1 7 2 03 D I P F 1 C 1 0 C ) , * 1 7 3 04 Z F U 1 ( 1 0 0 ) , Z F 0 1 ( 1 0 0 ) , X F U 1 ( 1 0 0 ) , Y F U 1 ( 1 0 0 ) , X F D 1 ( 1 0 0 ) , * 1 7 4 0

175 5 Y F 0 K 1 0 0 ) , 0 I P F 2 ( 1 0 0 ) , * 1 7 5 06 Z F U ? ( 1 0 0 ) , Z F D 2 ( 1 0 0 ) , X F U 2 ( 1 0 0 ) , Y F U 2 ( 1 0 0 ) , X F 0 2 ( 1 0 0 ) , Y F 0 2 ( 1 0 0 ) , D C 3 , 3 ) * 1 7 6 07 , * Z F 3 ( 1 0 0 ) , * Z F 4 ( 1 0 0 ) » I F R O F S ( I O ) * 1 7 7 0

R E * L N O R M K , N C R M L , K K K K , I F 6 I L * 1 7 8 0C READ NUMRER OF FRACTURE P L ANE S , AND THE U N I T WEIGHT OF THE * 1 7 9 0

180 C ROCK ( L 5 / F T 3 ) . * 1 8 0 0READ 1 1 0 , ' M .U N I TW T A 1 8 1 0

1 1 0 FORMAT ( 1 5 , F 1 0 . 2 ) * 1 8 2 0C READ THE MEAN AND STANDARD D E V I A T I O N VALUES FOR COHESION AND * 1 8 3 0C F R I C T I O N * 1 8 4 0

185 READ 1 2 0 , XMNC, S D C, X M NF , SDF * 1 8 5 01 2 0 FORMAT ( 4 F 1 C . 0 ) * 1 8 6 0

N»3 * 1 8 7 0C READ AZI MUTH AND D I P OF ROCK TOPOGRAPHIC SURFACES, PLUS L OCATI ON * 1 6 8 0C AT WHICH A Z I MUTH AND D I P WAS TAKEN * 1 8 9 0

190 READ 1 3 0 , ( A Z S < I ) , D I P S < I ) , S T S A ( I ) , S T S D < I ) , D I S S < I ) , I « 1 »N ) * 1 9 0 01 30 FORMAT ( 2 F 5 . 1 , 2 F6 . 2 , F 1 0 . 2 ) * 1 9 1 0C READ AZI MUTH A NO DI P OF FRACTURES, PLUS LOCATI ON AT WHICH * 1 9 2 0C A ZMI TH AND D I P WAS TAKEN * 1 9 3 0

READ 1 4 0 , ( a Z F ( J ) , D I P F ( J ) , S T F A ( J ) , S T F D ( J ) , D I S F ( J ) , J - 1 , M > * 1 9 4 0195 1 4 0 FORMAT ( 2 F 5 . 1 , 2 F 6 . 2 # F 1 0 . 2 ) • * 1 9 5 0

IF S * 0 A I 9 6 0I F A I L = 0 . • * 1 9 7 0XME A N - - 1 . * 1 9 8 0STOEV — 1 . * 1 9 9 0

2 0 0 K K KKa O. • A 2 0 0 0DO 1 5 0 1 * 1 , 1 0 A 2 0 1 0I F R Q F S t I ) * 0 * 2 0 2 0

1 50 CONTINUE * 2 0 3 0C P R I N T HEADI hG . • * 2 0 4 0

2 0 5 P R I NT 1 6 0 ' * 2 0 5 01 6 0 FORMAT ( 1 H 1 , 3 1 X , 2 8HROCK SLOPE S T A B I L I T Y P R O G R A M , / / ) * 2 0 6 0C P R I N T I NPUT * 2 0 7 0

P R I N T 1 70 * 2 0 8 01 70 FORMAT ( 1H , 3 8 X » 1 0 H I N P U T D A T A / ) * 2 0 9 0

2 1 0 C P R I N T I NPUT C AT* ( M, UNI TWT #XMNC, S D C , S M N F , S D F > * 2 1 0 0P R I N T l A O , M , U N l T W T , X M . N C , S D C , X M N F , S D f * 2 1 1 0

1 8 0 FORMAT ( 1H , * T C T A L NUMBER OF F R A C T U R E S * * , I 4 / 1 X , + U N I T WEIGHT OF THE * 2 1 2 01 F O C K * * . F 1 C . Z / 1 X , * M E A N COHESION V AL UE- * , F 1 0 . 3 / 1 X , * S T A N DA R 0 DEVI A T I * 2 1 3 0 2 0 N FOR C O H E S I O N * * , F 1 0 . 3 / 1 X , * M E A N F P I C T I ON. V A L UE - * , F 1 0 . 2 / I X , * S T A N D A * 2 1 4 0

2 15 3RD D E V I A T I O N FOR F R I C T I ON* ♦ , F 1 0 . 2 ) * 2 1 5 0C P R I N T I NPUT DATA FOR EXTERIOR TOPOGRAPHIC SURFACES * 2 1 6 0

PRINT 1 90 * 2 1 7 01 9 0 FORMAT ( 1 H 0 , 3 1 X , * E X T E R I 0 R TOPOGRAPHIC S URF A CE S * ) ' * 2 1 8 0

P R I N T 2 0 0 * 2 1 9 02 2 0 2 0 0 FORMAT ( I K . <■ S T R IKE * , 7 X , 3 H D I P , 5 X , 16H LOC AT I ON A Z I M U T H , 5 X , 1 2 H L 0 C A T I 0 * 2 2 0 0

I N D I P , 5X» 2 4H L 0 CA T ION DISTANCE ( F E E T ) ) * 2 2 1 0P R I N T 2 1 0 , ( A Z S ( I ) , D I P S ( I ) , S T S A ( I ) , S T S D ( I ) , D I S S ( I ) , 1 * 1 , N) A 2 2 2 0

2 10 FORMAT ( 1 H , 1 X , F 5 . 1 » 6 X , F 5 . 1 , 8 X , F 6 . 2 , 1 3 X , F 6 . 2 , 1 1 X , F 1 0 . 2 ) A 2 2 3 0C P R I N T I NPUT DATA FOR FRACTURES * 2 2 4 0

2 2 5 P R I N T 2 2 0 * 2 2 5 02 2 0 FORMAT ( 1 H 0 , 3 7 X , * F R A C T U R E P L A N E S * ) * 2 2 6 0

PRI NT 2 3 0 * 2 2 7 02 3 0 FORMAT ( 1 H >* STP I KE*,7 X , 3 H DIP , 5 X , 16HLOCAT ION A Z I M U T H , 5X , 1 2 H L 0 C A T I 0 * 2 2 8 0

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I N D I P # 5 X , 2 4 H L C C A T I 0 N D IS TA N C E ( P E E D ) A 2 2 9 0230 P R I N T 2 4 0 , ( A 7 F ( J ) , 0 I P F ( J ) # S T F A ( J ) , S T F D ( J ) , D I S F ( J ) , J » 1 , M ) A 2 3 0 0

240 FORMAT ( 1 H , 1 X , F 5 . 1 , 6 X , F 5 . 1 , 8 X , F 6 . 2 , 1 4 X , F 6 . 2 , 1 0 X , F 1 0 . 2 ) A 2 3 1 0C ROTATE COORDINATE A X I S U N T I L THE MIDDLE SLOPE FACE HAS A AZ I MUT H A 2 3 2 0C OF 9 0 DEGREES A 2 3 3 0

I F ( A Z S ( 2 ) - 9 0 . ) 2 5 0 , 2 6 0 , 2 7 0 A 2 3 4 0235 2 50 R 0 T A « A Z S ( 2 ) + Z 7 0 . A 2 35 0

GO TO 2 3 0 A 2 3 6 0260 ROT A« 0 . A 2 3 7 0

GO TO 2 30 • A 2 3 8 0270 ROT A* AZ S ( 2 ) - 9 G . A 2 3 9 0

240 C ROTATE THE THREE TOPOGRAPHIC SURFACES AND THE A Z I MUT H A 2 40 0C SHOWING THE LOCATI ON OF THE R ES P E CT I V E PLANES A 2 4 1 02 80 DO 3 4 0 I ■ 1 , N A 2 4 2 0

I F ( R 3 T A - A Z S ( I ) ) 2 9 0 . 2 9 0 , 3 0 0 A 2 4 3 02 9 0 A Z S ( I > *AZS C I ) - R O T A A 2 4 4 0

245 GO TO 3 10 A 2 4 5 03 00 A Z S ( I ) - 3 6 0 . - ( R C T A - A Z S ( I ) ) A 2 4 6 03 10 I F ( R O T A - S T S A ( I ) ) 3 2 0 , 3 2 0 , 3 3 0 A 2 4 7 03 20 S TS A ( I ) « S T S A < I J - R O T A A 2 4 8 0

GO TO 3 40 A 2 4 9 0250 3 3 0 ST S A ( I ) * 3 6 0 . - ( R O T A - S T S A { I ) ) A 2 5 0 0

3 40 CONTINUE A 2 5 10C ROTATE THE FRACTURE PLANES AND T H E I R . R E S P E C T I V E L OC A T I ON AZI MUTHS A 2 5 2 0

DO 4 0 0 J * 1 , M A253-0I F ( RO T A - A Z F < J ) ) 3 5 0 , 3 5 0 , 3 6 0 A 2 5 4 0

255 3 50 A Z F ( J ) - A Z F ( J ) - R O T A A 2 5 5 0GO TO 3 7 0 A 2 5 6 0

3 60 A Z F I J )» 3 6 0 . ~ ( R O T A - A Z F ( J ) ) A 2 5 7 03 70 I F (ROT A- S T F A ( J ) ) 3 3 0 , 3 8 0 , 3 9 0 A 2 5 8 03 80 S T F A ( J ) « S T F A ( J ) - P Q T A A 2 5 9 0

260 GO TO 4 0 0 A 2 6 0 03 90 S T F A ( J ) * 3 6 0 . - ( R 0 T A - S T F A ( J ) ) A 2 6 1 04 0 0 CONTINUE A 2 6 2 0C CHANGE L OCATION COORDINATES FOR THE TOPOGRAPHIC SURFACE A 2 6 3 0C FEATURES FROM POLAR TO C ART ESI AN A 2 6 4 0

265 DO 4 1 0 1 * 1 , N A 2 6 5 0X S D ( I ) - D I S S ( I ) * C O S ( S T S D ( I ) * . 0 1 7 4 5 ) * S I N ( S T S A ( I ) * . 0 1 7 4 5 ) A 2 6 6 0Y S D ( I ) « 0 1 S S ( I ) * C 0 S ( S T S D ( I ) * . 0 1 7 4 5 ) * C 0 S ( S T S A ( I ) * . 0 1 7 4 5 ) # 2 6 7 0Z S D ( I ) = D I S S ( I ) * S I N ( S T S D ( I ) * . 0 1 7 4 5) A 2 6 8 0

410 CONTINUE A 2 6 9 0270 C FORMULATION OF D I R E C T I O N NUMBERS FOR TOPOGRAPHIC SURFACE FEATURES A 2 7 0 0

DO 4 2 0 1 * 1 , N A 2 7 10X S U ( I ) - l . * S I N ( D I P S ( I ) * . 0 1 7 4 5 ) * s I N ( < A Z S ( I ) * 9 0 . ) * . 0 1 7 4 5 ) A 2 72 0Y S U ( I ) - 1 . * S I N ( D I P S ( I ) * . 0 1 7 4 5 ) * C O S < ( A Z S ( I ) * 9 0 . ) * . 0 1 7 4 5 ) A 2 7 3 0Z S U ( I ) * 1 . * C C S ( D I P S ( I ) * . 0 1 7 4 5 ) A 2 7 4 0

275 420 CONTINUE A 2 7 5 0C FORMULATION OF D I R E C T I O N NUMBERS AND CART E S I A N COORDINATE LOCATI ON A 2 7 6 0C OF S T R I K E AND D I P READINGS FOR FRACTURES - ALSO THE GROUPING OF A 2 7 7 0C FRACTURES INTO TWO GROUPS WI TH AZI MUTHS OF 2 7 0 - 9 0 AND 9 0 - 2 7 0 A 2 7 8 0

K * 0 A 2 7 9 0280 L * 0 A 2 8 0 0

DO 4 5 0 J * 1 , M A 2 P 1 0C S E R I ES OF ( I F ) STATEMENTS USED TO GROUP FRACTURES I NTO TWO GROUPS A 2 8 2 0

I F ( A Z F U ) . L E . 9 0 . ) GO TO 4 3 0 A 2 8 3 0I F ( A Z F U ) . I E . 2 7 0 . ) GO TO 4 4 0 A 2 8 4 0

285 430 K « K * 1 A2850

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KMsk 4 2 8 6 0A Z F 3 ( K ) « A Z F ( J ) A 2 8 7 0D I P F 1 ( K ) - O I P F ( J ) A 2 8 6 0X F D 1 ( K ) » O I S F ( J ) * C O S ( S T F D ( J ) * . 0 1 7 4 5 ) * S I N ( S T F A ( J ) * . 0 1 7 4 5> A 2 8 9 0Y F 0 1 ( K ) a D I S F ( ' ) * C 0 S ( S T F 0 ( J ) * . 0 1 7 4 5 ) * C O S ( S T F A ( J ) * . 0 1 7 4 5 ) A 2 9 0 0Z F D 1 C K ) » O I S F ( J ) * S I N ( S T F D ( J ) * . 0 1 7 4 5 ) A 2 9 1 0X F U 1 ( K ) = 1 . * 5 I N ( O I P F ( J ) * . 0 1 7 4 5 ) * S I N ( ( A Z F ( J ) * 9 0 . ) * . 0 1 7 4 5 ) A 2 9 2 0Y P U 1 ( K ) » 1 . * S I N ( D I P F ( J ) * . 0 1 7 4 5 ) * C O S ( ( A Z F ( J > * 9 0 . ) * . 0 1 7 4 5 ) A 2 9 3 0Z F U 1 ( K ) * 1 . * C 0 S < D I P F ( J ) * . 0 1 7 4 5 ) . A 2 9 4 0GO TO 4 5 0 • A 2 9 5 0L ' L + 1 A 2 9 6 0L M - L A 2 9 7 0A Z F 4 ( L ) - A Z F ( J ) A 2 9 8 0D I P F 2 ( L ) = D I F F C J ) A 2 9 9 0X F 0 2 ( L ) » 0 I S f ( J ) * C O S ( S T F D ( J ) + . 0 1 7 4 5 ) * S I f ; ( S T F A ( J ) * . 0 1 7 4 5 ) A 3 0 0 0Y F D 2 ( L ) - D I S F ( J ) * C O S ( S T F 0 ( J ) * . O 1 7 4 5 ) * C O S ( S T F A ( J ) * . O 1 7 4 5 ) A 3 0 1 0Z F 0 2 ( L ) * D I S F ( J ) * S I N ( S T F 0 < J ) * . 0 1 7 4 5 ) A 3 0 2 0X F U 2 ( L ) » 1 . * S I N ( D I P F ( J ) * . 0 1 7 4 5 ) * S I N ( ( A Z F ( J ) * 9 0 . ) * . 0 1 7 4 5 ) A 3 0 3 0Y F U 2 ( L ) » 1 . * S I N ( O I P F ( J ) * . 0 1 7 4 5 ) * C O S ( ( A Z F ( J ) * 9 0 . ) * . 0 1 7 4 5 ) A3 04 0Z F U 2 ( L ) » 1 . * C O S C D I P F ( J ) * . 0 1 7 4 5 ) A 3 0 5 0C C N T I H J E A 3 0 6 0P R I N T 4 6 0 A 3 0 7 0FORMAT ( 1 H 1 , 4 0 X > * R F S U L T S * ) A 3 0 8 0P R I N T 4 7 0 / KM/ L M • A 3 0 9 0FORMAT ( 1 H 0 . * T C T A L NUMBER OF (K ) GROUP F R A C T U R E S * * / I 4 / 1 X / * T O T A L NU A 3 10 0

1M3EP OF ( L ) GROUP F R A C T U R E S " * / 1 4 / / ) A 3 1 1 0PRI NT 4 9 0 A 3 1 2 0FORMAT ( l H 0 / * N U H B E R * / 6 X / * F A C T 0 R * / 4 X z * S T » I K E * / 5 X , * S T R I K E * / 5 X / * L I N E A 3 1 3 0

10F * / 4 X . * L I N E O F * , 6 X / * T E T R A H E D R O N * , 7 X / * C O H E S I O N * / 3 X / * C 0 H E S I O N * / 3 X / * A 3 1 4 02 F R I C T I C N * , 3 X , * F R I C T I 0 N * / 3 X , * 0 F * , 1 0 X , * 0 F * , 7 X , * Q F K * , 7 X , * 0 F L * , 5 X , * I A 3 1 5 03NTERS E C - I N T E R S E C - * # 6 X , * W E D G E * / 1 1 X , * F 0 R K * , 6 X , * F 0 R L * / 6 X , * F 0 R K * / A 3 1 6 04 6 X , * F OR L * / 1 X , * I T £ R A T I 0 N S S A F E T Y * , 4 X . * f R A C T U R E * / 3 X, * F R A C TU RE * / 4 X / A 3 1 7 05 * T I U N * / 7 X / * T I 0 N * / 1 1 X / * VOLUME* / 9 X , * F R A C T U R E * , 3 X , * F P A C T U R E * , 3 X , * F R A C A 3 1 8 06 T U P E * / 3 X , * F R A C T U R E * / 4 5 X , * A Z I M U T H * , ' 4 X , * P L U N G E * / ) A 3 1 9 0

K K K K » K i < < K * l . A 32 0 0KHOET * 0 A 3 21 0KNHDET * 0 A 3 2 2 0GENERATE RANDOM NUMBERS FOR K AND L FRACTURES, AND THEN SELECT A A 3 2 3 0

RANDOM K AND L FRACTURE A 3 2 4 0CALL RANOKL ( K M / K ) A 3 2 5 0CALL RANDKL ( L M, L ) A 3 2 6 0TEST TO DETERMINE I F THE TWO FRACTURES ARE EXACTLY 180 DEGREES A 3 27 0

APART . A 32 P 0I F ( A Z F 3 ( K ) . L E . 9 0 . ) GO TO 5 0 0 A 3 2 9 0G F F D - A Z F 3 ( K ) - 1 6 0 . A 3 3 0 0GO TO 5 10 A 3 3 1 0G F F D * A Z F 3 ( K > * 1 6 0 . A 3 3 2 0I F ( A Z F 4 ( L ) . EO. GFF D ) GO TO 4 9 0 A 3 3 3 0NORMALLY D I S T R I B U T E D RANDOM NU M3 E R GENERATOR FOR COHESION AND A 3 3 4 0F R I C T I O N VALUES • ' A 3 3 5 0CALL RANDCF <S DC, XMNC, COHES< > A 3 3 6 0I F ( C O H E S K . L T . O . O ) GO TO 5 20 A 3 3 7 0CALL RANDCF <S DC, XMNC, COMFSL) A 3 3 8 0I F ( C O H E S L . L T . O . C ) GO TO 5 3 0 A 3 3 9 0CALL RANDCF ( S 0 F , XMNF, FR I A N K ) A 3 4 0 0I F ( F R I A N K . L T . O . O ) GO TO 5 4 0 A3410CALL RANDCF ( S D F , X M N F , F R I A N L ) A3420

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I F ( F R I A N L . L T . 0 . 0 ) GO TO 5 5 0 A 3 4 3 0C AL C U L A T I ON OF I N T E R S E C T I O N P O I N T FCR THE TOP SURFACE AND A 3 44 0THE TWO FRACTURES A 3 45 0CALL S I M U L T <X S U # Y S U , Z S U » X S D » Y S D » Z S 0 . 1 # 1 # K > L * X F U 1 » Y F U 1 # Z F U 1 , XF O1 # Y A 3 4 6 0

l F D l , Z F 0 1 , X F ( . E , Y r U 2 , Z F U 2 # X F 0 2 , Y F D 2 # Z F D 2 , l , l . » l . » l . > l . # l . f X T » Y T > Z T , K A34 702 M , L M , K H 0 E T , k N H D E T ) A 3 4 8 0

I F ( K H O E T . E O . l ) GO TO 5 60 A 3 4 9 0I F ( K N H O E T . E C . 1 ) GO TO 5 60 A 3 5 0 0C ALC U LA T I O N CF I N T E R SE CT I ON P OI NT FOR THE MI DDLE SURFACE AND A 3 5 1 0THE TWO FRACTURES • A 3 5 2 0CALL SIMUCT ( X S U » Y S U » Z S U # X S D > Y S D » Z S D # 2 / 1 > K , L > X F U 1 > Y F U 1 » Z F U 1 > X F D 1 # Y A 3 5 3 0

1 F D 1 , Z F D 1 , X F U 2 , Y F U 2 , Z F U 2 , X F D 2 , Y F D 2 , Z F D 2 , 1 , 1 . , 1 . , 1 . , 1 . , I . , X M , Y M , Z M , K A 3 5 4 02 M , L r , X H D E T , K N H D E T ) A 3 5 5 0

I F ( K H O E T . E O . l ) GO TO 5 60 A 3 5 6 0I F ( K N H O E T . E O . 1 ) GO TO 5 60 A 3 5 7 0DETERMINE I F THE I N T E R SE CT I O N PLUNGES DOWN OR UP A 3 5 8 0I F ( Z M . G E . Z T ) GO TO 4 9 0 A 3 5 9 0I F - ( Y M . G E . Y T ) GO TO 4 9 0 A 3 6 0 0CALCULATE THE D I R E C T I O N PARAMETERS FOR THE V E R T I C A L I N T E R S E C T I O N A 3 6 1 0

PLANE A 3 6 2 0X U ' X M - X T A 3 6 3 0Y U « Y M - YT A 3 6 4 0U T » S 0 R T ( X U * * 2 + Y U * * 2 ) A 3 6 5 0X I U - - Y U Z U T ' A 3 6 6 0Y I U - X U / U T * A 3 4 7 0Z I U « 0 . A 3 6 8 0C AL C U L A T I ON OF I N T E R S E C T I O N PLANE WI TH MIDDLE ( S L O P E ) FACE AND A 3 6 9 0

BOTTOM FACE A 3 7 0 0CALL S I MUL T ( X S U , Y S U , Z S U , X S 0 , Y S D , Z S D , 2 , 0 , K , L , X F U 1 , Y F U I , Z F U 1 , X F D 1 , Y A 3 7 1 0

l F D l # Z F D l f X F L ' 2 » Y F U 2 » Z F U 2 * X F D 2 # Y F D 2 # Z F D 2 # 3 # X I U » Y I U # Z I U # X T » Y T * X M B I # Y M A 3 7 2 02 8 1 * ZM8 I » < M > L H> KHDET . KNHDET ) A 3 7 3 0

I F ( K H O E T . E O . l ) GO TO 5 60 A 3 7 4 0I F ( K N H O E T . E O . 1 ) GO TO 5 60 A 3 7 5 0METHOD OF D I SREGARDI NG VARIOUS COMBI NATI ONS OF FRACTURES WHICH DO A 3 7 6 0

NOT D A Y L I GHT ON SLOPE FACE AND L I M I T I N G THE FRACTURES A 3 7 7 0S TU D I E D TO THOSE WHICH ARE RELEVANT A 3 7 8 0

I F ( Z M 3 I . G T . Z M ) GO TO 4 9 0 A 3 7 9 0C ALCULA T I ON OF I N T E R SE C TI ON FOR V E R TI C A L I N T E R SE C T I O N PLANE WI TH A 3 8 0 0

THE TOP SURFACE AND THE MI DDLE ( S L O P F ) FACE A 3 31 0CALL S I MUL T ( X S U , Y S U , Z S U , X S D , Y S D , Z S D , 1 , 0 , K , L , X F U 1 , Y F U 1 , Z F U 1 , X F 0 1 , Y A 3 8 2 0

1 F D 1 > Z F D V , X F L 2 . Y F U 2 , Z F U 2 , X F 0 2 , Y F D 2 , Z F D 2 , 2 , X I U , Y I U , Z I U , X T , Y T , X T M I , Y T A 3 8 3 02 M I , Z T M I , K M , L N # K H P E T , X N H O E T ) A 3 8 4 0

I F ( K H D E T . E O . l ) GO TO 5 60 A 3 8 5 0I F ( K N H O E T . E C . 1) GO TO 5 60 A 3 P6 0METHOD OF D I SREGARDI NG VARIOUS COMBI NATI ONS OF FRACTURES WHICH A3P70

HAVE A I N T E R S E C T I O N L I N E WHICH INTERSECTS THE TCP SURFACE A 3 8 8 0ON THE SLOPE FACE S I D E OF Y T M I A 3 9 9 0

I F ( Y T M I . G E . Y T ) GO TO 4 90 A 3 9 0 0C ALC U LA T I O N CF I N T E R SE C T I O N FOR (K> FRACTURE WI TH TOP AND MI DDLE A 3 9 1 0

( F A C E ) SURFACES . ' A 3 9 2 0CALL S I MU LT ( X S U , Y S U , Z S L , X S D , Y S D , Z S D , 1 , 1 , K , 2 , X F U 1 , Y F U 1 , Z F U 1 , X F D 1 , Y A 3 9 3 0

1 F D 1 , Z F D 1 , X S U , Y S U , Z S U , X S D , Y S D , Z S D , K E , X I U , Y I U , Z I U , X T , Y T , X T M K , Y T M K , Z T A 3 9 4 02 M K , K M , L M , K H D E T , K N H D E T ) A 3 9 5 0

I F ( K H D E T . E C . 1 ) GO TO 560 A 3 9 6 0I F ( K N H D E T . E C . 1 ) GO TO 5 6 0 A 3 9 7 0C AL C U L A T I ON OF I N T E R SE C T I O N FOR ( L ) FRACTURE WITH TOP AND MIDDLE A 3 9 6 0

<FACF> SURFACES A 3 9 9 0

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654 0 0 CALL S I MU L T ( X S U , Y S U , Z S U , X S 0 , Y S D , Z S D , 1 , 1 , 2 , L , X S U , Y S U , Z S U , X S D , Y S D , Z A^OOO

l S 0 # X F I J 2 t Y F U 2 , Z F U 2 , X F D 2 t Y F D 2 , Z F D 2 , K E » X l U , Y I U , Z I U # X T , Y T , X T M L > Y T M L > Z T A<.010 2 M L # K M , L ‘1»KHC£T#KNHDET) A 4 0 2 0

I F ( K H O E T . E O . 1 ) GO TO 5 60 A403CI F ( K M H D E T . E C . 1 ) GO TO 5 6 0 A<,040

4 0 5 GO TO 6 7 0 A4C505 6 0 I F ( ( 3 6 0 . - 9 G T A ) - A Z F 3 ( K ) ) 5 7 0 , 5 7 0 , 5 9 0 A 4 0 6 05 70 A Z F 3 K » A Z F 3 ( K > - { 3 6 0 . - R O T A ) * A 4 0 7 0

GO TO 5 90 . A 4 0 8 05 6 0 A Z F 3 K » R O T A > A Z F 3 ( K ) A 4 0 9 0

4 1 0 \ 5 90 I F ( ( 3 6 0 . - R O T A ) - A Z F 4 ( L )> 6 0 0 , 6 0 0 , 6 1 0 A 4 1 0 06 0 0 A Z F 4 L * A Z F 4 ( l ) - ( 3 6 0 . - R O T A ) A 4 1 1 0

GO TO 6 2 0 A 4 1 2 06 1 0 A Z F 4 l * a O T A + A Z F 4 ( I ) A 4 1 3 06 2 0 I F ( K H O E T . E O . 1 ) GO TO 6 40 A 4 1 4 0

4 1 5 P R I N T 6 3 0 , A Z F 3 K , A Z F 4 L A 4 1 5 06 3 0 FORMAT ( 1 H 0 , * A Z F 3 K * * , F 5 . 1 , 2 X , * A Z F 4 L ■ * F 5 . 1 , 3 X , * NC UNI QUE SOL UT I ON F A 4 1 6 0

10R T H I S FRACTURE S E T * ) A 4 1 7 0GO TO 6 6 0 A 4 1 8 0

6 4 0 P R I N T 6 5 0 , A Z F 3 K , A Z F 4 L A 4 1 9 04 2 0 6 5 0 FORMAT ( 1 H 0 , * A Z F 3 K - * , F 5 . 1 , 2 X , * A Z F 4 L ■ * F 5 . 1 , 3 X , * ON L Y A T R I V I A L SOLUT A 4 2 0 0

I I O N x - Y - Z - 0 FOR T H IS HOMOGENEOUS FRACTURE S E T* > A 4 2 1 06 6 0 GO TO 4 9 0 ' A 4 2 2 0C C A LC U LA T I O N OF VOLUME AND SURFACE AREA OF TETRAHEDRON A 4 2 3 0C C A LC U LA T I O N OF THE VECTOR COMPONENTS A 4 2 4 0

4 25 6 7 0 D ( 1 , 1 ) * X T * K - X M . A 4 2 5 00 ( 1 , 2 ) = Y T M K - Y M A 4 2 6 00 ( 1 , 3 ) * Z TMK-Z M A 4 2 7 0D ( 2 , 1 ) * X T —XM A 4 2 8 00 ( 2 , 2 ) " YT- YM A 4 29 0

4 3 0 D ( 2 , 3 ) * Z T - Z M A430C0 ( 3 , 1 ) * XTML-XM A 431CD ( 3 , 2 > = YTML-YM . A 4 3 2 0D ( 3 , 3 ) * Z T M L - Z M ' . A 4 33 0

C C A L C U L A T I ON OF SURFACE AREA ( DKS I S SURFACE AREA FOR K FRACTURE A 4 3 4 04 35 C AND ELS I S SURFACE AREA FOR L FRACTURE) A4 35 0

D K S « S Q R T ( ( D ( 2 , 1 ) * D ( 2 , 1 ) + 0 ( 2 , 2 ) * D ( 2 , 2 ) * D ( 2 , 3 ) * D ( 2 , 3 ) ) * ( D ( 1 , 1 ) * D ( 1 , 1 A 4 3 6 01 ) + D ( 1 , 2 ) * D ( 1 , 2 ) + D ( 1 , 3 ) * D ( 1 , 3 ) ) - ( D ( 2 , 1 ) * D ( 1 , 1 ) + D ( 2 , 2 ) * D ( 1 , 2 ) + D ( 2 , 3 ) A43 702 * C < 1 , 3 ) ) * * 2 ) A 4 3 8 0

E L S = S Q R T ( ( C ( 2 , 1 ) * D ( 2 , 1 ) * D ( 2 , 2 ) * 0 ( 2 , 2 ) * D ( 2 , 3 ) * D ( 2 , 3 ) ) * ( D ( 3 , 1 ) * D ( 3 , 1 A4 39 04 4 0 1 ) 4 0 0 , 2 ) < - 0 ( 3 , 2 ) * 0 ( 3 , 3 ) * D ( 3 # 3 ) ) - t D ( 2 , l > * 0 ( 3 , 1 H D ( 2 , 2 1 * 0 ( 3 , 2 ) * 0 ( 2 , 3 ) A 4 4 0 0

2 * 0 ( 3 , 3 ) ) * * 2 ) A 4 4 1 0D K S - A 8 S ( O K S / 2 . ) . A 4 4 2 0E L S ' A S S ( E L S / 2 . ) A 4 4 3 0

C C A L C U L A T I ON FOR VOLUME OF TETRAHEDRON A 4 4 4 04 45 V C L * D ( 1 , 1 ) * D ( 2 , 2 ) * D ( 3 , 3 ) 4 0 ( 1 , 2 ) * 0 ( 2 , 3 ) * D { 3 , 1 ) 4 0 ( 1 , 3 ) * 0 ( 2 , 1 ) * D ( 3 , 2 ) A4 4 5 0

1 - D ( 3 , 1 ) * 0 ( 2 , 2 ) * 0 ( 1 . 3 ) - 0 ( 3 , 2 ) * 0 ( 2 , 3 ) * 0 ( 1 , 1 ) - 0 ( 3 , 3 ) * 0 ( 2 , 1 ) * 0 ( 1 , 2 ) A 4 4 6 0V O L ' A d S ( V O L / 6 . ) A 4 4 7 0

C C A LCULA T I ON FOR WEIGHT OF TETRAHEDRON , ' A 4 4 8 0W T * U N I T WT*VOL A 4 4 9 0

4 5 0 C WEIGHT COMPONENT OF TETRAHEDRON I N NORMAL D I R E C T I O N TO FRACTURE A 4 5 0 0C PLANE A 4 5 1 0

A N 3 N . ( S I N ( D I P F 1 ( K ) * . 0 1 7 4 5 ) ) * ( S I N ( O I P F 2 ( L ) * . 0 1 7 4 5 ) ) * ( C O S ( ( ( A Z F 3 ( K ) * A 4 5 2 01 9 C . ) - ( A Z F 4 ( L ) * 9 0 . ) ) * . 0 1 7 4 5 ) ) 4 ( C O S ( O I P F 1 ( K ) * . 0 1 7 4 5 ) ) * ( C O S ( D I P F 2 ( L ) * A 4 5 3 02 . 0 1 7 4 5 ) ) A 4 5 4 0

4 55 N O R M K * ( ( ( A N B N * ( —C O S ( D I P F 2 ( L ) * » 0 1 7 4 5 ) ) ) * C 0 S ( 0 I P F 1 ( K ) * . 0 1 7 4 5 ) ) / ( 1 . - A A 4 5 5 01 N 8 N * * 2 ) >*WT A 4 5 6 0

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66N C R M l « ( ( ( A N P N * ( - C 0 S ( D I P F l ( K ) * . 0 1 7 A 5 ) ) ) * C 0 S ( D I P F 2 ( L ) * e 0 1 7 A 5 ) ) / ( l , - A A 4 5 7 0

1 N B N * * 2 ) ) * W T A 4 5 6 0C C A L C U L A TI ON OF COHESIVE RESI STANCE ( THE 1 4 4 . CHANGES FROM INCHES ' A 4 5 9 0

460 C SCUARED TO FEET SQUARED) A 4 6 0 0C O H OK S = C OH ES K* D K S * 1 4 4 . A 4 6 1 0C 0 H E L S « C 0 M E S L * E L 5 * 1 4 4 . A 4 6 2 0

C C A L C U L A T I O N OF THE TOTAL R E S I S T I N G SHEARI NG FORCE A 4 6 3 0RES I S T *C OH D K S > C OH EL S *N OR MK *T AN ( F P I A N K r . 0 1 7 4 . 5 ) > N OP M L *TA N ( F R I A N L * . 0 1 A 4 64 0

465 1 7 4 5 ) A 4 6 5 0C CA L C U L A T I ON OF AZ I MUTH AND PLUNGE FOR L I N E OF I N T E R S E C T I O N OF ' A 4 6 6 0C FRACTURES A 4 6 7 0

X I N T - X 1 - X T " A 4 6 6 0Y I N T « YM- YT A 4 6 9 0

470 Z I N T - Z M - Z T A 4 7 0 0X Y I N T » SQ R T( ( X I N T ) * * 2 * ( Y I N T ) * * 2 ) A4 7 10D I P 1 N T - 5 7 . 3 * A T A N ( A B S ( Z I N T ) / A B S ( X Y I N T ) ) A4 7 2 0

C # I F # TEST TO DETERMINE AZ I MUTH OF L I N E OF I N T E R S E C T I O N * A 4 7 3 0T H E " T A - 5 7 . 3 * A T A N ( A B S ( X I N T ) / A 3 S ( Y I N T > ) A 4 7 4 0

475 I F ( X I N T ) 6 o C , 6 9 0 , 7 0 0 A 4 7 5 06 8 0 I F ( V I N T ) 7 1 C , 7 2 0 , 7 3 0 4 4 7 6 0690 I F ( Y I N T ) 7 4 0 , 7 5 0 , 7 7 0 A 4 7 7 07 0 0 I F ( Y I N T ) 7 8 0 , 7 9 0 , 6 0 0 A 4 7 8 07 1 0 A Z I N T " 1 9 0 . + T H E T A A 4 7 9 0

480 GO TO 8 10 . A 4 8 0 07 2 0 A Z I N T a 2 7 0 . A 4 8 1 0

GO TO 6 1 0 A 4 9 2 07 3 0 A Z I N T » 3 6 0 . - T H E T A • A 4 6 3 0

GO TO 8 1 0 A4 R4 0485 7 4 0 A Z I N T » 1 8 0 . . A 4 85 0

GO TO 8 1 0 A 4 8 6 07 5 0 P R I NT 7 6 0 A 4 8 7 07 6 0 FORMAT ( 29HSTRANGE CASE; XT- XM AND Y T - Y M ) A 4 6 9 0

GO TO 8 10 A 4 3 9 0490 7 7 0 A Z I N T ' O . • ‘ A 4 90 0

GO TO 8 1 0 A 4 9 1 07 8 0 A Z I N T * 1 8 0 . - T H E T A A 4 9 2 0

GO TO 6 1 0 A 4 9 3 07 9 0 A Z I N T - 9 0 . A 4 9 4 0

495 GO TO 8 1 0 A 4 9 5 08 0 0 A Z I N T - T H E T A A 4 9 6 0C ROTATION OF A X I S BACK TO O R I G I N A L P O S I T I O N A 4 9 7 06 1 0 I F ( ( 3 6 0 . - R C T A ) - A Z I N T ) 6 2 0 , 8 2 0 , 8 3 0 A 4 9 6 08 20 A Z I N T - A Z I N T - ( 3 6 0 . - R O T A ) A49QC

500 GO TO 8 4 0 * A 5 0 0 08 3 0 A Z I N T » R 0 T A * A 7 I N T A 5 0 1 08 4 0 I F ( ( 3 6 0 . - R 0 T 4 ) - A Z F 3 ( K ) ) 8 5 0 , 8 5 0 , 8 6 0 A 5 0 2 08 5 0 A Z F 3 K * A Z F 3 ( K ) - < 3 6 0 . - R Q T A ) A 5 0 3 0

GO TO 8 7 0 A 5 0 4 0505 8 6 0 A Z F 3 K « R Q T A + A Z F 3 ( K ) . A 5 0 5 0

8 7 0 I F ( < 3 8 0 . - R G 7 A ) - A Z F 4 ( L ) ) 8 8 0 , 8 8 0 , 6 9 0 1 A 5 0 6 08 8 0 A Z F 4 L » A Z F 4 ( L ) - ( 3 6 0 . - R O T A ) A 5 0 7 0

GO TO 9 0 0 A s 0 6 08 9 0 A Z F 4 L » R 0 T A * A Z R 4 ( L ) A 5 0 9 0

5 1 0 C C A L C U L A TI ON OF TOTAL D R I V I N G FORCE A 5 1 0 0900 D R I V E F - V T * S I N ( D I P I N T * . 0 1 7 4 5 ) A 5 1 1 0C C A L C U L A T I ON Or FACTOR OF SAFETY A 5 1 2 0

I F S - I F S + 1 A 5 1 3 0

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67FS( I F S ) * R E S I S T / D R I V ? F A 5 1 4 0

515 C DETERMINE FREQUENCY OF OCCURRENCE FOR VARIOUS FACTORS OF SAFETY A515CI F ( F S ( I F S ) . L E . 0 . 5 > GO TO 9 1 0 A 5 1 6 0I F ( F S ( I F S ) . L E . l . ) GO TO 9 20 A 5 1 7 0I F ( F S ( I F S ) . I E . 1 . 5 ) GO TO 9 3 0 A 51 80I F <FS( I F S ) . I E . 2 . ) GO TO 9 40 A 51 90

5 20 I F (FSC I F S ) . I E . 3 . ) GO TO 9 5 0 A 5 ? 0 CI F ( F S ( I F S ) . L E . 4 . ) GO TO 9 6 0 A 5 2 1 0I F ( F S ( I F S 1 . L E . 5 . ) GO TO 9 7 0 A 5 2 2 0I F ( F S ( I F S ) . L E . I O . ) GO TO 9 8 0 A 5 2 3 0I F ( F S d F S ) . L E . 2 0 . ) GO TO 9 9 0 A 52 40

5 2 5 I F R C F S ( 1 0 ) - I F R Q F S C 1 0 ) * l A 5? 50GO TO 1 0 0 0 A 5 2 60

9 1 0 I F R 0 F S ( 1 ) » I F R Q F S ( 1 ) + 1 A 5 2 7 0GO TO 1 0 0 0 A 5 2 8 0

9 2 0 I F R 0 F S ( 2 ) * I F P Q F S ( 2 ) + 1 A 5 2 9 05 30 GO TO 1 0 0 0 A 5 3 0 0

9 30 I FROFS ( 3 ) - I F R Q F S ( 3 ) - H A 5 3 1 0GO TO 1 0 0 0 A 5 3 2 0

9 4 0 I F R C F S ( 4 ) - I F P Q F S C 4 ) - f - l A 5 3 3 0GO TO 1 0 0 0 A 5 3 40

535 9 5 0 I F R C F S ( 5 ) « I F F 3 F S ( 5 ) * 1 A 5 3 5 0GO TO 1 0 0 0 A 5 3 6 0

9 6 0 I F R C F S ( 6 ) « I F R Q F S ( 6 ) + 1 A 5 3 7 0GO TO 1 0 0 0 A 5 3 8 0

9 7 0 I F R 0 F S ( 7 ) - I F P Q F S ( 7 ) t l A 5 3 9 05 4 0 GO TO 1 0 0 0 A 5 4 0 0

9 8 0 I F R 0 F S ( 8 ) » I F R Q F S ( 3 ) + 1 A 5 4 1 0GO TO 1 0 0 0 A542 0

9 9 0 I F R C F S ( 9 ) « I F R Q F S ( 9 ) + 1 A 5 4 3 01 0 0 0 CONTINUE A 54 40

545 C DETERMINE IF THE FACTOR OF SAFETY I S LESS THAN ONE A 5 4 5 0I F ( F S d F S ) . G E . l . ) GO TO 1 0 1 0 A546CI F A I L - I F A l L + 1 . A 5 4 7 0

1 0 1 0 CONTINUE A5 48 0P R I N T 1 02 0 * I F S . F S ( I F S ) » A Z F 3 K * A Z F 4 L , A Z I N T . 0 I P I N T t V O L * C O M E S K , C O H E S L A 5 4 9 C

5 5 0 1 , F R I A N K , F R I A N L A 5 5 0 01 0 2 0 FORMAT ( 1 H , 1 5 * 6 X , F 6 . 2 , 6 Y * F 5 . 1 , 6 X , F 5 . 1 , 6 X , F 5 . 1 * 6 X * F 4 . 1 , 2 X , £ 1 6 . 4 , 1 0 A551C

1 X * F 5 . 2 * 6 X , F 5 . 2 » 6 X » F 5 . 1 , 6 X » F 5 . 1 ) A552CI F ( I F S . I T . 1 6 0 ) GO TO 4 90 A553C

C EST I MA T E OF THE P R O B A B I L I T Y THAT F A I L U RE IS K I N E M A T I C A L L Y POSSI BLE A 5 5 4 0555 P O S F A L * I F S / k k k k A 5 5 50

C E STI MATE OF THE P R O B A B I L I T Y OF F A I LU R E FOR THOSE P AI RS OF A 55 60c FRACTURES IN WHICH F A I LU R E I S K I N E M A T I C A L L Y P OSSI BL E A 5 5 7 0p c f l - I F a u / i f s A 5 5 8 0c DETERMINE THE STANDARD D E V I A T I O N FOR THE P O S S I B I L I T Y OF F A I L U R E A 5 5 9 0

5 60 c AND THE P R O B A B I L I T Y OF F A I L U R E A 5 6 0 0• S I G 1 * S Q R T ( ( K C S F A L M l . - P O S F A L ) ) / * K K K ) A 56 1 0

S I G 2 - S Q R T ( ( P C F L * ( 1 . - ° C F L ) ) / I F S ) A 5 6 2 0c DETERMINE THE RANGE OF THE MAXIMUM AND MINIMUM TOTAL P R O B A B I L I T Y A 5 6 3 0c CF F A I L U R E C ON S I DE RI NG A 95 PERCENT CONFIDENCE LE V E L A 5 6 4 05 65 P l M A X = P 0 S F A l * l . c > 6 * S I G l A 5 6 5 0

P 1 M I N - P 3 S F A L - 1 . 9 6 * S I G 1 A 5 6 6 0P 2 M A X . P C F L + 1 . 9 6 * 5 IG2 A 5 6 7 0P 2 M I N » P C F L - 1 . 9 6 * S I G 2 A 5 6 6 0T 0 T P F 1 - A 6 S ( P 1 M A X * P 2 M A X ) A 5 6 9 0

570 T 0 T P F 2 - A R S ( P 1 M I N * R 2 M I N ) A 57C0

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68

575

5 80

585

590

595

6 0 0

C C A L C U L A TI ON OF MEAN FACTOR OF SAFETY AND STANDARD D E V I A T I O N A 5 7 1 0CALL MNSTO ( I F S , F S , % M E A N , S T O E V ) . A572CP R I NT 1 0 3 0 A573C-

1 0 3 0 FORMAT ( 1 H 1 , ♦ A C C U M U L A T I V E NUMBER OF FACTORS OF SAFETY FOR VARIOUS A 5 7 AC1RANGES+) A 5 7 5 0

PR I N T 1 0 4 0 , ( I F R O F S ( I ) , 1 * 1 , 1 0 ) A576C1 0 4 0 FORMAT ( I K , * F . S . ( 0 - . 5 M , 5 X , I 4 / l X , e F . S . ( . 5 - 1 . 0 M , 3 X , I 4 / l X , ^ F . S . ( l . A577G

1 0 - 1 . 5 ) + . 2 X , I 4 / l X , + F . S . ( 1 . 5 - 2 . 0 ) + , 2 X , I 4 / l X , 4 F . S . ( 2 - 3 ) + , 6 X , I 4 / l X , + F . A57B02 S . ( 3 - 4 ) ^ , 6 X , I 4 / 1 X , ♦ F . S . ( 4 - 5 ) * , 6 X , I 4 / 1 X , * F . S . ( 5 - 1 0 ) ^ , 5 X , I 4 / l X , * F . S . A579C3 ( 1 0 - 2 0 ) * , 4 X » I 4 / 1 X , + F . S . ( ♦ 2 0 ) * , 6 X , 1 4 ) A5 8 00

PRI NT 1 0 5 0 , XMEAN. STDEV A 5 8 1 01 0 5 0 FORMAT ( 1 H 0 , * “. E t N FACTOR OF S A F E T Y - * , 3 9 X , F 6 . 2 / / 1 X , * S T A N D A R 0 D E V I AT A582C

I I ON FOR THE FACTOR OF S A f E T Y - * , 1 3 X , F 1 0 . 2 ) A 5 8 3 0PR I NT 1 0 6 0 , I F A I L , I F S , K K K K , P C S F A L , P C F L A 5 8 4 0

1 0 6 0 FORMAT ( 1 K 0 , * N U M 3 E S OF I T E R A T I O N S WI TH A FACTOR OF SAFETY LESS THA A535C I N 0 N E » * » F 8 . 1 / / 1 X , ♦ N U M B E R OF I T E R A T I O N S WITH A K I N E M A T I C P O S S I B I L I T A586G 2 Y OF F A I L U R E - * , I 6 / / 1 X , * N U M B E R OF TOTAL I T 6 R A T I O N S - * » 3 2 X , F 8 . 1 / / 1 X , * A 5 8 7 03 E S T I MA T E OF THF P R O B A B I L I T Y THAT F A I LU R E I S K I N E M A T I C A L L Y * / 1 X , * POS A 5 8B 0 4 S I B L E « * , 4 6 X , F 1 0 . 4 / / 1 X , * E S T I M A T E OF THE P R O B A B I L I T Y OF F A I LU R E FOR A 5 8 9 05TH0SE P A I R S GF* / 1 X , * F R A C T U RE S IN WHICH FA I LU R E I S K I N E M A T I C A L L Y PO A 5 9 0 06 S S I B L E - * , 4 X , F 1 0 . 4 ) A 5 9 1 0

P R I N T 1 0 7 0 , S I G 1 , S I G 2 , T 0 T P F 1 . T 0 T P F 2 A 5 9 2 01 0 7 0 FORMAT ( 1 H 0 , * S T A N D A R D D E V I A T I O N FOR THE P R O B A B I L I T Y THAT F A I L U R E * / A 5 9 3 0

1 1 X , * I S K I N E M A T I C A L L Y P O S S I B L E - * , 3 1 X , F * 1 0 . 3 / / 1 X , *STANDARO D E V I A T I O N A 5 9 4 02F0R THE P R O F A B I L I T Y OF F A I L U R E f o r THOSE* / I X , * ? A IRS OF FRACTURES I A 5 9 5 0 3N WHICH F A I L U R E I S K I N E M A T I C A L L Y * / 1 X , * ? O S S I B L E - * , 4 8 X , F 1 0 . 3 / / I X , * UP A 5 96 0 4PER L I M I T FOR THF TOTAL P R O B A B I L I T Y OF FA I L U RE C O N S I D E R I N O * / 1 X, ♦ A A597C595 PERCENT CONFIDENCE L E V E L - * , 2 7 X , F 1 0 . 4 / / I X , *LOWER L I M I T FOR THE T A5 9 6C6 0 T A L P R O B A B I L I T Y OF F A I L U RE CONS I D E R I N G * / I X , * A 95 PERCENT CONFIDEN A599C7CE L E V E L - * , 2 7 X , F 1 0 . 4 ) A 6 0 0 0

END A 6 0 1 0

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691 SUBROUTINE S I M U L T ( X S U , Y S U , Z S U , X S D , Y S O , Z S D , K B , K D , K , L , X F U 1 , Y F U 1 , Z F U B 10

1 1 , X F D 1 , Y F D 1 , 2 F D 1 , X F U 2 , Y F U 2 , Z F U 2 , X F 0 2 , Y F 0 2 , Z F D 2 , K E , X I U , Y I U , Z I U , X T , Y B 202 T , X , Y , Z » K M 1 , L M 1 , K H P E T , K N * D E T ) B 3 0

C T H I S SUBROUTINE I S USED TO CALCULATE THE I N T E R SE C TI ON P OI NT FOR B 405 C THREE OLANES. THREE EQUATIONS ARE SOLVED SI MULTANEOUSLY TO e 50c DETERMINE THE I N T E R S E C T I O N P O I N T . THE GAUSS JORDAN E L I M I N A T I O N 9 60c METHOD WI TH BACK S U B S T I T U T I O N I S USED. B 70

D IM E N S I ON X S U ( 3 ) , Y S U ( 3 ) , Z S U < 3 ) , X S n ( 3 ) , Y S 0 ( 5 > # Z S D ( 3 ) . B 801 X F U 1 ( K M ) , Y F U 1 (KM1 ) > Z F U 1 ( K M 1 ) , X F C 1 ( K M 1 ) , Y F D 1 ( K N 1 ) , B 90

10 . 2 Z F 0 1 ( K M l ) i X F U 2 ( L M l ) , Y F U 2 ( L M 1 ) , Z F U 2 ( L M 1 ) > X F 0 2 ( L M 1 ) , Y F 0 2 ( L H 1 ) # B 1 003 Z F D 2 ( L M 1 ) , A ( 3 , 3 ) , C ( 3 ) B 110c B E GI NN I N G THE FORMULATION OF A 3 BY 3 MATRIX B 1 20c D I R E C T I O N NUMBERS FOR A SURFACE PLANE P 1 30

A ( i , i ) - x s i m e ) ■ e 1 4015 A ( 1 , 2 ) - Y S U ( K P ) B 150

A ( 1 , 3 ) = Z S U ( K B ) B 1 60C ( 1 ) = X S U ( K B ) * X S D ( K 9 ) * V S U ( K B ) * Y S D ( K B ) * Z S U ( K B ) * Z S D ( K B ) e 1 7 0c TEST TO SEE I F I N T E R SE C T I O N OF I N T E RS E C TI ON PLANE AND TWO SURFACE B 1 80c PLANES IS TO BE CALCULATED B 1 90

20 I F ( K D . E O . O ) GO TO 10 B 2 0 0c D I R E C T I O N NUMBERS FOR A K FRACTURE PLANE P 2 1 0A ( 2 , 1 ) « X F U 1 ( K ) B 2 20A ( 2 , 2 ) * Y F U 1 ( K) B 2 3 0A ( 2 , 3 ) * Z F U 1 ( K ) P 2 4 0

25 C ( 2 ) = X F U 1 C K ) * X F D 1 ( K ) * Y F U 1 ( K ) * Y F D 1 ( K ) * Z F U 1 ( K ) * Z F D 1 ( K ) B 2 5 0. c D I R E C T I O N NUMBERS FOR A L FRACTURE PLANE B 2 60

A ( 3 , l ) - X F U 2 ( l > B 2 7 0A ( 3 , 2 ) - Y F U 2 ( L ) B 2 6 0A ( 3 , 3 ) s Z F U 2 ( L ) B 2 90

30 C ( 3 ) « X F U 2 ( L ) * X F D 2 < L ) + Y F U 2 ( L ) * Y F D 2 ( L ) + Z F U 2 ( L ) * Z F D 2 ( L ) B 3 00GO TO 20 B 3 1 0c PART OF SUBROUTINE TO CALCULATE I N T E RS E C T I O N OF I N T E R S E C T I O N PLANE 8 3 2 0c AND TWO SURFACES PLANES B 3 30

c D I R E C T I O N NUMBERS FOR A SURFACE PLANE B 3 4035 10 A < 2 , 1 > « X S U ( K E > B 3 50

A ( 2 , 2 ) » Y S U ( K E ) P 3 6 0A ( 2 , 3 ) * Z S U ( K E ) B 3 7 0C ( 2 ) - X S U ( K E ) * X S D ( K F ) * Y S U ( K E ) * Y S D ( K E ) * Z S U ( K E ) * Z S D ( K E ) B 3 80c D I R E C T I O N NUMBERS FOR A I N T E R S E C T I O N PLANE B 3 9 0

40 A ( 3 , l ) * X I U B 4 0 0A ( 3 , 2 > * Y I U B 4 10A ( 3 , 3 > » Z I U B 4 2 0C ( 3 ) » X I U * X T * Y I U * Y T P 4 3 0c DETERMINE IF THERE I S A UNI QUE SOLUTI ON FOR THE THREE PLANES P 4 4 045 2 0 I F ( C ( l ) . N E . O . ) GO TO 30 B 4 5 0I F ( C m . N E . O . ) GO TO 30 B 4 6 0I F ( C m . N E . O . ) GO TO 30 B 4 7 0c DETERMINE I F DETERMINANT i s ZERO FOP HOMOGENEOUS SYSTEM OF P 4 80c EQUATIONS P 4 90

50 D « A ( l , l ) * A ( 2 , 2 ) * A ( 3 . 3 ) * A ( l , 2 ) * A ( 2 , 3 ) * A ( 3 , l ) - f A ( l , 3 ) * A ( 2 , l ) * A ( 3 , 2 ) - A B 5 001 ( 3 , 1 ) * 4 ( 2 , 2 ) * A ( 1 , 3 ) - A ( 3 , 2 ) « - A < 2 , 3 ) * A ( 1 , 1 ) - A ( 3 , 3 ) * A ( 2 , 1 ) * A ( 1 , 2 ) B 5 10

I F ( D . E O . O . ) GO TO 40 P 5 2 0K H DE T - 1 B 5 30GO TO 1 10 e 5 4 0

55 c DETERMINE IF DETERMINANT IS NOT EQUAL TO ZERO FOR THE B 5 50c NCNHOMQGENECUS SYSTEM OF EQUATIONS B 5 6 030 D " A ( 1 , 1 ) * A ( 2 , 2 ) * A ( 3 , 3 ) + A ( 1 , 2 ) * A ( 2 , 3 ) * A ( 3 , 1 ) + A ( 1 , 3 ) * A ( 2 , 1 ) * A ( 3 , 2 ) - A 8 5 7 0

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701 ( 3 # 1 > * A < 2 , 2 ) * A ( 1 # 3 ) - A ( 3 # 2 ) * A 1 2 , 3 ) * A ( 1 , 1 ) - A ( 3 » 3 ) * A ( 2 , 1 ) * A ( 1 , 2 > B 5 6 0

I F ( D . N 6 . 0 . ) GO TO 40 B 5 9060 KNHD 6 T » 1 B 6 0 0

GO TO 1 10 6 6 1 0C B E GI NN I NG OF E L I M A T I O N METHOD WHICH RESULTS IN S E T T I N G TO ZERO ALL P 6 20C THE ELEMENTS BELOW THE P R I N C I P A L DIAGONAL B 6 3 04 0 N - 3 P 6 4 0

65 DO PO J * 1 , N B 6 5 0B B « 1 . / A ( J , J > P 6 6 0

C SET EACH* P R I N C I P A L DIAGONAL ELEMENT EQUAL TO ONE B 6 7 0DO 50 K 5 - 1 , N P 6 8 0

50 A( J # K 5 ) - A ( J , K 5 ) * B B B 6 9 070 C ( J ) « C ( J ) * B B B 7 0 0

DO 70 L 5« 1 # N B 7 10I F ( L 5 . L E . J ) GO TO 70 B 7 2 0D D * A ( L 5 » J > B 7 3 0

C SET EQUAL TO ZERO EACH COLUMN OF ELEMENTS BELOW EACH P R I N C I P A L ( K ) E 7 4 075 C DIAGONAL ELEMENT 6 7 5 0

DO 60 K 5 * 1 » N B 7 6 06 0 A ( L 5 , K 5 ) s A ( L 5 # K 5 ) - A ( J , K 5 ) * D D B 7 70

C ( L 5 ) « C ( L 5 ) - C ( J ) * D D B 7 8 07 0 CONTINUE B 7 9 0

60 8 0 CONTINUE B BOOC B E GI NN I NG OF BACK S U B S T I T U T I O N METHOD FOR ALL ELEMENTS ABOVE THE e 8 1 0

DO 1 0 0 1 * 2 , K e 9 3 0. J * H - I + 1 B 8 4 0K 5 - I - 1 B 9 5 0

85 DO 9 0 L 5 * 1 » K 5 B 8 6 0M « N + 1 - L 5 B 8 7 0C ( J ) « C ( J ) - A ( J » M ) * C ( M ) B 8 80

9 0 CONTINUE B 8 9 01 0 0 CONTINUE B 9 00

90 X * C ( 1 ) B 9 1 0Y * C ( 2 ) B 9 2 0Z * C < 3 ) B 9 3 0

1 10 CONTINUE B 9 4 0RETURN 6 9 5 0

95 END B 9 6 0

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711 SUBROUTINE RANDCF ( SDCF, XMNCF, R NC F 1) C 10

C NORMALLY D I S T R I B U T E D RANDOM NUMBER GENERATOR FOR COHESION AND C 20c F R I C T I O N VALUES C 30

R A- G R AN D ( 1 ) C 405 RB-ORAND <1 ) C 50

V » ( - 2 . 0 * A L 0 G ( R A ) ) * * 0 . 5 * C 0 S ( 6 . 2 8 3 * R B ) C 6 0R NC F1 - V * SD C F> X M N CF C 70RETURN C 80END C 90-

1 SUBROUTINE RANDKL ( N T , N M I N ) 0 10D I M E N SI ON A C (1 CO) 0 20

C RANDOM NUMBER GENERATER FOR K AND L FRACTURES D 30> AC (1 J - R M I N - R A N F ( 0 . 0 ) 0 40

5 N M I N - 1 0 50DO 1 0 1 * 2 # NT D 60AC ( I J - R A N F ( O . O ) 0 70I F ( R K I N . L E . A C ( I ) ) GO TO 10 0 80R MIN * A C ( I ) 0 9 0

10 N M I N - I 0 1 0 010 CONTINUE 0 1 1 0

RETURN 0 1 2 0END 0 1 3 0 -

1 SUBROUTINE MNS TO ( I F S , F S , XMEAN, S T O E V ) E 10D I ME N SI ON F S ( I F S ) £ 20

C SUBROUTINE TO CALCULATE THE MEAN AND STANDARD D E V I A T I O N £ 30S U M - 0 . E 40

5 SUMSQ- O. E 50DO 1 0 J - l f I F S E 60S U M - S U M + F S U ) E 70S U M S O = S U M S O * ( F S ( J ) * * 2 ) E 8 0

1 0 CONTINUE E 901 0 X M E A N - S U M / I F S E ICO

S T O E V - S O R T C ( S U M S 0 - ( S U M * * 2 Z I F S ) ) / ( I F S - l ) ) E 1 1 0RETURN E 1 20END E 1 3 0 -

FUNCTION DRANO( I S S S ) F 10D R A N D - R A N F ( 0 . 0 ) F 20RETURN F 30END F 40-

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APPENDIX D

EXAMPLE PROBLEM INPUT DATA

72

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73I N P U T DATA

TOTAL NUMBER OF FRACTURES® 58 U N I T WEIGHT OF THE ROCK= 1 6 0 * 0 0 MEAN COHESION VALUE® 3 * 3 0 0STANDARD D E V I A T I O N FOR COHESION® 1 . 7 0 0MEAN F R I C T I O N VALUE STANDARD D E V I A T I O N

1 0 . 1 0FOR F R I C T I O N ® 4 . 0 0

EXTERIOR TOPOGRAPHIC SURFACESS T R I K E D I P L OC A T I ON AZ I MUTH L OC A TI O N D I P . L O C AT I O N D I S T A N C E

9 5 o 0 5 . 0 0 . 0 0 5 1 . 3 4 1 2 8 . 0 085 ©0 6 5 . 0 0 . 0 0 5 1 . 3 4 1 2 8 . 0 0

2 9 0 , 0 , 3 3 4 0 . 0 0 - . 5 0 2 5 . 0 0

S T R I K E D I PFRACTURE PLANES

L OCATI ON AZ I MUTH L OCAT I ON D I P L O C AT I O N D I S T A N C E1 5 , 0 4 5 . 0 3 5 7 . 0 0 3 8 . 5 7 1 5 0 . 0 01 8 , 0 4 8 . 0 8 . 0 0 3 9 , 5 7 1 4 6 . 0 02 0 , 0 4 0 . 0 3 5 3 . 0 0 3 8 . 5 7 1 4 1 . 0 02 5 , 0 3 9 . 0 . 3 5 0 . 0 0 3 B.o 0 0 1 3 9 . 0 0

2 , 0 6 0 . 0 3 5 7 . 0 0 3 8 . 5 7 1 4 5 . 0 08 , 0 5 4 . 0 3 5 3 . 0 0 3 8 . 5 7 1 3 5 . 0 0

3 5 4 . 0 2 0 . 0 1 0 . 0 0 3 7 . 5 7 1 2 9 . 0 02 1 , 0 4 5 . 0 3 5 0 . 0 0 3 8 . 5 7 1 4 2 , 0 01 4 , 0 5 0 . 0 3 5 7 . 0 0 3 8 . 5 7 1 5 0 . 0 01 7 . 0 4 2 . 0 3 5 6 . 0 0 4 0 . 5 7 1 4 8 . 0 01 9 . 0 3 7 . 0 3 5 6 . 0 0 4 0 . 5 7 1 5 5 . 0 02 9 . 0 5 6 . 0 8 . 0 0 4 0 . 5 7 1 5 6 b 0 01 1 , 0 5 0 . 0 7 . 0 0 3 8 . 5 7 1 4 2 . 0 0

8 , 0 4 6 . 0 3 5 2 . 0 0 4 0 . 5 7 1 5 3 , 0 05 . 0 4 2 . 0 3 5 3 . 0 0 3 8 . 5 7 1 4 2 . 0 0

1 7 . 0 4 9 . 0 3 5 6 . 0 0 4 0 . 5 7 1 6 2 . 0 08 . 0 4 1 . 0 3 4 9 . 0 0 4 0 . 5 7 - 1 4 6 . 0 0

1 6 . 0 4 8 . 0 1 2 . 0 0 4 0 . 5 7 1 5 9 . 0 01 5 . 0 5 0 . 0 4 . 0 0 4 0 . 5 7 1 4 3 . 0 01 0 . 0 4 7 . 0 - 3 5 7 . 0 0 3 7 . 5 7 1 3 5 . 0 02 2 . 0 4 3 . 0 3 5 0 . 0 0 3 8 . 5 7 1 4 5 . 0 02 0 . 0 3 8 . 0 3 5 3 . 0 0 3 7 . 5 7 1 2 6 . 0 01 7 . 0 3 9 . 0 3 5 2 . 0 0 4 0 . 5 7 1 4 1 . 0 016b 0 4 4 . 0 8 . 0 0 4 0 . 5 7 1 4 4 . 0 01 2 . 0 4 8 . 0 0 . 0 0 3 7 . 5 7 1 2 5 , 0 01 1 . 0 5 1 . 0 3 4 9 . 0 0 * 4 0 . 5 7 1 4 3 . 0 01 7 . 0 4 1 . 0 4 . 0 0 4 8 . 5 7 1 4 1 . 0 0

1 4 5 , 0 7 0 . 0 3 . 0 0 3 7 . 5 7 1 3 3 . 0 01 4 0 . 0 6 0 . 0 8 . 0 0 4 0 . 5 7 1 5 5 . 0 01 3 7 . 0 7 3 . 0 3 . 0 0 3 8 . 5 7 1 4 6 . 0 01 4 8 , 0 6 9 . 0 1 1 . 0 0 4 0 . 5 7 1 5 1 , 0 01 5 3 . 0 7 5 . 0 7 . 0 0 3 8 . 5 7 1 3 6 . 0 01 4 4 . 0 6 2 . 0 3 4 9 . 0 0 4 0 . 5 7 1 4 5 , 0 01 5 2 . 0 8 0 . 0 4 . 0 0 4 0 . 5 7 1 7 0 . 0 01 3 5 . 0 5 9 . 0 7 . 0 0 3 8 . 5 7 1 3 8 , 0 01 3 9 . 0 5 5 . 0 3 5 6 . 0 0 4 0 . 5 7 1 5 6 . 0 01 4 9 . 0 6 7 . 0 1 0 . 0 0 3 8 . 5 7 1 3 8 . 0 01 5 0 . 0 7 5 . 0 1 0 . 0 0 3 8 . 5 7 1 3 2 . 0 01 . 4 7 , 0 7 1 . 0 4 . 0 0 4 0 . 5 7 1 5 3 . 0 01 3 7 . 0 6 7 . 0 1 1 . 0 0 4 0 . 5 7 1 4 9 . 0 01 4 1 . 0 7 7 . 0 3 5 7 . 0 0 3 7 . 5 7 1 2 5 . 0 01 3 8 . 0 7 0 . 0 3 5 2 . 0 0 4 0 . 5 7 1 5 7 , 0 01 3 0 . 0 6 0 . 0 7 . 0 0 3 7 . 5 7 1 2 9 . 0 01 6 0 . 0 7 3 . 0 3 5 3 . 0 0 3 7 . 5 7 1 2 7 . 0 01 5 4 . 0 6 9 . 0 0 . 0 0 3 7 . 5 7 " " 1 2 7 . 0 01 3 8 . 0 7 5 . 0 8 . 0 0 4 0 . 5 7 1 4 8 . 0 01 4 9 . 0 6 2 . 0 1 0 , 0 0 3 7 . 5 7 1 3 8 , 0 01 4 8 , 0 8 0 . 0 1 1 . 0 0 4 0 , 5 7 1 5 8 . 0 01 5 2 . 0 5 9 . 0 8 . 0 0 4 0 . 5 7 1 6 4 . 0 01 6 0 . 0 5 5 . 0 1 2 . 0 0 4 0 , 5 7 1 6 3 . 0 01 4 3 . 0 6 7 . 0 4 . 0 0 4 0 . 5 7 1 6 9 . 0 01 4 6 . 0 6 1 , 0 4 . 0 0 3 9 . 5 7 1 4 6 , 0 01 5 1 . 0 5 8 . 0 8 . 0 0 4 0 . 5 7 1 6 6 , 0 01 4 9 . 0 6 7 . 0 4 , 0 0 3 9 , 5 7 1 4 8 . 0 01 4 5 . 0 7 2 . 0 3 . 0 0 3 7 . 5 7 ' 1 3 1 , 0 01 4 6 . 0 7 1 . 0 1 2 . 0 0 4 0 . 5 7 1 6 7 . 0 01 4 5 . 0 7 8 . 0 3 4 8 . 0 0 4 0 . 5 7 1 6 0 . 0 01 4 2 . 0 68 o 0 7 . 0 0 3 7 . 5 7 1 3 2 , 0 0

Page 81: STABILITY ANALYSIS OF WEDGE TYPE Ly - Open …arizona.openrepository.com/arizona/bitstream/10150/347880/1/AZU_TD... · STABILITY ANALYSIS OF WEDGE TYPE ROCK SLOPE ... major concern

. ■ . APPENDIX E

"RESULTS OF EACH ITERATION IN THE EXAMPLE PROBLEM

)

74

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RESULTS

TOTAL NUMBER OF W U GROUP FRACTURES* 27 TOTAL NUMBER OF (It GROUP FRACTURES* 31

NUMBER FACTOR STRIKE . STRIKE LINE OF LINE OFOF OF OF K OF L INTERSECT INTERSEC­

ITERATIONS SAFETY FRACTURE FRACTURE TIONAZIMUTH

TION • PLUNGE

1 1.27 ’ 8,0 149.0 163,2 30.02 I. 24 21.0 147,0 160,0 33,23 1.16 10,0 140,0 150,7 . 29.14 ,94 .20,0 153,0 161.1 * 27.85 1.06 15.0 152.0 158,0 31,06 1.50 17.0 139.0 159.3 26,47 1.02 5.0 146,0 154,9 24.38 .94 15,0 145.0 157.7 31,29 1.99 8,0 142,0 153,4 26,2

10 1,24 2.0 149,0 , 164,8 27.111 2.14 12.0 . • .160,0 173,9 19,012 * ,88 21.0 130,0 154,6 35.913 1.01 17.0 149.0 164.3 31.914 .95 25.0 154,0 165.4 27,315 ' 1*39 14.0 152,0 158,9 34.416 ii.ee . 16.0 144,0 161.1 * 28.9

! 1? 2.63 15.0 151.0 169,6 27.118 1,78 10.0 : 138.0 151.9 33.519 1.17 21,0 135.0 158,8 33.920 4.30 •354.0 . 142.0 145.9 9,721 1.18 8.0 .146,0 153,0 26.522 .94 11,0 150.0 159.8 32.623 1.56 • 18.0 145.0 . 159.5 34,624 4.25 2.0 144,0 162.2 30.425 1.34 8,0 148,0 161.6 31,526 1,05 12,0 153,0 161.6 29.327 1.65 15,0 139.0 161,6 28.828 1.51 8,0 145.0 157.9 34.629 . 1.14 21,0 145.0 157.8 34,430 1.89 16,0 135.0 158.8 33.931 1.28 , 21,0 153,0 162.5 31,932 1,55 17,0 139,0 164,5 31,733 1.32 8.0 151.0 168,0 25,234 2.34 11.0 145.0 158,4 32,635 1,10 25,0 146,0 163,3 28,336 1.26 18.0 138.0 150.6 39.237 1.45 17,0 140.0 162.2 33,338 1.73 . 15.0 153.0 161.5 28.939 ' 1,42 16,0 140.0 159.4 29.940 1,65 11.0 147.0 159,4 32,041 <» 7 8 29,0 135.0 169,5 43,342 - ' 1.39 11.0 152.0 168*1 24.9

TETRAHEDRONWEOGEVOLUME

COHESION COHESION FRICTION FRICTIONFOR K FOR L FOR K FOR L

FRACTURE FRACTURE FRACTURE FRACTURE

.5 797E+05 2.75 5,27 7.0 6.4

.72345+05 2.68 4.21 17.3 12.6,5715E+05 3.63 5.82 3.2 1.8,6321E+05 2.80 1.47 4.9 14.2,43935+05 3.69 ,75 9.2 9.4, 183 3 E+ 05., 3.82 2,78 10.4 3.2,112 IE + 06 2.92 .96 10.1 • 7.7,28475+05 ,93 2.60 15,8 3.4,60656+05 4.78 5,78 13.3 13.0,62656+05 2.16 3.63 8.1 7.6,62856+05 3.45 4.59 10.3 7,9,88156+05 2.32 3.78 13.7 12.2,38536+05 2.63 1.12 16.4 6,7,55296+05 1.55 1.16 13.3 11.5,49136+05 5,63 2.43 12.9 4.9,25686+01 .54 3.32 10.6 10.7,14206+05 4.67 5,19 17.2 10.6.14796+05 5,33 3.56 10.8 7.0,65386+05 4,19 4.65 14,1 11.9,68316+05 3.40 5.43 7.6 3.1,10676+06 1.20 4,84 8,3 13.4,92046+05 .71 2.46 12.2 15.1.59676+04 ' 2.05 2.95 13.8 14.6*11036+04 4,66 4,31 5.4 10.5,10116+06 • 4,07 5,17 9,1 10,6,51306+05 2.96 2,17 7,1 7,7,11346+05 2.56 2,94 15.6 12.6,55476+05 5,42 5.01 6,0 12.0,66456+05 4.16 3.85 11.2 9,6,36766+04 1,87 5,32 5.0 5.6.84616+05 3.06 4.01 15.5 12.0.78006+04 ,53 5,24 11,1 9.6,66586+05 2,86 2.68 10,1 10.1,79886+04 5,41 4.60 9.3 4,4,70516+05 .2.6.8 4.30 11.4 5,7.16146+05 5,61 1.54 6.3 11.7,33306+05 3.81 ■ 4.03 17.0 10.8.36246+05 4.51 1.37 18.6 12*1,10886+05 2,01 2.94 15,4 • 7*8o8690E+04 2,78 3,io 11,7 8.0.17926+05 2.80 2.38 9,4 6*0«1153E+09 1,79 1,32 12,5 10ol

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43 3 * 6 2 1 6 * 0 1 4 2 , 0 1 5 8 , 0 3 4 . 344 . 6 4 1 0 , 0 1 3 6 . 0 ’ 1 4 8 , 9 3 5 . 245 , 8 7 1 0 ,0 1 4 2 . 0 1 5 6 . 0 3 0 * 94 6 1 , 2 3 1 5 * 0 1 4 6 * 0 1 6 0 * 5 2 9 , 947 3 , 1 4 1 7 , 0 1 6 0 , 0 1 6 7 , 1 2 2 .048 ■ 1 , 2 9 1 4 . 0 1 4 1 , 0 1 5 1 . 6 3 8 * 749 1 ,0 2 8 , 0 1 4 9 , 0 1 6 5 . 3 2 7 . 95 0 1 . 5 2 2 5 . 0 1 5 4 , 0 1 6 5 . 4 2 7 . 351 3 . 2 9 1 6 . 0 1 4 9 . 0 1 6 2 . 1 2 8 , 352. . 8 7 1 7 , 0 1 4 5 , 0 1 5 8 , 4 3 5 , 653 2 . 2 0 " - - 8 , 0 1 5 4 . 0 1 6 2 , 3 - ' 2 0 . 7 '54 *88 2 2 . 0 1 5 2 . 0 1 6 9 . 5 2 6 , 6 '55 , 9 8 1 7 . 0 1 - 37 . 0 1 5 1 , 2 3 0 . 156 1 . 5 1 2 . 0 1 5 2 , 0 1 6 7 , 3 2 3 . 757 1 * 2 9 1 7 , 0 1 3 7 . 0 1 5 5 . 7 3 7 . 25 8 1 . 4 8 1 2, 0 1 4 2 , 0 1 5 6 . 9 3 2 . 559 1 . 4 2 1 0 . 0 1 5 1 , 0 1 6 6 , 5 2 3 * 260 1 . 2 7 1 8 . 0 1 4 7 . 0 1 6 0 . 4 3 4 , 161 2 , 9 3 1 7 . 0 1 5 2 . 0 1 6 7 . 1 2 3 . 462 1 . 6 6 1 4 . 0 1 5 1 . 0 1 6 9 . 2 . 2 6 . 663 1 . 9 7 1 6 , 0 1 5 1 , 0 1 6 9 . 2 2 6 , 664 4 . 1 1 1 7 . 0 1 4 9 . 0 1 6 1 . 4 2 6 . 865 1 . 4 6 8 . 0 1 4 5 . 0 1 5 1 . 3 2 7 . 466 5 , 6 2 1 7 . 0 1 4 1 , 0 1 4 9 . 5 3 2 , 76 7 1 , 0 5 2 9 , 0 1 4 0 . 0 1 7 1 . 4 4 2 . 168 1 . 9 3 1 7 . 0 1 5 4 . 0 1 6 3 . 8 2 3 . 96 9 • 1 , 3 6 1 7 , 0 1 4 5 . 0 1 5 8 . 4 . 3 5 . 6 .70 , 9 9 8 . 0 1 4 3 . 0 1 5 9 , 3 3 3 , 571 2 . 2 9 1 6 . 0 1 4 3 . 0 1 5 9 . 3 3 3 . 572 . 9 4 12 .0 1 3 6 . 0 1 5 2 , 6 3 5 . 173 1 . 8 2 1 5 , 0 1 3 7 , 0 1 5 1 , 5 3 9 , 474 1 , 0 9 2 1 . 0 1 4 0 , 0 1 6 1 , 5 3 2 , 475 1 . 1 8 1 5 . 0 ' 1 5 3 . 0 1 6 1 , 5 2 8 , 976 * 6 5 2 1 , 0 1 3 8 . 0 1 5 0 . 0 3 7 . 877 1 , 4 8 1 5 . 0 1 4 5 . 0 1 5 6 . 6 3 1 . 87 8 , 8 2 1 5 . 0 1 3 7 . 0 1 4 9 . 5 3 5 . 579 1*02 1 7 . 0 1 3 7 , 0 1 5 1 . 2 3 0 . 180 1 . 3 2 8 . 0 1 4 5 . 0 1 5 6 . 4 2 8 . 581 4 , 1 9 1 6 , 0 1 5 1 , 0 1 6 9 . 2 2 6 . 6

.82 2 . 2 3 3 5 4 . 0 1 4 6 , 0 1 4 9 , 5 8 . 683 1 , 2 6 1 1. 0 1 4 8 . 0 1 6 1 . 5 3 1 . 36,4 1 * 5 6 1 7 , 0 1 4 5 . 0 1 5 9 . 7 3 4 * 985 1 . 1 9 1 4 . 0 1 4 9 * 0 1 6 3 , 7 3 1 , 086 2 . 0 6 1 7 , 0 1 6 0 . 0 1 7 6 . 4 2 2, 087 2 , 3 2 1 5 . 0 1 4 9 . 0 1 6 2 . 2 2 8 . 488 2 . 4 5 1 7 . 0 1 4 2 . 0 1 5 5 . 4 3 0 , 069 2 ,2 2 1 9 . 0 1 4 3 . 0 1 5 5 * 6 2 7 * 390 . : 2 . 5 1 1 9 , 0 1 4 6 . 0 1 6 0 . 9 2 4 * 991 1 . 4 3 1 8 . 0 1 3 0 . 0 1 5 5 . 6 3 6 * 8 .92 2 . 6 2 1 8 . 0 1 5 1 . 0 1 7 0 . 0 2 7 . 593 1 * 5 1 5 . 0 1 4 8 . 0 1 5 7 , 2 2 2 . 79 4 1 * 6 5 1 9 , 0 1 4 6 * 0 1 5 6 * 1 2 7 * 19 5 1*68 1 5 * 0 1 5 2 * 0 1 6 9 , 7 2 6 * 996 2 . 3 9 8 . 0 1 4 5 . 0 1 5 4 , 3 ' 3 7 . 3

, 1 7 8 4 E + 04 2 o 4 4 800 9 1 2 e 3 1 3 . 4. 7 2 0 8 E + 0 5 . 9 6 2 . 9 4 1 7 . 5 3 . 8. 5 8 0 9 F + 0 5 . 3 . 6 4 1 . 9 1 7 . 1 4 . 6o 5 2 9 1 E * 0 5 3 . 5 3 3 . 4 1 9 . 5 8 . 2. 1 1 0 2 E + 05 3 •■94 5 . 4 9 6 . 4 1 4 . 8. 1 3 4 2 E + 0 5 1 . 5 0 4 . 9 2 1 0 . 4 9 . 5. 9 8 3 4 F + 0 5 3 . 4 6 3 . 4 2 6 . 7 5 . 1. 5 5 2 9 F + 0 5 2 . 9 2 3 . 9 0 1 6 . 2 1 2 . 6. 4 8 0 7 E + 0 4 4 . 7 3 5 . 6 5 1 2 . 2 1 0 . 8. 2 0 5 7 E + 0 5 2 . 7 4 . 6 5 1 0 . 5 9 . 8

' . 3 2 9 0 E + 0 5 4 . 6 9 . 9 8 ' 1 2 . 6 1 2 . 7. 7 366E t 0 5 * 2 . 0 6 1 . 9 7 3 . 7 1 6 . 0. 6 6 6 6 E + 0 5 1 . 1 1 2 . 8 1 1 6 . 6 1 0 . 9. 3 8 8 8 E + 0 5 3 . 9 1 1 . 3 9 1 3 . 0 4 . 0. 4 7 2 0 E + 0 5 6 . 2 0 2 . 6 9 9 , 3 1 8 . 8. 5 4 9 9 E + 0 5 2 . 3 9 4 . 9 7 1 8 . 2 1 4 . 9. 5 2 2 3 F <05 4 . 8 6 . 1 . 4 0 1 0 , 0 4 . 9. 6 4 3 5 E + 0 4 1 . 0 6 3 . 2 2 6 . 3 1 7 . 7. 2 54 7E + 04 3 . 4 4 2 . 5 9 1 1. 1 6 . 7. 3 6 9 5 E<05 2 . 9 3 6 . 1 3 6 . 6 . 7 . 9. 1 0 1 4 E + 0 4 1 . 3 5 1 . 4 4 1 2 . 5 1 3 . 20139 3 F <04 5 . 7 3 2 . 8 9 3 . 3 1 1 . 9» 7 3 5 3 E < 0 4 1 . 8 6 2 , 7 4 6 . 0 7 . 1. 4 8 5 5 F < 0 2 2 . 3 6 2 . 7 0 1, 1 1 1 , 9» 1 5 1 9 E < 0 5 5 , 1 2 . 6 9 1 2 . 4 1 1 , 8, 2 7 9 3 F < 0 5 4 . 7 8 1 . 7 3 , 1 3 , 8 7 . 3. 2 0 5 7 E + 0 5 4 . 5 8 2 . 8 5 9 . 8 . 8 . 1. 7 2 7 9 E + 0 5 . 2 . 0 5 3 . 9 5 4 . 8 1 5 . 2. 1 4 7 8 E + 0 4 . 3 , 1 1 2 . 7 2 1 0 . 1 7 . 3« 1 3 7 5 E < 0 5 1 . 7 7 2 . 0 8 1 1 . 6 5 , 6o l 8 6 7 E < 0 5 5 , 3 1 6 . 9 2 7 . 3 7 . 9• 8 3 5 1E + 05 3 , 0 1 . 5 , 2 9 1 1 . 3 9 , 2, 3 6 2 4 E < 0 5 4 . 1 8 1 . 2 0 . 6 . 0 9 , 1. 1 0 6 9 E <06 5 . 2 7 1 , 0 5 6 , 0 3 . 0. 2 7 1 4 E < 0 5 3 . 9 5 3 . 4 9 1 1 . 9 9 . 3. 4 04 PE <05. • 2 , 5 5 3 . 0 5 3 , 0 1 2 . 8. 6 8 6 6E < 0 5 1 . 9 6 2 . 1 1 1 4 , 2 1 6 . 5o 3-15 2£ < 05 3 . 9 5 2 . 5 7 8 . 5 5 , 1. 1 0 1 4 E + 0 4 2 . 8 9 5 . 0 7 6 . 6 1 4 , 101 308E' <06 1 , 0 3 3 , 1 1 4 , 4 7 . 6. 1 0 2 2 E < 0 6 3 , 3 2 4 . 2 9 1 0 , 2 1 4 . 1. 2 1 6 7 F < 0 5 5 . 7 8 3 . 1 9 8 , 7 9 , 1

. ® 5 254 E <05 2 , 7 1 2 . 9 5 1 5 . 3 7 . 4, 3 2 8 7 E < 0 5 2 , 9 8 4 , 0 6 1 4 , 7 8 , 5, 2 9 1 7 E < 0 5 6 , 7 2 4 . 2 9 1 5 . 6 1 1 , 2

. e 3 7 2 2 E < 0 4 1 5 . 2 9 1 , 4 0 1 4 , 8 9 , 2, 2 4 2 2 E < 0 5 6 . 0 9 5 , 2 0 1 0 , 6 7 . 8, 1 8 1 5 E < 0 5 6 , 5 7 2 . 3 7 1 4 . 3 1 3 . 7, 1 7 1 5 F < 0 5 3 . 6 4 5 . 1 4 7 . 4 1 1 , 1. 8 5 6 2 E < 0 4 3 , 4 9 ■ 4 . 9 8 1 6 . 7 1 3 . 80 81 3 4 E < 0 5 2 . 6 1 7 , 4 9 , 4 8 . 0o 4 8 6 6 E < 0 5 . ‘ . 4 014. 4 . 6 6 1 3 , 0 6 , 4o ! 4 1 5 E < 0 5 2 * 3 4 3 , 9 2 5 * 0 1 3 , 9 OXe8870E. .<04 3 . 6 9 5 * 4 7 1 4 , 0 1 8 . 6

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9 7 1 ©409 8 , l o l l9 9 1 , 0 7

1 00 1 , 5 01 01 1 , 7 8102 * 1 , 1 71 03 • 851 04 1 »02105 2 . 5 7106 2 © 36107 1 , 5 9108 10,67109 2 . 6 81 10 2 . 7 0111 1 . 8 51 12 1 . 9 1

, 11 3 1 , 7 81 14 1 * 5 6115 3 . 1 8116 3 , 4 1117 1 . 1 2118 1 . 3 91 19 1 . 3 21 20 6 . 6 71 21 1 . 9 41 22 2 . 0 2123 1 . 3 7124 , 5 4125 , 9 6126 1 , 0 11 27 1 . 3 81 28 2 , 4 01 29 . 651 30 1 . 3 1131 1 . 1 2132 • 1 . 1 01 33 2 . 6 8134 . 8 3135 4 , 9 31 36 1 . 8 4137 , 8 3138 ©871 39 1 . 2 91 40 1 © 03141 1 . 2 11 42 1 , 0 7143 1 9 , 2 2144 1 , 8 91 45 1 . 3 61 46 . . 9 61 4 7 • 1 , 7 314 8 1© 5 41 4 9 ©681 5 0 4 . 9 7

1 5 , 0 1 3 6 , 02 2 . 0 1 3 8 , 01 1 . 0 1 4 0 . 0

2 . 0 1 4 1 . 02 5 . 0 1 6 0 . 01 7 , 0 1 5 2 . 01 0 . 0 1 4 6 . 0• 8 . 0 1 4 0 . 01 7 , 0 1 3 7 . 01 7 , 0 1 4 4 , 01 8 . 0 1 4 6 , 01 1 . 0 1 4 1 , 0

8 , 0 1 5 2 . 01 5 . 0 1 3 9 , 01 7 . 0 1 4 1 , 01 9 . 0 1 4 5 . 0

5 . 0 1 4 8 . 01 5 . 0 1 4 1 , 01 7 . 0 ' 1 6 0 . 01 7 . 0 1 4 9 . 0

8 , 0 1 5 0 . 08 . 0 1 3 7 . 08 . 0 1 5 2 . 0

3 5 4 . 0 1 4 9 , 08 . 0 1 4 6 . 0

2 0 , 0 1 6 0 , 02 . 0 1 3 5 , 08 . 0 • 1 3 7 . 08 , 0 1 4 9 . 0

1 8 . 0 1 3 8 . 01 6 . 0 1 4 1 . 01 6 . 0 1 3 8 . 01 2 . 0 1 4 5 . 0

8 . 0 1 4 6 . 02 2 . 0 1 5 2 . 02 1 , 0 1 4 9 , 01 7 , 0 1 4 1 . 01 5 . 0 1 3 7 . 01 6 . 0 1 4 9 , 01 0 . 0 1 4 7 . 0

2 . 0 1 3 7 , 02 5 , 0 1 4 2 . 02 0 . 0 1 5 0 , 01 6 , 0 . 1 4 5 , 02 1 . 0 1 4 2 . 02 0 . 0 1 4 9 , 01 6 , 0 1 3 9 , 01 0 . 0 1 5 4 , 0

5 , 0 1 3 0 , 02 0 . 0 1 4 7 . 01 7 , 0 1 4 7 , 01 4 , 0 1 4 9 . 02 2 , 0 1 3 0 , 0

3 5 4 . 0 1 4 3 , 0

15 0 , 8 9 9 , 71 5 2 . 8 3 5 . 21 6 0 . 9 3 1 , 81 5 2 , 4 4 0 . 61 7 5 . 9 2 1 . 51 7 0 . 1 2 7 , 41 5 7 . 4 3 0 , 01 5 5 . 6 2 5 , 01 5 2 . 1 3 1 . 51 6 3 , 6 3 2 . 31 6 5 , 3 3 0 , 91 5 1 , 1 3 7 . 41 6 5 . 6 2 1 . 51 6 4 , 2 3 1 , 41 5 1 . 8 3 9 . 21 5 5 . 8 2 7 , 31 5 7 . 2 2 2 , 71 5 0 . 3 3 5 . 11 6 7 . 1 2 2 . 01 6 1 . 4 2 6 , 81 5 7 . 9 . 2 7 , 41 4 8 . 6 3 3 , 31 6 8 , 2 2 5 , 01 5 2 , 2 7 . 71 6 4 , 0 2 9 , 21 7 4 . 6 1 9 , 81 5 9 , 0 3 4 , 11 5 0 . 1 2 8 . 11 6 5 . 3 2 7 . 91 5 0 , 6 3 9 . 21 5 0 . 2 3 4 . 71 5 2 . 1 3 3 . 81 5 3 , 4 3 4 . 71 5 6 . 6 2 8 . 31 5 8 . 5 3 2 , 71 6 3 , 8 3 1 , 11 4 9 , 7 3 3 . 51 5 3 , 3 3 3 , 61 6 3 , 6 3 0 . 71 5 8 . 2 2 9 , 51 5 2 , 2 4 0 . 71 5 6 . 2 . 3 1 . 31 5 8 , 5 2 9 . 01 5 6 , 5 3 1 . 61 5 8 , 0 3 4 , 31 6 4 , 1 2 6 , 21 6 3 , 6 3 0 , 71 6 4 , 3 2 5 , 01 4 8 . 1 2 8 . 41 5 8 . 1 2 9 . 31 5 7 , 8 2 8 , 71 6 6 , 2 2 9 . 11 5 3 , 7 3 4 . 91 4 7 . 0 9©4

. 2 5 4 6 H 0 5 4 . 9 2 2 . 6 5 1 7 . 1 1 2 . 6

. 2 9 5 9 F * 0 5 3 , 1 0 2 , 8 4 1 2 . 3 1 2 . 6

. 7 5 3 3 F * 05 3 . 0 1 2 , 5 3 1 2 . 6 1 3 . 8o l 0 6 6 E 4 - 0 5 4 . 7 5 , 1 . 3 8 1 1 , 4 1 0 , 7. 9 8 1 3 6 * 0 5 2 . 8 1 5 . 6 4 1 3 . 0 1 1 . 8• 2 6 7 3 6 * 0 5 2 . 7 4 2 . 2 2 8 . 7 7 , 2. 1 0 8 0 6 * 0 6 2 . 5 1 3 . 8 9 8 , 8 . 6, 6 0 6 6 6 * 0 5 2 . 6 4 1 . 5 2 1 2 . 3 4 , 6. 7 7 3 5 6 * 0 4 • 4 . 3 2 6 . 9 5 1 1 . 6 6 . 6. 1 3 4 8 6 * 0 4 2 , 5 0 3 . 3 4 7 . 9 9 . 0. 6 4 5 1 6 * 0 4 2 . 0 9 3 . 1 9 1 0 , 4 1 1 . 5, 1 6 3 8 6 + 0 4 1 . 0 4 3 . 3 4 1 0 , 7 5 , 3, 4 1 8 8 6 + 0 5 6 , 2 5 4 . 9 2 1 4 , 4 1 2 . 1, 3 1 1 5 6 + 0 4 3 , 1 2 5 . 1 3 9 . 8 1 4 . 3, 8 5 8 6 6 + 0 4 5 . 4 9 4 . 2 6 1 . 5 1 1 . 9. 1 7 2 1 6 + 0 5 3 . 1 8 5 . 1 3 9 . 4 1 3 , 4, 8 1 3 4 6 + 0 5 4 , 3 4 . 3 , 2 2 6 o,5 1 5 . 9. 1 2 4 1 6 + 0 5 2 . 5 1 3 , 7 3 1 7 , 3 9 , 1, 1 1 0 2 6 + 0 5 4 , 4 2 4 . 2 6 1 4 , 1 6 . 9, 4 5 9 0 6 + 0 4 5 , 5 0 - 4 . 1 2 1 4 . 2 1 0 , 5, 5 0 7 3 6 + 05 • 2 . 9 9 1 . 8 9 1 0 , 7 3 , 6, 4 3 7 4 6 + 0 5 4 . 8 8 3 . 9 3 1 1 . 1 7 . 3. 6 8 0 4 6 + 0 5 3 . 1 9 2 . 4 9 1 0 . 0 8 , 8. 1 2 2 3 6 + 0 6 • 3 . 9 4 5 . 6 3 1 2 . 6 9 . 9. 5 3 6 7 6 + 0 5 7 , 3 4 4 . 3 0 1 2 , 5 1 0 . 1. 7 4 7 8 6 + 0 5 5 . 8 9 2 , 7 7 . 1 3 , 8 2 . 2. 4 6 6 8 6 + 0 5 6 , 2 1 4 . 2 8 5 , 9 7 , 6, 7 6 7 6 6 + 0 5 2 . 4 1 . 8 1 * 3 . 6 5 , 0. 9 8 3 4 6 + 0 5 1 . 7 1 3 . 1 8 1 0 . 9 5 . 6. 1 6 1 4 6 + 0 5 1 . 7 5 2 . 6 4 . 1 3 . 6 1 2 . 0, 7 7 2 9 6 + 0 3 1 . 4 0 1 , 7 4 1 . 6 4 , 6. 5 4 1 3 6 + 0 3 1 . 5 5 2 . 7 1 1 3 . 3 1 2 . 4. 5 2 4 3 6 + 0 4 1 . 0 5 . 4 7 5 . 7 9 , 0, 8 7 7 9 6 + 0 5 4 , 2 3 4 . 0 3 5 , 7 1 3 . 0, 9 2 9 9 6 + 0 5 3 . 7 0 2 . 0 8 , 1 2 , 4 1 5 , 6, 1 0 0 0 6 + 0 6 5 . 1 5 3 , 1 1 8 . 2 9 , 4, 1 0 0 4 6 + 0 5 5 . 5 5 5 . 8 8 1 8 . 2 1 1 . 5, 5 5 7 6 6 + 0 5 2 . 8 1 2 . 0 7 9 , 3 9 , 8, 3 7 0 5 6 + 0 3 6 . 1 0 1 . 2 2 7 . 9 1 4 . 2, 4 5 5 4 6 + 0 5 4 . 3 9 5 . 5 5 1 1 . 1 1 3 . 6, 4 7 7 7 6 + 0 5 1 . 8 1 2 . 4 3 1 3 , 2 7 , 2, 9 0 1 6 6 + 0 5 3 . 3 5 2 . 4 8 6 . 1 1 5 , 4. 7 2 9 2 6 + 0 5 2 . 9 1 4 . 0 6 1 1 . 5 1 3 , 8, 4 1 9 2 6 + 0 4 1 , 6 5 . 8 0 8 . 2 1 0 , 4, 8 7 0 8 6 + 0 5 6 . 1 3 3 . 7 2 1 1 , 4 8 . 9. 6 3 2 1 6 + 0 5 6 , 2 9 5 . 2 4 1 1 . 7 1 3 , 6, 1 1 7 2 6 + 0 2 5 , 2 9 5 , 5 5 1 3 , 7 9 . 7. 2 8 0 1 6 + 0 5 3 , 3 1 4 . 1 6 . 3 , 3 1 7 , 5. 6 1 2 5 6 + 0 5 3 . 9 7 3 . 4 9 1 4 . 6 8 . 6, 5 4 1 1 6 + 0 5 1 , 3 7 4 , 9 1 7 , 7 6 . 3. 1 4 9 5 6 + 0 4 2 . 7 8 , 6 7 6 , 2 5 , 9. 5 7 1 9 6 + 0 5 4 . 1 3 4 , 5 0 1 2 . 0 1 0 . 1. 8 4 0 6 6 + 0 5 1 , 1 4 2 . 7 5 1 3 . 9 7 . 0. 6 9 1 4 6 + 0 5 4 . 2 2 3 . 0 9 9 . 5 1 5 . 8

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. .. , APPENDIX F

SUMMARIZATION.OF THE RESULTS IN THE EXAMPLE PROBLEM

78

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ACCUMULATIVE NUMBER OF FACTORS OF SAFETY FOR VARI OUS RANGESF»S.(0-.5)F o S o(o 5-1o 0) FeSeCldO-lo5>F o S o(1« 5-2 o 0 ) F.S.(2-3) F.S.(3-4)FoSo(4-5)F.S.(5-10)F•S o(10-20> FoSo 020)

02 4573321

56 2 2 0

MEAN FACTOR OF S AF ET Y= • l o 9 0

STANDARD D E V I A T I O N FOR THE FACTOR OF S A F E T Y * 1 , 9 2

NUMBER OF I T E R A T I O N S WI TH A FACTOR OF SAFETY LESS THAN ONE* 2 4 . 0

NUMBER OF I T E R A T I O N S WI TH A K I N E M A T I C P O S S I B I L I T Y OF F A I L U R E * 1 50

NUMBER OF TOTAL I T E R A T I O N S * 1 5 6 . 0

ESTI MATE OF THE P R O B A B I L I T Y THAT F A I L U R E I S K I N E M A T I C A L L Y P O S S I B L E * o 9 6 1 5

E ST I MAT E OF THE P R O B A B I L I T Y OF F A I LU R E FOR THOSE P A I R S QF FRACTURES I N WHICH F A I L U R E I S K I N E M A T I C A L L Y P O S S I B L E * . . 1 6 0 0

STANDARD D E V I A T I O N FOR THE P R O B A B I L I T Y THAT F A I L U R E I S K I N E M A T I C A L L Y P O S S I B L E * . 0 1 5

STANDARD D E V I A T I O N FOR THE P R O B A B I L I T Y OF F A I L U R E FOR THOSEP AI RS OF FRACTURES I N WHICH F A I LU R E I S K I N E M A T I C A L L YP O S S I B L E * . 0 3 0

UPPER L I M I T FOR THE TOTAL P R O B A B I L I T Y OF F A I LU R E C ONSI DERI NG A 95 PERCENT CONFIDENCE L E V E L * ' . 2 1 6 9

LOWER L I M I T FOR THE TOTAL P R O B A B I L I T Y OF F A I LU R E C ON S I DE RI NG .A 9 5 PERCENT CONFIDENCE L EV E L * . 0 9 4 4

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Appendix g

-PROGRAM FLOW CHART

80

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Start ^

•Read number of fracture planes and the unit weight of rock

Read the mean and standard deviation for cohesion and friction

Read azimuth and dip of the. three exterior topographic surfaces, plus their location azimuths and dips ____________________

Read azimuths and dips of fractures, plus location azimuths and dips

Print all values read

1Rotate the polar coordinate axis so that theslope face wil have an azimuth of 90°

iChange the exterior surface location coordinates from polar to cartesian

VFormulate direction numbers for fractures and planer surface features

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82

Fracture azimuth / / D i v i d e X ^ Fracture azimuthis between 0° -90° — -<rf actures inw>— , is betweenor 270° -360° x̂ fcwo groups/f y S \ 90° -270°(K) ' (L)

820)

IChange fracture location coordinates from polar to cartesian ,

Print number of (K) fractures and (L) fractures ____ .

Counter to count the total number of iterations attempted

iCallRANDKL

IGenerate random numbers for K and L fractures, and then select a random K and L fracture

CallRANDCF

A subroutine which selects from a normally distributed random number generator a value for cohesion on the K fracture

CallRANDCF

A; subroutine which selects from a normally distributed random number generator a value for cohesion on the L fracture

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Call RANDCF

A subroutine which selects from a normally distributed random number generator a value for friction on the K fracture

CallRANDCF

A; subroutine which selects from a normally distributed random number generator a value for friction on the L fracture

CallSIMULT

ICalculate intersection point for top surface plane and the two fracture planes (K and L)

CallSIMULT

Calculate intersection point for the slope face and the two fracture planes (K and L)

Test to determine if intersection is dipping up or down

down

Calculate the direction parameters for the vertical intersection plane

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Call SIMULT

Call SIMULT

Call SIMULT

Call SIMULT

Calculate intersection point for intersection plane and the slope face and "bottom surface planes-

Determine if the fractures intersection daylights on the slope face

Calculate intersection point for intersection plane and the slope face and top surface planes

Determine if YTMI is greater or equal to YT

Calculate intersection point of K fracture with the top and slope face surface planes

Calculate intersection point of L fracture with the top and slope face surface planes

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9Calculation of volume and surface area for the.tetrahedron

85

Calculate the weight of the tetrahedron and the shear resisting force along the fractures

Print the following: number of iterations,factor of safety, strike of K and L fractures line of intersection azimuth and plunge, tetrahedron wedge volume, cohesion for K and L fractures, friction for K and L fractures •

Rotation of axis back to original position

Calculation of driving force and the factor of safety

Calculate azimuth and dip for line of intersection of fractures

Determine the frequency of occurrence for various factors of safety

For each iteration determine if the factor of safety is less than one and keep a count of those iterations with a factor of safety less than one_______________ .

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CallMNSTD

Print the accumulative number of factors of safety for various r a n g e s ______ j

Determine if 150 iterations have occurred

Calculation of mean factor of safety and standard deviation

Estimate of the probability that failure is kinematically possible

Determine the range of the maximum and minimum total probability of failure considering a 95$ confidence level

Estimate of the probability of failure for those pairs of fractures in which failure is kinematically possible______ ■

Determine the standard deviation for the possibility of failure and the probability of failure

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Print the following: mean factor ofsafety, standard deviation for the factor of safety, number of iterations with a factor of safety less than one, number of iterations with a kinematic possibility of failure, number of total iterations, estimate of the probability that failure is kinematically possible, estimate of the probability of failure for those pairs of fractures in which failure is kinematically possible, standard deviation for the probability that failure is kinematically possible, standard deviation for the probability of failure for those pairs of fractures in /which failure is kinematically possible, /upper limit for the total probability of /failure considering a 95^ confidence level, / lower limit for the total probability of / failure considering a 95$ confidence level /

STOP

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REFERENCES

Golden, J . T ., ±965, Fortran IV Programming andComputing,. Prentice Hall, Inc., Englewood Cliffs, New Jersey.

Goodman, R. E., and Taylor, R. L., 1967? Methods of Analysis for Rock Slopes and Abutments'', A Review of Recent Developments, Failure and Breakage of Rocks, Edited by C. Fairhurst,AIME, New York.

Heuze, F. E.., and Goodman, R. E. , 1971, Three-Dimensional Approach for the Design of Cuts in Jointed Rock, Proc. 13th Symposium on Rock Mechanics, Urbana, Illinois.

Hoek, E., and Bray, J. W., 1974, Rock Slope Engineering, institution of Mining and Metallurgy, London.

Jennings, J.E., 1970, A Mathematical Theory for the' Calculation of the Stability of Slopes in Open Cast Minesi- Planning Open Pit Mines,Edited by P. W. J. Van Rensburg, Johannesburg.

John, K. W., 1968, Graphical Stability Analysis of Slopes in Jointed Rock, Journal of Soil Mechanics and Foundations Division , ASCE, v. 94, no. SMB, p. 497-526.

John,. K. W., 1970, Three-Dimensional Stability Analyses of Slopes in Jointed Rock; Planning Open Pit Mines, Edited by P. W. J . Van Rensburg., Johannesburg.

Kreyszig, E ., 1967, Advanced Engineering Mathematics, John Wiley and Sons, Inc., New York.

Londe, P., Vigier, G. and Vormeringer, R., 1969,Stability of Rock Slopes, a Three-Dimensional Study, Journal of Soil Mechanics and Foundations Division, ASCE, v. 95, ho. SMI, p. 235-262.

88

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89

Londe, P., Vigier, G. and Vormeringer, R., 1970,Stability of Rock Slopes - Graphical Methods, Journal of Soil Mechanics and Foundations Division, ASCE, v. 96, no. SM4, p. 1411-1434.

McMahon, B. K., 1971> A Statistical Method for theDesign of Rock Slopes, Proc. 1st Australian- New Zealand Geomechanics Conference, Melbourne.

Peterson, T. S., i960, Elements of Calculus, Harper and Row, Publishers, New York, 2nd ed.

Spiegel, M. R ., 1968, Mathematical Handbook ofFormulas and Tables, McGraw-Hill, New York.

Subcommittee on Program Documentation of the Committee on Computer Applications, 1973> Engineering Computer Program Documentation Standards ,Journal of Soil Mechanics and Foundations Division, ASCE, v. 99, no. SM3, p. 249-266.

Wittke, W. W., 1965* Method to Analyse the Stability of Rock Slopes with an without Additional Loading , Felsmechanik und Ingenieurgeologie, Supp. II, v. 30, p. 52-79•

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