@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Mining Engineering, Keevil Institute of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Smith, Hubert Rodney"@en ; dcterms:issued "2010-01-29T17:37:14Z"@en, "1975"@en ; vivo:relatedDegree "Master of Applied Science - MASc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The research for this thesis was carried out to investigate the effects of anisotropy on stress and strain measurements made by the Leeman "doorstopper" technique. The stress field in rock is influenced by many different sources. These complicate the methods for obtaining a stress tensor which is a representive model of the in-situ stress condition. Research has been carried out to determine stress concentrations for the Leeman doorstopper technique in isotropic ground, but previously, no values were known to exist for anisotropic conditions. A three-dimensional finite-element computer model was used to investigate the effect of anisotropy on stress and strain concentration factors. Displacement data and strain concentration factors obtained from this analysis, although not quantitatively accurate, showed that the doorstopper technique can indicate erroneous stress levels in anisotropic ground unless the appropriate corrections can be made."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/19323?expand=metadata"@en ; skos:note "A STUDY OF STRESS AND STRAIN CONCENTRATION FACTORS IN A TRANSVERSELY ISOTROPIC MEDIUM RELEVANT TO THE LEEMAN DOORSTOPPER TECHNIQUE by HUBERT RODNEY SMITH B.A.Sc, University of B r i t i s h Columbia, 1971 A Thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements for the degree of Master of Applied Science i n the Department of Mineral Engineering We accept t h i s thesis as conforming to the require^'standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1975 In present ing th i s thes i s in p a r t i a l f u l f i lment of the requirements f o r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le fo r reference and study. I fur ther agree that permission for extensive copying o f t h i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h i s representat ives . It i s understood that copying or p u b l i c a t i o n o f th i s thes i s f o r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. H. RODNEY SMITH Department of M i n e r a l Engineering The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date September 1975 ABSTRACT The research f o r t h i s t h e s i s was c a r r i e d out t o i n v e s t i g a t e the e f f e c t s of anisotropy on s t r e s s and s t r a i n measurements made by the Leeman \"doorstopper\" technique. The s t r e s s f i e l d i n rock i s i n f l u e n c e d by many d i f f e r e n t sources. These complicate the methods f o r o b t a i n i n g a s t r e s s tensor which i s a r e p r e s e n t i v e model of the i n - s i t u s t r e s s c o n d i t i o n . Research has been c a r r i e d out to determine s t r e s s concentrations f o r the Leeman doorstopper technique i n i s o t r o p i c ground, but p r e v i o u s l y , no values were known to e x i s t f o r a n i s o t r o p i c c o n d i t i o n s . A three-dimensional f i n i t e - e l e m e n t computer model was used to i n v e s t i g a t e the e f f e c t of anisotropy on s t r e s s and s t r a i n c o n c e n t r a t i o n f a c t o r s . Displacement data and s t r a i n c o n c e n t r a t i o n f a c t o r s obtained from t h i s a n a l y s i s , although not q u a n t i t a t i v e l y accurate, showed t h a t the doorstopper technique can i n d i c a t e erroneous s t r e s s l e v e l s i n a n i s o t r o p i c ground unless the appropriate c o r r e c t i o n s can be made. i i i TABLE OF CONTENTS Page LIST OF TABLES V LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i i INTRODUCTION . . . . . . . . . . . 1 STRESS IN ROCK . 5 MEASUREMENT OF STRESS . . . . 7 THE LEEMAN DOORSTOPPER TECHNIQUE . . . . . . . . . . . . . . lb PROCEDURE IN THE FIELD . . . . . . . . . . . . . 11 STRESS CONCENTRATION VALUES 15 REVIEW OF CONCENTRATION FACTORS IN ISOTROPIC MATERIAL . . . 16 METHOD OF ANALYSIS 25 SUMMARY OF THE FINITE-ELEMENT ANALYSIS PROCEDURE 26 D i s c r e t i z a t i o n of the Continuum 26 Selection of the Displacement Model 28 Derivation of the Element S t i f f n e s s Matrix 29 Assembly of the Algebraic Equations for the Overall . . . Di s c r e t i z e d Continuum . . . 31 Setting of Boundary Conditions . 32 Solution f o r the Unknown Displacements 34 RESULTS . 34 General 3 4 F o l i a t i o n plane p a r a l l e l to borehole bottom 3 6 F o l i a t i o n plane at 3 0 ° to borehole bottom 37 F o l i a t i o n plane at 6 0 ° to borehole bottom 38 F o l i a t i o n plane perpendicular to borehole bottom . . . . . 38 i v TABLE OF CONTENTS Page DISCUSSION . . . . . . . . . . . . . . . 40 General 40 D i s c r e t i z a t i o n of the Continuum .'4-1 S e l e c t i o n of the Displacement Model . . . . . 42 D e r i v a t i o n of the Element S t i f f n e s s M a t r i x . . . . . . . . 43 Assembly of the A l g e b r a i c Equation f o r the O v e r a l l > D i s c r e t i z e d Continuum . . . . 43 S e t t i n g of Boundary Conditions . . . . 43 S o l u t i o n f o r the Unknown Displacements 43. Accuracy of I n - S i t u S t r e s s Measurements 44 CONCLUSIONS . . 45 RECOMMENDATIONS • . 46 BIBLIOGRAPHY ' . 49 APPENDICES . . . . 7 2 A. Mathematical Formulation f o r F i n i t e Element Technique . . . 72 .B. Computor Program f o r S t r a i n . A n a l y s i s of the Borehole Bottom i n a Transversely I s o t r o p i c Body . . . . . . . . . . . . . 81 C. Nodal Displacements and Loads 92 D. S t r a i n from C e n t r a l Node to F i r s t Nodal Ring . . .. . . 10;9, LIST OF TABLES Table 1 Some Methods of Measuring S t r e s s i n Rock v Page 8 Table 2 Summary of S t r e s s Concentration Fa c t o r s f o r I s o t r o p i c M a t e r i a l s 2 4 v i F i g u r e 1 F i g u r e 2 F i g u r e 3 F i g u r e 4 F i g u r e 5 F i g u r e 6 F i g u r e 7 F i g u r e 8 F i g u r e 9 F i g u r e 10 F i g u r e 11 F i g u r e 12 F i g u r e 13 F i g u r e 14 F i g u r e 15 F i g u r e 16 F i g u r e 17 LIST OF FIGURES S t r e s s Tensor A c t i n g on a Cube I l l u s t r a t i o n of E l a s t i c Constants f o r a Tran s v e r s e l y I s o t r o p i c Medium The Doorstopper and I n s t a l l i n g Tool The F i e l d Procedure f o r the Doorstopper Values of S t r e s s and S t r a i n Concentration F a c t o r s \"a\" and \"A\" from v a r i o u s Authors Values of S t r e s s and S t r a i n Concentration F a c t o r s \"b\" and \"B\" from v a r i o u s Authors Values of S t r e s s and S t r a i n C o n c e ntration F a c t o r s \"c\" and \"C\" from v a r i o u s Authors F i n i t e Element G r i d Used t o Model Borehole I t e r a t i v e Loop f o r D i r e c t S t i f f n e s s Method F o l i a t i o n Angles Considered i n the F i n i t e Element A n a l y s i s .. _. . .._ The Four Load Cases A p p l i e d t o the Boundary of the Model S t r a i n Concentration F a c t o r \"A^\" F o l i a t i o n Angles S t r a i n Concentration F a c t o r \"A 2\" F o l i a t i o n Angles S t r a i n Concentration F a c t o r \"A^\" F o l i a t i o n Angles S t r a i n Concentration F a c t o r \"A 4\" F o l i a t i o n Angles S t r a i n Concentration F a c t o r \"A,-\" F o l i a t i o n Angles S t r a i n Concentration F a c t o r \"Ag\" F o l i a t i o n Angles f o r V a r y i n g f o r V a r y i n g f o r V a r y i n g f o r V a r y i n g f o r V a r y i n g f o r V a r y i n g Page 5(4 5 5 5 6 5 7 5 8/ 5 9 60 61 .'62 63 64 65 6 6 67 6 8 a 68b 6'9 LIST OF FIGURES - continued F i g u r e 18 S t r a i n Concentration F a c t o r \"A^\" f o r an I s o t r o p i c Medium Fi g u r e 19 S t r a i n Concentration F a c t o r \"A2\" f o r an I s o t r o p i c Medium Fi g u r e 20 S t r a i n Concentration F a c t o r \"A3\" t o \"Ag\" f o r an I s o t r o p i c Medium V I X ACKNOWLEDGEMENTS The research d e s c r i b e d i n t h i s t h e s i s was c a r r i e d out i n the Department of M i n e r a l Engineering, U n i v e r s i t y of B r i t i s h Columbia,;ainder the d i r e c t i o n of Doctor I . Weir-Jones. The a s s i s t a n c e provided by Doctor Weir-Jones i n the i n i t i a t i o n of the p r o j e c t , and guidance through to the completion of the p r o j e c t are most g r a t e f u l l y acknowledged. Without the help and dedicated work of Dr. H. Kiyama, t h i s p r o j e c t c o u l d not have been i n i t i a t e d . His he l p i s most g r a t e f u l l y acknowledged. The expert help of many f r i e n d s d u r i n g p r e p a r a t i o n o f the t e x t i s a l s o g r a t e f u l l y acknowledged. INTRODUCTION In r ecent years numerous attempts have been made t o develop techniques to e f f e c t i v e l y monitor both absolute s t r e s s and s t r e s s changes i n rock. Yet, i n s p i t e of a l l the work accomplished to date, i t appears t h a t a v e r s a t i l e and conven-i e n t technique has y e t to be developed. Moreover, many i n v e s t i g a t o r s have i n d i c a t e d t h a t making s t r e s s measurements merely d e t r a c t s from a program of making deformation measure^ ments, which i n many cases can be of gre a t e r v a l u e . However, others f e e l t h a t development of i n - s i t u s t r e s s measuring devices i s e s s e n t i a l f o r the development of rock mechanics as a science (see/ f o r example, Roberts, 1968, p. 158), and such measurements are e s s e n t i a l aspects of c e r t a i n design procedures i n rock engineering p r a c t i c e . In the summer of 1971, the author spent three months p a r t i c i p a t i n g i n a f i e l d i n s t r u m e n t a t i o n program a s s o c i a t e d w i t h the design and c o n s t r u c t i o n of a l a r g e underground powerhouse. P a r t of t h i s program i n v o l v e d the c o l l e c t i o n of f i e l d data on i n - s i t u rock s t r e s s e s using the Leeman technique (Leeman, 1964). The powerhouse was s i t u a t e d i n metamorphic r o c k s , c o n s i s t i n g almost e n t i r e l y of mica s c h i s t s and q u a r t z f e l d s p a r g n e i s s e s . Upon r e t u r n i n g from the f i e l d , c a l c u l a t i o n s were c a r r i e d out to determine the s t r e s s f i e l d present a t each of the measurement s t a t i o n s . I t was apparent t h a t the rock i n 2. the f i e l d was not i s o t r o p i c as had been assumed i n the i n i t i a l a n a l y s i s o f the data. Metamorphic rocks such as s c h i s t and gneiss almost always have t r a n s v e r s e l y i s o t r o p i c p r o p e r t i e s . Berry and F a i r h u r s t (1966) have shown t h a t s i g n i f i c a n t e r r o r s can be intro d u c e d i n such c a l c u l a t i o n s by i g n o r i n g a n i s t r o p y i n an a n a l y s i s . A m a t e r i a l i s c a l l e d t r a n s v e r s e l y i s o t r o p i c when i t s p h y s i c a l p r o p e r t i e s are constant f o r any d i r e c t i o n i n a g i v e n plane, but change i n d i r e c t i o n s t h a t i n t e r s e c t t h a t plane. Thus, the rock considered i n t h i s study i s a t r a n s -v e r s e l y i s o t r o p i c e l a s t i c m a t e r i a l f o r which the e l a s t i c p r o p e r t i e s are i n v a r i a n t w i t h respect to r o t a t i o n s about a s i n g l e a x i s o n l y . This type of m a t e r i a l i s s i m i l a r to t h a t d e s c r i b e d by t e t r a g o n a l symmetry ( B i s p l i n g h o f f e t a l , 1965, p; 178) except t h a t there are f i v e independent e l a s t i c constants i n s t e a d of the s i x r e q u i r e d t o d e f i n e the c o n d i t i o n s f o r t e t r a g o n a l symmetry. The f i v e independent e l a s t i c constants r e q u i r e d t o d e f i n e a t r a n s v e r s e l y i s o t r o p i c m a t e r i a l are d e s c r i b e d l a t e r . F o l l o w i n g the i n i t i a l a n a l y s i s of the f i e l d data and i n view of the obvious a n i s o t r o p i c nature of the rock i t was decided to develop an a n a l y s i s technique t h a t would account f o r a t r a n s v e r s e l y i s o t r o p i c e l a s t i c m a t e r i a l . This i n v o l v e d p r i m a r i l y o b t a i n i n g m o d i f i e d s t r e s s - c o n c e n t r a t i o n f a c t o r s which c o u l d be a p p l i e d to the normal measurements usi n g the Leeman technique. S t r e s s c o n c e n t r a t i o n s are the r e s u l t of the r e d i s t r i b u t i o n of a p r e - e x i s t i n g s t r e s s f i e l d when an 3. a l t e r a t i o n i s made to the shape of the stressed body. For the case of small openings i n a continuous rock media the stress i s generally increased adjacent to a small c y l i n d r i c a l opening. This increase i n stress i s usually given i n terms of stress concentration factors which are a d i r e c t function of the o r i g i n a l stress before the opening was present. Many papers have been published which determine stress concentration factors to be used with the Leeman technique i n i s o t r o p i c rocks and the r e s u l t s of several of these analyses are presented and discussed i n t h i s t h e s i s . Stress concentration factors have also been determined f o r anisotropic ground, but not f o r the Leeman technique. These include the work by Berry (in 1968 and 1970), who considered s t r e s s determination i n transversely i s o t r o p i c mediums using the c y l i n d r i c a l overcoring method and using the instrumented c y l i n d r i c a l i n c l u s i o n method respectively. Transversely i s o t r o p i c conditions have also been discussed by F a i r h u r s t (1968). This thesis provides a b r i e f general introduction on stress d i s t r i b u t i o n i n rock, and the methods used to measure i t . The Leeman doorstopper technique i s described i n d e t a i l to help acquaint the reader with t h i s method. A review of stress concentration factors f o r the Leeman doorstopper technique i n i s o t r o p i c ground i s presented. F i n a l l y , the research which was c a r r i e d out i n the course of t h i s project i s presented. The research work involved analysing stress or s t r a i n c o n c e n t r a t i o n f a c t o r s f o r the bottom of a borehole i n a t r a n s v e r s e l y i s o t r o p i c medium. In order to c a r r y out t h i s work, a three-dimensional f i n i t e - e l e m e n t model and computer program were developed by Kiyama (1972) and these are presented i n Appendices A and B. This computer program was developed f o r the s o l u t i o n of the w r i t e r ' s research problem and was used e x t e n s i v e l y by the w r i t e r i n the p r e p a r a t i o n of t h i s t h e s i s . The program was used to o b t a i n displacement of nodal p o i n t s on the bottom of a borehole w i t h d i f f e r e n t f i e l d s t r e s s e s and f o l i a t i o n angles. This displacement data was then converted to s t r a i n c o n c e n t r a t i o n f a c t o r s f o r comparison w i t h s t r e s s and s t r a i n c o n c e n t r a t i o n f a c t o r s i n i s o t r o p i c ground. The s t r a i n c o n c e n t r a t i o n values are presented and the r e s u l t s are d i s c u s s e d . S t r e s s c o n c e n t r a t i o n values were not determined as i t was e s t a b l i s h e d t h a t the computer model was not s u f f i c i e n t l y r e f i n e d to warrant these computations. Several methods are then discussed f o r c o n s t r u c t i o n of an improved computer model. Conclusions and recommendations are then made regarding present and f u t u r e use of the doorstopper i n t r a n s v e r s e l y i s o t r o p i c ground. 5. STRESS IN ROCK The s t r e s s f i e l d w i t h i n a rock mass can be expressed as a second-order tensor t h a t i s de f i n e d by the nine components of s t r e s s a c t i n g a t a p o i n t w i t h i n the rock mass. The p h y s i c a l meaning and s i g n convention of these s t r e s s components are pre-sented i n F i g u r e 1 (Page 51). Since some of the shear s t r e s s e s are dependent upon other s t r e s s components the s t a t e of s t r e s s a t a p o i n t can be completely s p e c i f i e d by s i x independent com-ponents. The s t a t e of s t r e s s a t any p o s i t i o n i n a rock mass i s the cumulative e f f e c t of a l l f o r c e s t h a t are a c t i n g and have acted i n the g e o l o g i c past a t t h a t p o i n t . Forces a f f e c t i n g the s t r e s s tensor i n c l u d e : 1) G r a v i t y induced f o r c e s caused by the weight of the overburden -a) D i r e c t - S t r e s s caused by overburden weight i s almost always present and can g e n e r a l l y be c a l c u l a t e d i f the u n i t weight, y, of the overburden i s known and i f the ground suface i s r e l a t i v e l y uniform. In the general case the v e r t i c l e s t r e s s , c = yh z where h i s equal to the depth of cover. b) I n d i r e c t - L a t e r a l s t r e s s induced by the overburden. This v a r i e s w i t h the type of m a t e r i a l and w i t h the l a t e r a l con-s t r a i n t d u r i n g the g e o l o g i c h i s t o r y of the rock. Hence i t i s a f u n c t i o n of the s t r e s s h i s t o r y of the rock. The maximum l a t e r a l s t r e s s due to overburden t h a t can be present i s a s t r e s s 6. equivalent to the passive shearing resistance of the rock (see item 3 below). Under conditions of hydrostatic stress i t can be equal to the weight of the overburden. Under conditions of e l a s t i c f i r s t - t i m e loading i t i s d i r e c t l y r e l a t e d to the e l a s t i c moduli of rock and generally l i e s i n the range of 0.25 to 0.50 times the overburden s t r e s s . The minimum of zero i s reached with absence of l a t e r a l constraint. 2) Tectonic stresses due to c r u s t a l movement - These are often termed r e s i d u a l stresses and may be due to e i t h e r large-scale plate movement, or l o c a l warping, f o l d i n g and f a u l t i n g . They are, therefore, found mainly i n g e o l o g i c a l l y active areas, rather than i n f l a t - l y i n g sediments. 3) S i g n i f i c a n t modifications i n the overburden stress due to recent and unequilibrated geologic unloading — I t is. p o s s i b l e foe the ef-f active stresses to be less than a n t i c -ipated i f consolidation of the sediments i s incomplete or i f anomalously high pore f l u i d pressures are present i n the rock. Geologic unloading caused by r i v e r erosion or due to melting of g l a c i e r s can cause exceptionally high l a t e r a l stress conditions which can reach the passive shearing resistance of the s o i l or rock. These have been discussed by Hendron'(1963) and Brobker and Ireland (1965). 4) Stress modifications due to bridging - When two materials of varying e l a s t i c properties both support a load, the s t i f f e r material c a r r i e s more of the load and hence i s under greater s t r e s s . 5) Stress modifications due to chemical changes - Swelling, s o l u t i o n , d e s s i c a t i o n , cementation, l e a c h i n g and r e c r y s t a l l i z a -t i o n can cause volume and hence s t r e s s changes i n a rock mass. 6) S t r e s s due t o \"engineering\" a c t i v i t i e s - Surface and underground removal or placement of m a t e r i a l w i l l a f f e c t the s t r e s s f i e l d by the r e d i s t r i b u t i o n of weight, l o s s of support, • -and by the i n t r o d u c t i o n of s t r e s s c o n c e n t r a t i o n s . These and other c o n d i t i o n s complicate the methods f o r o b t a i n i n g a s t r e s s tensor which i s a r e p r e s e n t a t i v e model of the i n - s i t u s t r e s s c o n d i t i o n . MEASUREMENT OF STRESS The s t r e s s f i e l d can only be determined by measuring i t s e f f e c t on the rock mass. The most common method i s t o measure displacements o r s t r a i n s and convert these measurements i n t o e q u i v a l e n t s t r e s s e s or s t r e s s changes. Many d i f f e r e n t methods o f e s t a b l i s h i n g the s t r e s s f i e l d have been devised and many of these have been summarized and are given on Table 1. Approximate f i e l d methods i n c l u d e observations of the d i s c i n g of rock core and h y d r a u l i c f r a c t u r i n g . These, however, only show extreme s t r e s s r a t h e r than to determine q u a n t i t a t i v e v a l u e s . The methods which r e l y on sensing displacements or s t r a i n s can be d i v i d e d i n t o s t r e s s - r e l i e v i n g and stress-compensation methods. S t r e s s - r e l i e v i n g methods i n v o l v e instrumenting a s e c t i o n of s t r e s s e d rock, t a k i n g an i n i t i a l measurement, s t r e s s r e l i e v i n g METHOD Stress r e l i e f by Overcoring Stress Re l i e f on Excavation Walls S t a t i c Equ i l i b r ium Method Stress compensation Method Rock Fracture Stress Other Methods Sonic Method E l e c t r i c R e s i s t i v i t y Method Ca lo r imet r i c Method X-Ray Methods TYPE OF INSTRUMENT Borehole Deformation Gauge Rig id Inclusion Stressmeter S t ra in Gauges on Borehole Bottom S t ra in gauges on Borehole Walls S t ra in Gauges Photoe las t l c Discs Rig id Inclusion or Hydraul ic l^tress Gauges F l a t j a ck Curved Jack Core Discing Hydraul ic f r ac tu r ing TABLE 1: SOME METHODS OF MEASURING STRESS IN ROCK REFERENCE Panek (7966) Leeman (1964) Leeman (1964) Leeman (1967) Lieurance (1932) Hawkes (1968) Obert and Duval 1 (1967, P-427) Hoskins (1966) Jaeger and Cook (1964) Leeman (1964) Fairhurst (1968) Obert and Duval 1 (1967) Leeman (1964) Voight (1966) Voight (1966) OO 9. the instrumented rock by c u t t i n g i t f r e e from i t s surroundings, then measuring the displacement or s t r a i n t h a t took p l a c e . With a knowledge of the e l a s t i c p r o p e r t i e s of the m a t e r i a l s (assuming the m a t e r i a l was w i t h i n i t s e l a s t i c range), and any s p e c i a l c o n d i t i o n s imposed at the measuring s i t e , the s t r e s s c o n d i t i o n w i t h i n the rock can be c a l c u l a t e d . With the s t r e s s compensation methods, the rock i s instrumented, i n i t i a l readings taken, then the m a t e r i a l i s r e l i e v e d by s l o t t i n g ( i n one d i r e c t i o n o n l y ) . The s l o t i s then s t r e s s e d w i t h a f l a t jack u n t i l i n i t i a l readings are reproduced. The s t r e s s r e q u i r e d t o reproduce the i n i t i a l c o n d i t i o n s r e f l e c t s the s t r e s s e s t h a t were present i n the rock mass p r i o r t o s l o t t i n g . Examples of instruments using these methods are l i s t e d i n Table 1. Of these, the Leeman doorstopper method ( s t r a i n gauges on f l a t t e n e d end of borehole) has o f t e n been used i n p r a c t i c e . Reasons f o r i t s use i n c l u d e ; 1. I t can be p l a c e d w e l l o u t s i d e the zone of c o n s t r u c -t i o n i n f l u e n c e . 2. I t r e q u i r e s only a BX borehole. 3. I t i s quick to i n s t a l l . 4. I t does not r e q u i r e cables passing through the d r i l l rods, and 5. I t can be used i n ground w i t h a higher f r a c t u r e d e n s i t y than a d i a m e t r a l deformation gauge can be used i n . I t s major disadvantages i n c l u d e ; l b . IV A measurement success r a t i o (number of s u c c e s s f u l doorstoppers compared to number of attempted doorstoppers) of 70% i s considered good i n rock of e x c e l l a n t q u a l i t y . 2. Three holes are r e q u i r e d f o r a three dimensional s o l u t i o n . 3. The sampling area i s very s m a l l compared to any opening s i z e and a l s o i n d i v i d u a l gauge lengths are o f t e n s i m i l a r i n dimensions to the g r a i n s i z e of the rock mass. This means ' t h a t the q u a l i t y of the r e s u l t s i s very much r e l i a n t on rock mass q u a l i t y . 4. Since the doorstopper i s cemented a t the end of the borehole, the q u a l i t y of the r e s u l t s i s dependent upon the a b i l i t y of the glue to t r a n s m i t s t r a i n over the measurement p e r i o d w i t h no creep i n an o f t e n unfavorable environment. The use of the doorstopper appears to be a c c e p t i b l e and t h i s view i s shared by others w i t h experience i n the f i e l d , n o t a b l y M a r t i n e t t i (1970) and Hoskins (1973). THE LEEMAN DOORSTOPPER TECHNIQUE The s t r a i n c e l l used by Leeman (1964) i s c a l l e d a \"doorstopper\". This i s i n reference to i t s appearance o n l y and has no s c i e n t i f i c meaning. The doorstopper i s a r u b b e r - f i l l e d p l a s t i c p l ug which has a plug w i t h p i n contacts on one end and a three or f o u r gauge s t r a i n gauge r o s e t t e on the other. The 11. s t r a i n gauge r o s e t t e i s cemented to the f l a t t e n e d end of a BX borehole and then overcored. Gauge readings taken before and a f t e r o v e r c o r i n g are used to determine the s t r a i n which occurred i n the rock due to the s t r e s s r e l i e f . A diagram of the door-stopper c e l l and the i n s t a l l i n g t o o l used i s shown i n F i g u r e 3 • (Page 53). I f the rock q u a l i t y i s a p p r o p r i a t e , r e l i a b l e f i e l d readings can be made, and e l a s t i c p r o p e r t i e s and s t r e s s concen-t r a t i o n f a c t o r s can be obtained, then standard a n a l y t i c a l t e c h -niques and s t a t i s t i c a l methods r e a d i l y y i e l d v alues f o r the s t r e s s f i e l d and r e l a t e d confidence l i m i t s . To o b t a i n the complete s t r e s s f i e l d a t a f i e l d measurement s t a t i o n , the doorstopper must be used i n three d i f f e r e n t boreholes which have d i f f e r e n t o r i e n -t a t i o n s . Optimum l o c a t i o n s and o r i e n t a t i o n s of s t r a i n c e l l s have been d e f i n e d by Gray and Toews (1967). PROCEDURE IN THE FIELD The procedure used to o b t a i n doorstopper readings i s shown s c h e m a t i c a l l y on a s e r i e s of s e q u e n t i a l diagrams i n F i g u r e 4 (Page 54). D e t a i l s of t h i s procedure f o l l o w a s i m i l a r sequence and are present below: 1) A standard BX borehole i s d r i l l e d t o the depth i n the rock where the s t r e s s determination i s d e s i r e d . The d r i l l i n g must be done without l u b r i c a n t s . Besides other c o n s i d e r a t i o n s , the depth of the borehole w i l l depend upon the nature of the zones of d e s t r e s s i n g and b l a s t induced f r a c t u r e s caused by 12. excavation of the rock surface a t the d r i l l s t a t i o n and by the width of the zone of s t r e s s c o n c e n t r a t i o n , i f any, t h a t has been induced by the d r i l l s t a t i o n . In p r a c t i c e , the maximum depth of the borehole i s i n f l u e n c e d by the s e t t i n g time of the s t r a i n gauge cement used, but otherwise the depth i s e s s e n t i a l l y u n l i m -i t e d i n theory. 2) The end of the borehole i s then ground f l a t w i t h a square-faced b i t and can be p o l i s h e d w i t h a diamond impregnated b i t . The b i t s must be checked f r e q u e n t l y as the centres of the b i t s wear very q u i c k l y . This c e n t r a l r e g i o n i s very important as i t i s i n the c e n t r a l p o r t i o n of the borehole bottom t h a t the gauges are placed. The end of the borehole i s then cleaned, i n s p e c t e d , and d r i e d . I f f r a c t u r e s are found on the prepared rock s u r f a c e then the doorstopper c e l l i s not i n s t a l l e d and the hole must be ad-vanced p a s t the f r a c t u r e zone. The doorstopper can be s u p e r i o r to many s t r e s s - r e l i e f measuring instruments i n t h i s r e s p e c t as a much higher d e n s i t y of f r a c t u r e s can be t o l e r a t e d w h i l e s t i l l o b t a i n i n g a s u c c e s s f u l measurement (although the q u a l i t y o f any measurement must be questioned i n h i g h l y f r a c t u r e d ground). Cleaning and d r y i n g of the hole i s done w i t h e i t h e r a l i n t - f r e e c l o t h and acetone, or hot a i r . The t e s t r e q u i r e s a p e r f e c t l y dry hole bottom as cements of s t r a i n gauge q u a l i t y are not a v a i l a b l e which w i l l bond to wet rock. 3) The doorstopper i s i n s t a l l e d . The c e l l i s p l a c e d on the i n s t a l l i n g t o o l , the cement i s mixed, and then the assembly i s moved to the end of the borehole where i t i s placed and 13. pressure i s a p p l i e d . Several precautions must be e x e r c i s e d a t t h i s p o i n t i n the procedure. F i r s t , the cement must be t e s t e d t o ensure t h a t there i s s u f f i c i e n t time a v a i l a b l e t o reach the bottom of the h o l e , o r i e n t the gauges, and apply the pressure t o the c e l l before the cement has begun t o s e t . A l s o , the cement must be of s u f f i c i e n t q u a l i t y t o be e f f e c t i v e i n the g e n e r a l l y unfavourable environment a t the end of a borehole. S e l e c t i n g the s t r a i n gauge cement i s o f t e n a t r a d e o f f between s e t t i n g time and q u a l i t y . Secondly, the s t r a i n gauge r o s e t t e must be s e t as c l o s e to the centre of the hole as p o s s i b l e . In the centre of a hole bottom i n an i s o t r o p i c medium shear s t r e s s e s a c t i n g i n the coordinate system d e f i n e d by the borehole l e n g t h , the gauge l e n g t h , and the t h i r d a x i s have no e f f e c t on the s t r a i n measure-ments. However, i f the gauge i s o f f c e n t r e , t h i s i s no longer t r u e (See Gray and Toews, 1967). T h i r d l y , the o r i e n t a t i o n of the gauges must be a c c u r a t e l y known. This can be done by u s i n g t o r s i o n a l l y s t i f f i n s t a l l i n g rods o r , i n holes w i t h s u f f i c i e n t h o r i z o n t a l component, by using a mercury s w i t c h . L a s t l y , an app r o p r i a t e pressure must be put on the system which corresponds to t h a t recommended by the s t r a i n gauge cement manufacturer.. This i s done q u i t e e a s i l y i n the i n s t a l l i n g t o o l by a l l o w i n g a s p r i n g to deform by an amount which w i l l g ive the r e q u i r e d f o r c e . This f o r c e must be a p p l i e d u n t i l the cement i s completely s e t . 4) I n i t i a l readings are taken of the s t r a i n gauge. A l l equipment used, i n c l u d i n g dummy gauges, i n s t a l l i n g t o o l connec-t i o n s , s w i t c h i n g boxes, and read-out boxes should be checked 14. a g a i n s t a standard gauge or standard deformation both before and a f t e r t a k i n g a reading. In a t y p i c a l borehole, moisture and dust o f t e n i n f i l t r a t e equipment t h a t i s supposedly h e r m e t i c a l l y s e a l e d . This c a l i b r a t i o n check should be simple but thorough. 5) I n s t a l l i n g rods and t o o l are removed from the b o r e h o l e . 6) The doorstopper i s overcored using a standard BX b i t and the rock core w i t h the doorstopper attached i s removed. Enough i n t a c t core must remain cemented to the doorstopper so t h a t the sample i s f r e e of end e f f e c t s (a l e n g t h of core of about 4 times the core diameter i s r e q u i r e d ) . The core must have no f r a c t u r e s through the gauged area. A bead of s t r a i n gauge cement should e x i s t around the perimeter of the doorstopper. 7) The s t r a i n r e l i e f due to the s t r e s s r e l i e f i s measured. Due c o n s i d e r a t i o n must be given to the c a l i b r a t i o n check o u t l i n e d i n step 5. In order to achieve reasonable r e s u l t s , the same equipment must be used to measure t h i s f i n a l s t r a i n r e a d i n g as was used f o r the i n i t i a l reading. 8) The procedure i s repeated i n two more d r i l l h o l e s l o -cated i n the same general r e g i o n of the rock mass. Using other methods to determine the e l a s t i c moduli o f the rock, and by a p p l y i n g the appropriate theory,: as demonstrated i n Appendices A and B, the s t r e s s changes i n the rock can be determined and the o r i g i n a l f r e e f i e l d s t r e s s i n the rock c a l c u -l a t e d . S t r a i n readings may vary from doorstopper t o doorstopper due to s t r e s s f i e l d changes at d i s c o n t i n u i t i e s and other f a c t o r s . However, using s t a t i s t i c a l procedures, these readings can o f t e n 15. be made acceptable. STRESS CONCENTRATION VALUES Str e s s c o n c e n t r a t i o n s w i l l occur around openings i n an e l a s t i c s t r e s s e d m a t e r i a l . These concentrations are due t o a r e d i r e c t i n g of s t r e s s t h a t would normally be c a r r i e d by m a t e r i a l w i t h i n t h a t opening. S t r e s s c o n c e n t r a t i o n s t y p i c a l l y r e s u l t i n s t r e s s e s 1.5 to 3 times the p r e - e x i s i t i n g s t r e s s l e v e l , but i t i s p o s s i b l e t o o b t a i n values over 4 around rock openings. T h i s means t h a t s t r e s s e s over f o u r times the i n i t i a l s t r e s s can e x i s t c l o s e t o an opening i n a rock mass. S t r e s s c o n c e n t r a t i o n values f o r three-dimensional open-ings i n an i s o t r o p i c homogeneous e l a s t i c media are reasonably w e l l e s t a b l i s h e d , van Heerden (1968) s t u d i e d the s t r e s s concen-t r a t i o n produced a t the end of a borehole and Crouch (1969) v e r i f i e d these r e s u l t s w i t h a complete f i n i t e - e l e m e n t a n a l y s i s . These r e s u l t s w i l l be d i scussed f u r t h e r i n the f o l l o w i n g s e c t i o n . Although these values are a v a i l a b l e f o r i s o t r o p i c homogeneous m a t e r i a l , i t appears t h a t no authors have explored s o l u t i o n s f o r doorstopper measurements i n a t r a n s v e r s e l y i s o t r o p i c medium such as i s u s u a l l y found i n metamorphic and t h i n l y bedded sedimentary r e g i o n s . I t was f o r t h i s reason t h a t t h i s study was undertaken to attempt to determine the e f f e c t s of t r a n s v e r s e l y i s o t r o p i c mediums on doorstopper measurements. REVIEW OF CONCENTRATION FACTORS IN ISOTROPIC MATERIAL Many authors have considered s t r e s s c o n c e n t r a t i o n s a t the end of a borehole. Each has made a c o n t r i b u t i o n towards a b e t t e r understanding of these values by approaching the problem from a d i f f e r e n t p o i n t of view. In an i s o t r o p i c medium, s t r e s s c o n c e n t r a t i o n values are expressed as a, b, and c, where the s t r e s s on the end of the borehole p a r a l l e l t o the gauge l e n g t h i s : a\" = ao* + ho + ca x x y z where a i s s t r e s s i n d i r e c t i o n of gauge l e n g t h away from the x i n f l u e n c e of the borehole a i s the s t r e s s p e r p e n d i c u l a r t o the gauge l e n g t h on the ^ plane of the borehole bottom away from the i n f l u -ence of the borehole a i s the s t r e s s p a r a l l e l to the borehole away from the i n f l u e n c e of the borehole Shear s t r e s s e s a c t i n g on the borehole bottom have no e f f e c t on the gauge length i f gauges are i n s t a l l e d a t the centre of the borehole bottom. This was demonstrated by Gray and Toews (1967). The i n f l u e n c e of the borehole can a l s o be expressed as s t r a i n c o n c e n t r a t i o n f a c t o r s A, B, and C, where the s t r a i n measured p a r a l l e l to the gauge l e n g t h , e v i s given by: 17. e' = 1 (Aa + Ba + Ca ) x g- x y z In an i s o t r o p i c medium, plane s t r e s s e l a s t i c theory shows t h a t : A = a - bv B = b - av C = c (1 - v) where v = Poissons r a t i o of m a t e r i a l , and E = Youngs modulus. G a l l e and Wilhout These i n v e s t i g a t o r s were i n t e r e s t e d i n e v a l u a t i n g the s t r e s s a t the bottom of the borehole i n s o f a r as i t has a p p l i c a -t i o n i n the o i l i n d u s t r y . However, t h e i r data has a p p l i c a t i o n to, a l l types- o f - boreholes . A three-dimensional p h o t o e l a s t i c study was c a r r i e d out usin g a model made of p h t h a l i c anhydride-cured epoxy r e s i n . T h e i r model f o r a x i a l l y symmetrical l o a d i n g ( p a r r a l l e l t o borehole) was a c y l i n d r i c a l specimen w i t h a h e i g h t of 11 inches and a diameter of 11% inches. A b o r i n g was made i n the c e n t e r of the c y l i n d e r , 2 inches i n diameter and 5h inches deep. The b i a x i a l l o a d i n g ( a x and a ) was c a r r i e d out on a 6 i n c h cube. The borehole was d r i l l e d i n the middle of a face, a t a diameter of 1.250 inches and a depth of 3 i n c h e s . The curves provided by G a l l e and Wilhout are presented w i t h the p r o v i s i o n t h a t they are t r u e only when the hole i s d r i l l e d i n the d i r e c t i o n of one of the p r i n c i p l e s t r e s s e s . The values of s t r e s s c o n c e n t r a t i o n a t the end of the boreholes from t h e i r curves a r e : a = 1.56 b « 0 c - - 1.04 Leeman (19 64) To determine the s t r e s s c o n c e n t r a t i o n f a c t o r s , samples were loaded i n the l a b o r a t o r y a f t e r being instrumented w i t h s t r a i n gauges. Leeman used cubes of g r a n i t e , s t e e l , and a r a l d i t e . The s i z e of these cubes was not given. A hole was d r i l l e d h a l f way i n t o each cube from the c e n t r e o f one of the f a c e s . Each cube was then loaded i n compression i n a t e s t i n g machine, the compressive l o a d a c t i n g i n a v e r t i c a l d i r e c t i o n , the a x i s of the borehole being h o r i z o n t a l . Loading was c a r r i e d QU£. p a r a l l e l , to _and perpendicular, to the gauge l e n g t h on the h o l e bottom. Although h i s l a b s t u d i e s were not completed a t t h i s time, Leeman f e l t t h a t the e f f e c t of s t r e s s p a r a l l e l t o the borehole c o u l d be neglected as long as i t was s m a l l com-pared w i t h the s t r e s s p a r a l l e l t o the gauge l e n g t h . Therefore, no value of \"c\" was g i v e n . Leeman's s t r e s s c o n c e n t r a t i o n v a l u e s were: a = 1.53 b = 0 c not given P a l l i s t e r (1967) P a l l i s t e r c a r r i e d out h i s l a b o r a t o r y t e s t s t o determine the f a c t o r s a and b on an 18 i n c h long, 6 i n c h diameter m i l d 19. s t e e l c y l i n d e r . The Youngs modulous was 30 x 10^ p s i and Poissons r a t i o was 0.3. A 2 3/8-inch diameter hole was machined 6 inches i n t o one end of the cyl i n d e r . A doorstopper was cemented to the bottom of t h i s hole. The c y l i n d e r was then loaded by placing across i t s diameters four i d e n t i c a l copper quadrantal jacks, 9h inches long, and f i t t i n g the whole assembly into a mild s t e e l annulus 12 inches x 12 inches x 9 inches i n dimension. As the r a t i o of the hole diameter to cylin d e r diameter was rather large, the tests were also run i n a one-inch diameter hole. To carry out tests to determine the value of c, an aluminum cylin d e r 15 inches long by 5 31/32 inches i n diameter was used. A 1%-inch diameter by 3-inch deep hole was machined into one end of t h i s c y l i n d e r and a s t r a i n gauge rosette cemented on the end. Care was taken to prepare the sample well f o r i d e a l loading conditions. A 12-inch long cylinder, machined to the same dimension as the instrumented section with a 1%-inch hole d r i l l e d f u l l length was placed on top of the instrumented cyl i n d e r . I t was considered that t h e i r combined length would ensure a uniform u n i a x i a l stress f i e l d f o r the gauges. The Youngs modulous f o r the c y l i n d e r was 6 10.54 x 10 p s i and Poissons r a t i o n was 0.343. The stress concentration values found were: a = 1.1 b = 0 c = -0.75 20 . Hiramatsu and Oka (1968) In t h i s paper, the method used f o r o b t a i n i n g the s t r e s s f i e l d on the f l a t t e n e d end of a borehole i s not presented. However, the r e s u l t o f t h e i r r e s e a r c h , f o u r data p o i n t s , are presented. The data i s presented as s t r a i n c o n c e n t r a t i o n f a c t o r s , but by simple e l a s t i c theory and assuming plane s t r e s s , the s t r e s s concentrations are: Poissons r a t i o , v = 0. .24 0. .29 0. .37 0. .44 a 1. .36 1. .32 1, .39 1. .42 b -0. .304 -0. .289 -0. .204 -0. .060 c -0. .69 - -0. .82 -0. .98 -1. .10 Bonnechere and F a i r h u r s t (196 8) These researchers d i d a l a b determination of the s t r e s s c o n c e n t r a t i o n values of the end of a borehole. The sample used was a 6-inch cube of p l e x i g l a s s w i t h s i d e s m i l l e d f l a t and p a r a l l e l . A 3/4-inch hole was d r i l l e d t o a depth of 3 inches normal to and a t the centre of a f a c e . The 3-element s t r a i n gauge r o s e t t e was cemented to the c e n t r a l p o r t i o n of the h o l e bottom. The sample was t e s t e d both between s t e e l p l a t e n s and between s i m i l a r cubes of p l e x i g l a s s . When t e s t e d between cubes of p l e x i g l a s s s t r a i n s 10 percent l a r g e r were obtained. The same cube was used to get \"c\" as \"a\" and \"b\", except t h a t i t was turned so t h a t the hole was v e r t i c a l i n s t e a d of h o r i z o n -t a l . A range of values f o r \"c\" was found by comparing ranges found i n s p h e r o i d a l c a v i t i e s , e l l i p t i c a l h o l e s , e t c . Values 2.1. found f o r the s t r e s s c o n c e n t r a t i o n f a c t o r s were: a = 1.25 b = 0 c = -0.75 (0.5 + v) Van Heerden (1968) van Heerden d i d e x t e n s i v e l a b o r a t o r y i n v e s t i g a t i o n s on both f l a t and s p h e r i c a l l y shaped boreholes. Two d i f f e r e n t experimental procedures were used to determine c o n c e n t r a t i o n f a c t o r s a and b. These were a three-dimensional p h o t o e l a s t i c study and t e s t i n g of b l o c k s and c y l i n d e r s of d i f f e r e n t materials i n u n i a x i a l compression. In the p h o t o e l a s t i c study, a model 12 inches h i g h , 6 inches wide and 5 inches deep was used. A 3/4-irtch diameter hole was d r i l l e d 2 inches i n t o one of the 12-inch by 6-inch f a c e s . Four u n i a x i a l compression t e s t s were conducted employing an aluminum c y l i n d e r and b l o c k s of s t e e l , sandstone, and n o r i t e . The c y l i n d e r was 12 inches l o n g and 6 inches i n diameter and two blocks were 12 inches long and 6 inches square. Into the centre of each was d r i l l e d a 3/4-inch diameter h o l e , 2 inches long, except f o r the sand-stone block where the hole diameter was 0.9 i n c h e s . In order t o t e s t s i z e e f f e c t , the n o r i t e block was made 24 inches long and 8 inches square. A 0.9-inch diameter hole was d r i l l e d i n t o the centre of one face to a depth of 3 i n c h e s . The s t r a i n gauge r o s e t t e was used to monitor s t r a i n on the hole bottoms. A t o t a l of 2 c y l i n d e r s (aluminum and s t e e l ) and one block ( n o r i t e ) were t e s t e d to determine the value of s t r e s s 2 2 . c o n c e n t r a t i o n f a c t o r \"c\". The bl o c k s and c y l i n d e r s were 12 inches i n l e n g t h , the c y l i n d e r s 6 inches i n diameter and the blo c k s 6 inches square. A hole 3/4 inches i n diameter was d r i l l e d i n t o the centre of the end of each sample t o a depth of 6 inches. The s t r a i n s on the hole bottoms were monitored u s i n g a s t r a i n gauge r o s e t t e . E x t r a i n s t r u m e n t a t i o n was used i n a l l samples so t h a t m a t e r i a l p r o p e r t i e s would be a c c u r a t e l y known. S t r e s s c o n c e n t r a t i o n values proposed by Van Heerden were: a = 1 . 2 5 b = 0 c = 0 . 7 5 ( 0 .645 + v) 'van Heerden a l s o proposed a c o r r e c t i o n f a c t o r f o r authors who had used cubic samples. Those c o r r e c t e d v a l u e s are presented i n Table 2. Crouch (1969 and 1970) Crouch used the f i n i t e - e l e m e n t method t o f i n d s t r e s s c o n c e n t r a t i o n values i n s t e a d o f l a b o r a t o r y techniques. . He uses the axisymmetric technique as de s c r i b e d by Wilson (1965) . His f i n i t e - e l e m e n t mesh had 8 t r i a n g u l a r elements i n the h a l f s e c t i o n of the borehole bottom. The s o l u t i o n assumes t h a t b = 0 . Values proposed by Crouch f o r s t r e s s c o n c e n t r a t i o n f a c t o r s are: Poissons r a t i o 0_ 0 .1 0 .2 0 . 3 0 . 4 a 1.22 1.23 1.25 1.25 1.21 b 0 0 0 0 0 c - 0 . 4 5 5 - 0 . 5 4 0 - 0 . 6 2 0 - 0 . 7 0 0 - 0 . 7 8 0 23. Coates and Yu (1970) Coates and Yu a l s o used the axisymmetric technique o f the f i n i t e - e l e m e n t method t o o b t a i n a s o l u t i o n f o r the s t r e s s c o n c e n t r a t i o n f a c t o r s . T h e i r f i n i t e - e l e m e n t mesh has th r e e t r i a n g u l a r elements i n the h a l f s e c t i o n of the borehole bottom. Values proposed by Coates and Yu are: a = 1.366 + 0 . 0 2 5 V + 0.502v 2 b = -0.125 + 0.154v + 0.390V 2 c = -0.520 - 1.331V + 0.886V 2 de l a Cruz arid R a l e i g h (1972) These values were a l s o found u s i n g the f i n i t e - e l e m e n t technique. However, d e t a i l s o f the model used are not presented. The values proposed are: a = 1.30 b = 0.085 + 0 . 1 5 V - v 2 c = 0.473 + 0.91v Although t h i s i s not a l l the c o n c e n t r a t i o n f a c t o r s determined, i t represents the spread of data t h a t i s a v a i l a b l e . Table 2 provides a summary of a l l c o n c e n t r a t i o n f a c t o r s presented by these authors. A p l o t o f these s t r e s s c o n c e n t r a t i o n values together w i t h the corresponding s t r a i n c o n c e n t r a t i o n f a c t o r s i s g i v e n i n F i g u r e s 5, 6 and 7 f o r v a r i o u s values of Poissons r a t i o . R e f e r e n c e Model Type Po i s-son s Rat io R e s u I t s Obta i ned a b c Gal lie & Wilhout (1962) Cube 0.48 1.56 0 -1.04 * 1.30 Leeman (1964) Cube - 1.53 0 not considered * 1.28 Pal l i s t e r (1967) Cy1i nder 0.3 1.1 0 0.343 -0.75 Hi ramatsu & Oka Not known 0.24 1.36 -0.304 -0.69 0.29 1.32 -0.289 -0.82 0.37 1.39 -0.204 -0.98 0.44 1 .42 -0.060 -1.10 Bonnechere & Fa i rhurst(1968) Cube 1.25 0 -0.75 (0.5 + v) van Heerden (1968) Cubes & Blocks 1.25 0 -0.75 (0.645 + v) Crouch (1969 & 1970) Axi symmetric 0 1 .22 0 -0.455 F i n i t e Element 0.1 1.23 0 -0.540 Method (F.E.M.) 0.2 1.25 0 -0.620 0.3 1.25 0 -0.700 0.4 1.21 0 -0.780 Coates 5 Yu (1970) Axi symmetric 1.366 -0.125 -0.520 - 1.331v F.E.M. +0.025V +0.154v +0.886v2 +0.502V2 +0.390vz de la Cruz & Raleigh (1972) F.E.M. .. 1.30 0.085 -(0.473 + 0.91v) +0.15v - V 2 * Correction of 1.195 applied to cubes by van Heerden TABLE 2: SUMMARY OF STRESS CONCENTRATION FACTORS FOR ISOTROPIC MATERIALS METHOD OF ANALYSIS None of the authors reviewed i n the p r e v i o u s s e c t i o n had considered the s t r e s s c o n c e n t r a t i o n s i n t r a n s v e r s e l y i s o t r o p i c m a t e r i a l s . Therefore, i t appeared to be d e s i r a b l e t o attempt t o i n v e s t i g a t e the a p p l i c a t i o n o f the doorstopper technique i n such m a t e r i a l s . Two methods were i n i t i a l l y a v a i l a b l e by which s t r e s s c o n c e n t r a t i o n values f o r a t r a n s v e r s e l y i s o t r o p i c medium c o u l d be analysed. The f i r s t of these was t o c o n s t r u c t p h y s i c a l models and t e s t these i n the l a b t o f i n d the reponse on the bottom of the borehole. A f t e r a l i m i t e d amount of r e s e a r c h , i t appeared t h a t t h i s would be an i m p r a c t i c a l program w i t h i n the l i m i t a t i o n o f t h i s r e s e a r c h program. A l s o , i t appeared t h a t an a n a l y t i c a l s o l u t i o n would permit more v a r i a b l e s t o be c o n s i d e r e d i n the time a v a i l a b l e . As no d i r e c t mathematical s o l u t i o n was a v a i l -a b l e , the f i n i t e - e l e m e n t method appeared t o be the b e s t method. At f i r s t , an axisymmetric s o l u t i o n t o the problem was attempted. Non-symmetric l o a d i n g and deformation were p o s s i b l e as t h i s had been solved by Wilson (1965) . However, w i t h any f o l i a t i o n c o n d i t i o n other than f o l i a t i o n p e r p e n d i c u l a r t o the borehole, the axisymmetric case was not a p p l i c a b l e . The o n l y o p t i o n l e f t appeared to be the use of the three-dimensional f i n i t e - e l e m e n t technique. The b a s i c format f o r t h i s technique i s w e l l known and 26. has been presented i n textbooks by Z i e n k i e w i t z and Cheung (1965), Z i e n k i e w i t z (1971), and Desai and Abel (1972). A general s i x -s i d e d element was chosen w i t h a l i n e a r displacement f i e l d . The a n a l y s i s procedure o u t l i n e d i n the f o l l o w i n g s e c t i o n and given i n g r e a t e r d e t a i l i n Appendices A and B was developed by Kiyama (1972) i n a s s o c i a t i o n w i t h the w r i t e r . Kiyama s e t up the mathematical model and computer program as an i n t e g r a l p a r t o f the w r i t e r ' s research program. This a n a l y s i s has not been p u b l i s h e d t o date except as summarized i n t h i s work. The w r i t e r used t h i s program e x t e n s i v e l y i n the p r e p a r a t i o n of t h i s t h e s i s . SUMMARY OF THE FINITE ELEMENT ANALYSIS PROCEDURE The summary below f o l l o w s the general o u t l i n e as pre-sented by Desai and Abel (1972). The step-by-step procedure r e q u i r e d to o b t a i n a s o l u t i o n i s : A) D i s c r e t i z a t i o n o f the continuum B) S e l e c t i o n o f the displacement model C) D e r i v a t i o n o f the element s t i f f n e s s m a t r i x D) Assembly o f the a l g e b r a i c equation f o r the o v e r a l l d i s c r e t i z e d continuum E) S e t t i n g of boundary c o n d i t i o n s F) S o l v i n g f o r the unknown displacements D i s c r e t i z a t i o n of the Continuum The continuum i s the s o l i d body which i s analysed. D i s c r e t i z a t i o n i s the a c t of s u b d i v i d i n g the continuum i n t o 27. many f i n i t e elements. Four important c o n s i d e r a t i o n s presented themselves when d e c i d i n g on how t o d i v i d e the model. These were: 1) What s i z e , number and arrangement of f i n i t e elements w i l l g i v e an e f f e c t i v e r e p r e s e n t a t i o n of the continuum f o r the p a r t i c u l a r problem considered? The g r e a t e r the s t r e s s g r a d i e n t through a r e g i o n , the more r e f i n e d the element mesh should become. So, i n areas of s t r e s s changes, there should be more elements than are present elsewhere i n the body. A l s o , n a t u r a l d i s c o n t i n u i t i e s i n the body must e x i s t as nodal p o i n t s i n the model. -2) Nodal p o i n t s must be s e l e c t e d and numbered t o minimize the band width of the r e s u l t i n g s t i f f n e s s , m a t r i x . This i m p l i e s t h a t when numbering the nodes w i t h i n the continuum, the i n t i m a t e neighbours to any given node must have the s m a l l e s t d i f f e r e n c e i n numerical value p o s s i b l e . 3) Nodal p o i n t s may be s e l e c t e d so t h a t s e l e c t i o n and numbering can be automated w i t h i n a computer program. 4) Although b e t t e r s o l u t i o n s r e s u l t from i n c r e a s i n g the number of elements, computation time and storage c a p a c i t i e s must be considered i n the a n a l y s i s . With these p o i n t s i n mind, the g r i d shown i n F i g u r e 8 (Page 58) was c o n s t r u c t e d . I t solved the four c r i t e r i a i n the f o l l o w i n g way: 1) S t r e s s c o n c e n t r a t i o n s e x i s t e d i n the v i c i n i t y of the borehole and e s p e c i a l l y on the borehole bottom i n the semi-c y l i n d e r . Therefore, as t h a t r e g i o n was approached, g r e a t e r numbers of smaller elements were constructed. 2) A f t e r attempting various combinations, i t appeared that the numbering system shown on Figure 8 produced the narrowest band width. 3) This method allowed f o r automated s e l e c t i o n of points. This means that input of nodal coordinates was not required and saved input time and e r r o r . 4) The g r i d presented appeared to be the most e f f i c i e n t g r i d with only as many elements as were required. However, many var i a t i o n s were not t r i e d and these are given further consideration i n the discussion. Selection of the Displacement Model The displacement model i s simply a function which represents the displacements within the element i n terms of the nodal displacements. The simplest model that i s commonly employed i s a l i n e a r polynomial. Three factors influenced the s e l e c t i o n of a displacement model. They were: 1) The type and degree of the displacement model. 2) The p a r t i c u l a r displacement magnitudes that describe the model. 3) The convergence requirements which the model must s a t i s f y . These are: a) The displacement models must be continuous within the elements, and the displacements must be compatible between adjacent elements. b) The displacement models must include the r i g i d body displacements of the element. 29. c) The displacement models must i n c l u d e the constant s t r a i n s t a t e s of the element. The displacement model chosen was from Z i e n k i e w i t z (1971). I t was a l i n e a r polynomial and the displacement magnitudes t h a t d e s c r i b e the model were the nodal displacements. Because of i t s s i m p l i c i t y , the model c l e a r l y s a t i s f i e d the convergence c r i t e r i a . The displacement model was of the form: u = + o^x + a^y + a^z v = + oigX + a 7 y + oigZ (1) w = a g + a 1 Qx•+a^ 1y + ^ 1 2 z Equating the values of the displacements a t the nodes, we evaluate a-^ t o a-^* A more g e n e r a l i z e d displacement f u n c t i o n (using g e n e r a l i z e d coordinates) i s i n Appendix A, Equation 4. D e r i v a t i o n of the Element S t i f f n e s s M a t r i x The element s t i f f n e s s m a t r i x r e l a t e s f o r c e s a t the nodes (F) w i t h displacements a t the nodes (s) by the r e l a t i o n -s h i p : F = Ks where K i s the element s t i f f n e s s m a t r i x . The element s t i f f n e s s m a t r i x depended upon: 1) The displacement model 2) The geometry of the element 3) The l o c a l m a t e r i a l p r o p e r t i e s or c o n s t i t u t i v e r e l a t i o n s . The displacement model and the geometry of the element 30. have been di s c u s s e d . The c o n s t i t u t i v e r e l a t i o n s or e l a s t i c p r o p e r i t e s f o r a t r a n s v e r s e l y i s o t r o p i c medium f o l l o w : {e} - [D] 1 { a } where [D] \" \" i s : E. -V -V 21 31 0 0 0 - V l 2 / E 2 E l / E 2 \" E 1 V 3 2 / E 2 0 0 0 - V l 3 / E 3 E 1 V 2 3 / E 3 E l / E 3 0 0 0 E l / G 1 2 0 0 0 E l / G 2 3 0 0 0 E 1 / G 3 1 In t h i s a n a l y s i s : E l / E 2 E l / E 3 v 12 V13 = V 2 3 E l / G 2 3 = E l / G 3 1 E l / G 1 2 1.0 3.0 0.25 0.15 4.2857 2.5 = 2 d + v 1 2 ) (2) This m a t r i x i s transformed u s i n g standard e l a s t i c theory as the coordinate axes are r o t a t e d about the \"x\" a x i s from 0 t o 90 degrees. 31. The element s t i f f n e s s m a t r i x was d e r i v e d u s i n g the f o l l o w i n g procedure: 1) The displacements were examined i n terms o f g e n e r a l i z e d c o o r d i n a t e s . 2) The s t r a i n v e c t o r was examined i n terms o f d i s -placement and the g e n e r a l i z e d c o o r d i n a t e s . 3) The s t r e s s v e c t o r was examined i n terms o f s t r a i n , then displacement, then g e n e r a l i z e d c o o r d i n a t e s . 4) The element s t i f f n e s s m a t r i x was formulated by the a p p l i c a t i o n of v i r t u a l work and m i n i m i z a t i o n of p o t e n t i a l energy. This process i s c a r r i e d through i n Appendix A — f i r s t as a general case and then i n more d e t a i l . Assembly of the A l g e b r a i c Equations f o r the O v e r a l l D i s c r e t i z e d Continuum The assembly procedure used was the most common assembly technique known—the d i r e c t s t i f f n e s s method. The b a s i s f o r the method i s t h a t a node i n the continuum, a l s o a node w i t h i n s e v e r a l d i f f e r e n t elements, must have the same displacement w i t h i n each element, and t h a t loads and s t i f f n e s s e s f o r the continuum nodes are summations of loads and s t i f f n e s s e s of the element nodes. This can be s t a t e d mathematically as: [K] = E E e=l (3) and {R} = Z E e=l (4) 32. where [K] = s t r u c t u r a l s t i f f n e s s m a t r i x E = t o t a l number of elements i n f l u e n c i n g a node e = element number {R} = s t r u c t u r a l l o a d v e c t o r {R_}= element l o a d v e c t o r [ K J = element s t i f f n e s s m a t r i x T h i s means assembly o f the s t i f f n e s s m a t r i x was s i m p l y a procedure o f adding s t i f f n e s s v a l u e s a f f e c t i n g any node w i t h i n an element i n t o the same l o c a t i o n f o r the s t r u c t u r a l s t i f f n e s s m a t r i x , u s i n g the d i r e c t s t i f f n e s s procedure. A b l o c k diagram showing a computer loop f o r the d i r e c t s t i f f n e s s p rocedure i s g i v e n i n F i g u r e 9 (Page 59). S e t t i n g o f Boundary C o n d i t i o n s There were s i x t e e n separate c o n d i t i o n s run on the computer. These i n v o l v e d f o u r l o a d cases w i t h each l o a d case run a t f o u r d i f f e r e n t f o l i a t i o n a n g l e s , 0°, 30°, 60° and 90° t o the plane o f i s o t r o p y . These cases are i l l u s t r a t e d on F i g u r e 10 (Page 60). c o n d i t i o n s , ( r e f e r t o F i g u r e l l ' v . Page 61) the boundary con-d i t i o n s i n . a l l cases were: e Usi n g c a r t e s i a n c o o r d i n a t e s f o r r e f e r e n c e on boundary 1) a t (x, y, h) displacements were (u, v, o) and f o r c e s were (0, 0, W) 2) a t (o, y, z) displacements were (0, v, w) and f o m e s were (U, 0, 0) 33. 3) a t (0, 0, 0) displacements were (0, 0, w) and f o r c e s were (O, V, 0) For the i n d i v i d u a l l o a d cases, the f o l l o w i n g c o n d i t i o n s were i n e f f e c t . These i n d i v i d u a l cases are i l l u s t r a t e d i n F i g u r e 11-Load Case 1 = (1, 1, 0, .0, 0, 0) at r = a displacements were (u, v, w) and f o r c e s were (cos 0, s i n 0, 0) Load Case 2 {02>T = ( 1 , - 1 , 0 , 0 , 0 , 0 ) at r = a displacements were (u, v, w) and f o r c e s were (cos 0, - s i n 0 , 0) Load Case 3 {03>T = (0, 0, 1, 0, 0, 0) at z = -h displacements were (u, v, w) and f o r c e s were (0, 0, -1) Load Case 4 {04>T = (0, 0, 0, 0, 1, 0) at r = a displacements were (u, v, w) and f o r c e s were (0, 0, sin0) and a t z = -h displacements were (u, v, w) and f o r c e s were (0, 1, 0) By the p r i n c i p l e of s u p e r p o s i t i o n (Timoshenko and Goodier (1970), pp. 8 and 9 ) , i t seems t h a t these f o u r l o a d i n g con-d i t i o n s can e f f e c t i v e l y model a l l s t r e s s s t a t e s , t h a t i s : + 0. T + 0. (0 - J-)T H A yz ^4 2 xz ^ 34. S o l u t i o n f o r the Unknown Displacements When the s t r u c t u r a l s t i f f n e s s m a t r i x was formed and boundary c o n d i t i o n s s a t i s f i e d , then a l l t h a t remained was t o determine the displacements of the nodes. This i s normally a s t r a i g h t f o r w a r d m a t r i x manipulation procedure. However, the s i z e of the ma t r i x i n v o l v e d i n t h i s study (4,368 degrees o f freedom) made normal d i r e c t approaches such as the Gaussian e l i m i n a t i o n expensive. I t was decided, t h e r e f o r e , t o use an i t e r a t i o n technique. Varga (1962) g i v e s the theory and p r a c t i c e of i t e r a t i v e methods of s o l u t i o n s of equations. The i t e r a t i v e procedure i s a s e r i e s of c o r r e c t i o n s t o an o r i g i n a l e s t i m a t e o f the unknown, the procedure being repeatedly c a r r i e d out u n t i l covergence w i t h a s o l u t i o n occurs. A copy of the computer program used can be found i n Appendix B. The output l i s t e d node numbers, displacements, and f o r c e s . RESULTS General The r e s u l t s were i n i t i a l l y p r i n t e d out on computer hard copy as displacements and f o r c e s of nodes w i t h i n the s t r u c t u r e . Appendix C gives nodal displacements f o r the v a r i o u s cases on the bottom of the borehole. The s t r a i n was c a l c u l a t e d from these values using two separate formulas. These formulas were: Method 1 ere = (/(Rcos8+u) Z+(Rsin8+v) 2+(Aw) 2 -R)/R Method 2 er9 = (ucos6+vsin9)/R \"(6) Although the f i r s t formula i s t e c h n i c a l l y more c o r r e c t , f o r ease of computation the m a j o r i t y of s t r a i n v a l u e s were c a l c u l a t e d u s i n g the second equation. This i s j u s t i f i e d because 2 of the s m a l l i n p u t of Aw i n the f i r s t equation. I t a l s o g i v e s s i g n i f i c a n t l y more s t a b l e r e s u l t s because of an uns t a b l e u) a t the c e n t r a l node. These s t r a i n (er0) values can be found i n Appendix D. S t r a i n s were then computed a t d i f f e r e n t angles around the boreholes and averaged across the borehole so t h a t : a(8,r=0.1) = e (6)+e(9+n) /2.0 ' (7) Or i n the h a l f space used i n t h i s a n a l y s i s a(8,r=o.l) = e(8)+e(-6) /2.0 (8) Then the s t r a i n c o n c e n t r a t i o n values were computed as: A l = ( a 1 + a 2 ) / ' 2 A 2 = ( a 1 ~ a 2 ) / 2 A 3 = a 3 ( 9 ) A 4 = a 2(8- n/4) A,. = a. 5 4 A 6 = a 4(8- n/2) where e(r,8) = 1 ( A 1 a x + A z a y + A 3 a z + A 4 T x y + A 5 T y z + A 6 T x a s y (10) E l These values can be seen i n Appendix E. The A^' th e r e f o r e , are dependent on y, r , 8 and m a t e r i a l p r o p e r t i e s . 3 6 . Figures 12 through 17 (Pages 62 through 67) show the p l o t s of A^ f o r f o l i a t i o n t o borehole bottom angles, y, of 0 ° , 3 0 ° , 6 0 ° , and 90° . These p l o t s show the s t r a i n c o n c e n t r a t i o n f a c t o r s A^ t o Ag versus the angle of the node 0 from the x a x i s (the s t r i k e d i r e c t i o n of the f o l i a t i o n on the borehole bottom). As the r e - . . s u i t s a l s o i n c l u d e the coordinate t r a n s f o r m a t i o n terms, the curves are s i n u s o i d a l . F i g u r e s 18 through 20 (Pages 68 through 70) are p l o t s of s t r a i n c o n c e n t r a t i o n f a c t o r s A, to A,, from r e s u l t s of i n v e s t i g a t o r s covered p r e v i o u s l y w i t h p r e s e n t a t i o n i n c l u d i n g the coordinate t r a n s f o r m a t i o n terms. The r e s u l t s are d i s c u s s e d below f o r each f o l i a t i o n angle. F o l i a t i o n plane p a r a l l e l to borehole bottom (y = 0°) A l l values of A^ should be i d e n t i c a l to the i s o t r o p i c case as we are e s s e n t i a l l y d e a l i n g w i t h an i s o t r o p i c plane on the borehole bottom. A^, however, has a higher magnitude curve. i s a s t r a i g h t l i n e a t approximately - 0 .8 . A l l these v a l u e s appear to be on the o u t s i d e l i m i t of the values found by o t h e r r e s e a r c h e r s , but c l o s e enough so t h a t the m a t e r i a l p r o p e r t i e s appear to be modelled reasonably w e l l . A,, and Ag are zero i n t h i s study as they are assumed to be i n an i s o t r o p i c case. T h i s case was c a r r i e d out f i r s t , i n p a r t , as a t e s t t o compare ag a i n s t i s o t r o p i c v a l u e s . Although the model does not converge to the best values i n t h i s case, i t does approach what c o u l d be considered reasonable v a l u e s . 37. F o l i a t i o n plane at 30° to borehole bottom (y '='• 30°) The value f o r A^ i s very s i m i l a r to the i s o t r o p i c case except that the amplitude of the curve increases. This i s also true f o r the value of These increases are due to the changing material properties, which at 30° s t i l l represents a small s t r a i n change. The curve f o r A^ changes r a d i c a l l y . Values f o r 0 = ± 90° are s i m i l a r to the i s o t r o p i c cases, but large increases can be seen f o r a l l other o r i e n t a t i o n s . At t h i s point i n the analysis, inspection of the analysis method was warranted as t h i s A^ value had an e x t r a o r d i n a r i l y high amplitude. Examination of the deformation data i n d i c a t e d that values of Aw (deformation i n the z direction) f o r the c e n t r a l point (13 separate points i n the analysis) varied excessively with 0. The forces u and v (u and v are held equal to 0.) also a t t a i n s i g n i f i c a n t l y high values i n order to maintain continuity, thereby imposing a d d i t i o n a l loads on the borehole bottom. Therefore, although t h i s curve probably has the r i g h t trend, the quantitative values are d e f i n i t e l y questionable. The value of A^ was found by taking values of Load Case Two but using them at 45 degrees from the p o s i t i o n at which they were computed. That i s 0 (A^) = 0 (JZ^) ~ 45°. Although t h i s method i s acceptable for the i s o t r o p i c case, i t now appears that i t i s erroneous i n anisotropic cases. The curves are presented as calculated. Values f o r A,- increase considerably from the i s o t r o p i c case of 0.0. This i s caused by the shear stress having a component along the f o l i a t i o n 3 8 . which causes the non-zero value. Although t h i s curve demon-s t r a t e s the t r e n d t h a t occurs, i n s t a b i l i t y of Aco as d e s c r i b e d p r e v i o u s l y and l a r g e f o r c e s r e q u i r e d a t the curve node to maintain c o n t i n u i t y imply t h a t the r e s u l t s are suspect. Values f o r A, were c a l c u l a t e d using v a l u e s of A,, but changing 0. That i s : 0 (Ag) = 0(# 4) - 90°. As i n the c a l c u l a t i o n s f o r A 4, t h i s method i s acceptable f o r the i s o t r o p i c case, but i t now appears t h a t i t i s erroneous i n a n i s o t r o p i c cases. F o l i a t i o n plane a t 6 0 ° t o borehole bottom (y = 6 0 ° ) A^ i s almost i d e n t i c a l t o the 3 0 ° case. A ^ r however, changes c o n s i d e r a b l y . This i s the most d r a s t i c change of curves seen i n the whole a n a l y s i s . However, the v a l i d i t y o f the magnitude of change i s again unknown as the f i x e d c e n t r a l node on the borehole bottom again caused imposed loads and an unstable Ato. A 3 i s a l s o s u b j e c t t o the same problem. A 4 i s erroneous f o r the same reason d i s c u s s e d f o r A 4 a t y = 3 0 ° . A,, may or may not be c o r r e c t , but again the v a l i d i t y o f the change i s unknown as an unstable Aw and l a r g e induced l o a d occur a t the centre of measured r e g i o n . Ag i s erroneous f o r the same reason as discussed f o r A. a t y = 3 0 ° case. b F o l i a t i o n plane p e r p e n d i c u l a r t o borehole bottom (y = 9 0 ° ) The curve of A^ values has a higher amplitude than any of the other curves. However, i t s t i l l does not vary g r e a t l y from i s o t r o p i c c o n d i t i o n s , and appears to be o n l y s l i g h t l y a f f e c t e d . 39. The curve of A 2 i s s i m i l a r to the 60° curve except f o r a s l i g h t p o s i t i v e displacement. This trend appears reasonable but as w i t h A 2 a t y = 60°, the values are i n doubt as t h e r e are i n s t a b i l i t i e s i n Aw and l a r g e induced f o r c e s a t the c e n t r a l node. The curve i s a l s o i n doubt f o r the same reason as f o r the A 2 curve. However, i t s t r e n d back towards the y = 0° case seems reasonable. The A 4 curve i s i n e r r o r f o r the same reasons as the Ad curve a t y = 30° and 60®. The A,, curve f a l l s on the 0, t h e o r e t i c a l l y c o r r e c t . This gives confidence i n A,- v a l u e s , as. t h i s had an u n s t a b l e Aw as w e l l . The Ag values are c o r r e c t as they should a l s o be zero a t 90°: The f a m i l y of curves i s r e p r e s e n t a t i v e of what occurs i n a t r a n s v e r s e l y i s o t r o p i c medium. However, the curves f o r A^ and Ag are based on wrong assumptions and except f o r the y = 0° case, should not be accepted as v a l i d . Of l e s s importance i s the e r r o r introduced by f i x i n g the c e n t r a l nodal r i n g i n the x and y d i r e c t i o n s . This i n t r o d u c e d some e r r o r i n the r e s u l t s as evidenced by induced f o r c e s on these nodes i n the x and y d i r e c t i o n . I t was necessary t o f i x these nodes to s a t i s f y c o m p a t i b i l i t y of the s t r u c t u r e . 40. DISCUSSION General The f o l l o w i n g p o i n t s can be made about the r e s u l t s from t h i s a n a l y s i s : 1) Data obtained f o r the y = 0 case i n t h i s a n a l y s i s shows s i m i l a r i t y to values obtained f o r the i s o t r o p i c case. T h i s i n d i c a t e s t h a t the program i s converging c l o s e t o the s o l u t i o n . 2) The curve f o r A c a t y = 90° f a l l s on zero throughout o i t s l e n g t h which i s t h e o r e t i c a l l y c o r r e c t . This shows t h a t d e s p i t e problems w i t h the c e n t r a l node of the s t r u c t u r e , reason-able values are obtained. 3) The s t r e s s concentrations obtained i n t h i s a n a l y s i s d i f f e r when the values f o r y = 0° d i f f e r from the i s o t r o p i c case by about 15 percent. With a change i n y c o n c e n t r a t i o n f a c t o r s vary much more than t h i s 15 percent (note graphs on Fi g u r e 12 through 17). This i n d i c a t e s t h a t the e f f e c t of a n i s o -tropy i s one which must be considered when computing the s t r e s s f i e l d i n a t r a n s v e r s e l y i s o t r o p i c medium. 4) A l l data i s based on deformation across one element. The displacement model i s l i n e a r , t h e r e f o r e the element can d i s p l a y only constant s t r a i n or s t r e s s w i t h i n i t s boundaries. B e t t e r data c o u l d be obtained w i t h more elements through the c e n t r a l r e g i o n . 5) The displacement of the c e n t r a l node i n the s t r u c t u r e (centre of the end of the borehole) was f i x e d i n the x and y d i r e c t i o n s . This was done to prevent an excessive d i s c o n t i n u i t y from developing a t t h i s p o i n t . However, t h i s caused loads t o develop a t t h i s p o i n t i n the x and y d i r e c t i o n s which may i n f l u - -ence deformations i n the surrounding elements. 6) The displacement of the c e n t r a l node i n the s t r u c t u r e was not f i x e d i n the z d i r e c t i o n . This allowed some d i s c o n t i n -u i t y t o occur a t t h i s c e n t r a l node, which i n f l u e n c e d deformation of the surrounding elements. 7) Values f o r and Ag were c a l c u l a t e d by u s i n g data from l o a d cases which do not apply t o these v a l u e s i n a n i s o t r o p i c c o n d i t i o n s . To c o r r e c t the i n a c c u r a c i e s introduced i n t h i s a n a l y s i s s e v e r a l steps could be taken when b u i l d i n g a f u t u r e m o d i f i e d f i n i t e - e l e m e n t model. The a n a l y s i s procedure o u t l i n e d e a r l i e r i n t h i s t h e s i s i s f o l l o w e d again i n order t o present ideas f o r improvement i n a l o g i c a l order. D i s c r e t i z a t i o n of the Continuum As mentioned e a r l i e r i n the d i s c u s s i o n , the g r i d c l o s e to the bottom of the borehole needs to be f i n e r . T o t a l con-vergence to the s o l u t i o n has occurred o n l y when a f i n e r g r i d than t h a t run causes no change i n the s o l u t i o n . I t a l s o appears t h a t a f u l l c y l i n d e r i s r e q u i r e d which w i l l double the number of elements. With methods p r e s e n t l y a v a i l a b l e on the computer, t h i s i s a very c o s t l y experiment. F o l l o w i n g are 42. some of the methods which could help t o i n c r e a s e the e f f i c i e n c y of t h i s step: 1) The coarse to f i n e s u b d i v i s i o n method i n v o l v e s u s i n g a very coarse g r i d over the t o t a l s t r u c t u r e ; o b t a i n i n g displacement i n f o r m a t i o n f o r a s m a l l e r r e g i o n ; and a p p l y i n g • • t h a t displacement data around the s m a l l e r r e g i o n w i t h a much f i n e r g r i d . 2) Another method which can be used i s the s u b s t r u c t u r e method. P h y s i c a l l y , t h i s method i n v o l v e s d i v i d i n g the s t r u c t u r e i n t o many sm a l l p a r t s , a n a l y s i n g each p a r t separately and then a n a l y s i n g the i n f l u e n c e of these p a r t s on one another. From the p o i n t of view of a n a l y s i s , i t means t h a t much s m a l l e r banded symmetric matrices can be analysed u s i n g common, l e s s expensive m a t r i x manipulation techniques, r a t h e r than the i t e r a t i v e approach. A d i f f e r e n t shape of element should be i n v e s t i g a t e d f o r the centre of the s t r u c t u r e so t h a t a d i s c o n t i n u i t y o r e q u i l i b r i u m problem does not a r i s e at t h i s l o c a t i o n as i t d i d i n t h i s study. S e l e c t i o n of the Displacement Model Desai and Abel (1972, p. 173) presented a d i s c u s s i o n of mesh refinement versus higher order elements. They suggested t h a t f o r a given problem each method must be t r i e d i n order t o evaluate which w i l l g ive an acceptable answer a t the lowest c o s t . So, although i n c r e a s i n g the order of the displacement model polynomial may give a b e t t e r answer a t 4 3 . reasonable c o s t , there i s no guarantee t h a t t h i s w i l l be the case s i n c e the problem has not been s t u d i e d d e f i n i t i v e l y . D e r i v a t i o n of the Element S t i f f n e s s M a t r i x This i s f a i r l y s t r a i g h t - f o r w a r d and would r e q u i r e adjustment only when the element shape was changed o r the order of the displacement model polynomial was changed. Assembly of the A l g e b r a i c Equation f o r the O v e r a l l D i s c r e t i z e d Continuum In the program used here, much of the storage was done i n core f o r ease of f i l i n g and r e c a l l . However, f o r a system as l a r g e as t h i s , other storage methods should be implemented, p l a c i n g i n core o n l y those values t h a t are r e q u i r e d a t any one time. This can be done w i t h ease i f the s u b s t r u c t u r e technique i s developed. S e t t i n g of Boundary C o n d i t i o n s The problems i n v o l v e d w i t h boundary c o n d i t i o n s are l i m i t e d t o the p i n n i n g of the centre of the s t r u c t u r e i n the x and y d i r e c t i o n . This can be sol v e d by u s i n g a d i f f e r e n t element shape as noted e a r l i e r . S o l u t i o n f o r the Unknown Displacements As was discussed e a r l i e r , t h i s should probably be done using a modified s u b s t r u c t u r e method which would a l l o w the use of a much cheaper method than the present i t e r a t i v e one. At the time t h i s study was c a r r i e d out (autumn 197 2) no s o l u t i o n e x i s t e d at the U n i v e r s i t y of B r i t i s h Columbia f o r t h i s s i z e of s t r u c t u r e . To the w r i t e r ' s knowledge, no such system e x i s t s y e t . However, i t would appear to be w i t h i n the s t a t e of the a r t t o c o n s t r u c t a system to s o l v e l a r g e g e n e r a l i z e d m a t r i x systems. D i s c u s s i o n s were h e l d w i t h A. Fowler of the Computer Science Department i n 1972 t o e x p l o r e p o s s i b l e development of t h i s program, but because of the time i n v o l v e d f o r development, i t was not pursued. Accuracy of I n - s i t u S t r e s s Measurements Bonnechere (19 69) a f t e r an e x t e n s i v e f i e l d study of c u r r e n t rock s t r e s s determination techniques claimed t h a t i n -s i t u s t r e s s e s cannot be r e l i e d upon w i t h i n twenty o r t h i r t y percent of the mean values i n i d e a l c o n d i t i o n s . F a i r h u r s t (196 8) says t h a t values b e t t e r than twenty-f i v e percent accuracy are not r e q u i r e d . The q u e s t i o n t o ask of a p a r t i c u l a r technique, t h e r e f o r e , i s \" W i l l t h i s method pr o v i d e , economically, i n f o r m a t i o n t h a t w i l l a s s i s t i n the p r a c t i c a l s o l u t i o n of the design problem faced?\" F a i r h u r s t (1968, p. 5) a l s o s t a t e s t h a t apart from measurement i n a c c u -r a c i e s , which can be minimized by c a r e f u l experimental technique, e r r o r s i n the s t r e s s determination are due e n t i r e l y to the e r r o r i n the assumed value of the m a t e r i a l p r o p e r t i e s , although i t would a l s o appear t h a t the u n c e r t a i n t y i n the a p p l i c a b l e s t r e s s c o n c e n t r a t i o n f a c t o r s approaches 10 t o 20 percent f o r i s o t r o p i c m a t e r i a l s . 45. Berry and F a i r h u r s t (1966) show t h a t the assumption o f i s o t r o p y i n simple cases of tran s v e r s e i s o t r o p y ( t y p i c a l o f sedimentary and f o l i a t e d metamorphic rocks) leads t o e r r o r i n the computed s t r e s s e s of as much as 50 percent. CONCLUSIONS In order t o design s a f e r and more e f f i c i e n t s t r u c t u r e s i n r o c k s , i t i s necessary t o have a working knowledge of the i n -s i t u s t r e s s e s . In many cases, the exact s t r e s s l e v e l i s not as important as approximate magnitudes, p r i n c i p a l s t r e s s d i r e c t i o n , and r a t i o s between p r i n c i p a l s t r e s s e s . In an a n i s o t r o p i c m a t e r i a l , many problems are encountered i n o b t a i n i n g a s o l u t i o n f o r the s t r e s s tensor u s i n g the Leeman technique. F i e l d techniques are w e l l e s t a b l i s h e d and although problems may occur, w i t h proper care these can be d e t e c t e d and c o r r e c t e d before they have a s e r i o u s i n f l u e n c e on the r e s u l t s . O btaining e f f e c t i v e m a t e r i a l property values f o r a t r a n s v e r s e l y i s o t r o p i c m a t e r i a l i s a very d i f f i c u l t t h i n g t o do once the core has been r e t r i e v e d . However, m a t e r i a l p r o p e r t i e s do have a l a r g e p a r t to p l a y i n determining the s t r e s s tensor and attempts should be made to determine t h e i r t r u e values r a t h e r than assum-i n g an i s o t r o p i c case. The f o l l o w i n g c o n c l u s i o n s can be drawn: 1) The s t r a i n c o n c e n t r a t i o n f a c t o r s obtained i n t h i s a n a l y s i s d i f f e r s i g n i f i c a n t l y from the i s o t r o p i c s t r a i n concen-t r a t i o n f a c t o r s . This suggests t h a t as i n other s t r e s s 4 6 . determination techniques, a n i s o t r o p i c rocks must e i t h e r be avoided or taken i n t o c o n s i d e r a t i o n i n the a n a l y s i s . Therefore, the Leeman doorstopper method should perhaps not be used i n such m a t e r i a l s u n t i l a more r e l i a b l e a n a l y s i s i s a v a i l a b l e . 2) An e v a l u a t i o n of the r e s u l t s obtained i n the a n a l y s i s , presented i n d i c a t e s t h a t they are q u a l i t a t i v e l y c o r r e c t but q u a n t i t a t i v e l y i n e r r o r . The e r r o r s present are due t o a f i n i t e -element mesh t h a t was too coarse i n the re g i o n of g r e a t e s t i n t e r e s t , and to boundary e f f e c t s imposed on the model i n order to s a t i s f y c o n t i n u i t y . 3) I t may be t h a t the r e s u l t s (except A^ and A g) are c l o s e r t o the true values t h a t the i s o t r o p i c values would be. They are a t l e a s t u s e f u l i n showing how the i s o t r o p i c v a l u e s may be a l t e r e d t o giv e a c l o s e r approximation t o the a n i s o t r o p i c c o n c e n t r a t i o n v a l u e s . At present i t appears t h a t use of i s o -t r o p i c values c o u l d produce e r r o r s i n the s t r e s s f i e l d by as much as 1 0 0 percent depending on the d i r e c t i o n a l r e l a t i o n s h i p between s t r e s s f i e l d , borehole, and e l a s t i c p r o p e r t i e s . RECOMMENDATIONS D i r e c t determination of the s t r e s s f i e l d should be c a r r i e d out only i f f i e l d o b s e r v a t i o n w i l l not y i e l d accurate enough i n f o r m a t i o n f o r the design problem on hand. I f the s t r e s s f i e l d i s determined using the Leeman technique i n an a n i s o t r o p i c 4 7 . rock mass, every attempt should be made t o conduct the t e s t i n the most i s o t r o p i c zone, or to d r i l l the borehole p e r p e n d i c u l a r to the plane of i s o t r o p y . The core from t h i s zone should l a t e r be t e s t e d i n the l a b o r a t o r y to ensure t h a t i t i s not e x c e s s i v e l y a n i s o t r o p i c . In order to make the doorstopper technique more u s e f u l i n t r a n s v e r s e l y i s o t r o p i c ground, the f o l l o w i n g recommend-a t i o n s are presented; 1) C a l c u l a t i o n s to determine the e f f e c t o f a n i s o t r o p i c s t r a i n c o n c e n t r a t i o n values versus i s o t r o p i c v a l u e s should be c a r r i e d out. As a s t a r t i n g base one co u l d use the s t r e s s con-c e n t r a t i o n values found i n t h i s study, tempered by the knowledge of the i s o t r o p i c v a l u e s . 2) A l a b o r a t o r y program to f i n d an e f f e c t i v e and accurate method of determining e l a s t i c p r o p e r t i e s e f f e c t i v e f o r the Leeman technique i n both i s o t r o p i c and t r a n s v e r s e l y i s o t r o p i c e l a s t i c m a t e r i a l should be i n i t i a t e d . 3) R e s u l t s obtained by researchers f o r s t r e s s d i s t r i b u -t i o n about simple c a v i t i e s i n t r a n s v e r s e l y i s o t r o p i c mediums should be used as a s t a r t i n g p o i n t f o r determining s t r e s s con-c e n t r a t i o n v a l u e s . Bonnechere and F a i r h u r s t (1968) show t h a t i n i s o t r o p i c m a t e r i a l , s t r e s s c o n c e n t r a t i o n value \"c\" a t the end of a borehole can be given bounds by examining c o n c e n t r a t i o n s around s l o t s and e l l i p t i c a l h o l e s . F u r t h e r f i n i t e - e l e m e n t work w i l l be simpler to i n t e r p r e t and have more use i f t h i s i s done. 4) With new computer methods becoming a v a i l a b l e every day, a more e f f i c i e n t program f o r handling l a r g e m a t r i c e s can be b u i l t than the one used i n t h i s a n a l y s i s . Therefore, i t , 4 8 . would seem t h a t i t w i l l soon be p o s s i b l e , f o r a reasonable c o s t , to o b t a i n b e t t e r values of a n i s o t r o p i c s t r e s s c o n c e n t r a t i o n f a c t o r s . These improved values would be u s e f u l i n extending the use of the Leeman doorstopper i n t o design problems i n t r a n s v e r s e l y i s o t r o p i c m a t e r i a l s . 49. BIBLIOGRAPHY Berry, D.S. and F a i r h u r s t , C. (1966). \"Influence o f Rock Anisotropy and Time-Dependent Deformation on The S t r e s s R e l i e f and High Modulous I n c l u s i o n Techniques of I n - S i t u S t r e s s Determination,\" T e s t i n g Techniques f o r Rock Mechanics, American S o c i e t y f o r T e s t i n g and M a t e r i a l s , STP 402, P h i l a d e l p h i a , Penn., pp. 190-206. Berry, D.S. (1968). \"The Theory of St r e s s Determination by Means of S t r e s s R e l i e f Techniques i n a Tr a n s v e r s e l y I s o t r o p i c Medium,\" Te c h n i c a l Report No. 5-68, M i s s o u r i R i v e r D i v i s i o n , U.S. Army Corps of Engineers, Omaha, Nebraska 68101. Berry, D.S. (1970). \"The Theory of Determination of S t r e s s Changes i n a Transversely I s o t r o p i c Medium, u s i n g an Instrumented C y l i n d r i c a l I n c l u s i o n , \" T e c h n i c a l Report MRD-1-70, M i s s o u r i R i v e r D i v i s i o n , U.S. Army Corps o f Engineers, Omaha, Nebraska 68101. B i s p l i n g h o f f , R.L., Mar, J.W. and P a i n , T.H.H. (1965). STATICS OF DEFORMABLE SOLIDS, Reading, Mass.: Addison-Wesley, 321 pp. Bonnechere, F. and F a i r h u r s t C. (1968). \"Determination o f the Regional S t r e s s F i e l d From Doorstopper Measurements,\" J o u r n a l of the South A f r i c a n I n s t i t u t e of Min i n g and M e t a l l u r g y , V o l . 68, No. 12, pp. 520-544. Brooker, E.W. and I r e l a n d , H.O. (1965). \"Earth Pressures a t Rest Related to St r e s s H i s t o r y , \" Canadian G e o t e c h n i c a l J o u r n a l , V o l . 2, No. 1, pp. 1-15. Coates, D.F. and Yu, Y.S. (1970). \"Three-Dimensional S t r e s s D i s t r i b u t i o n s Around a C y l i n d r i c a l Hole and Anchor,\" Proc. 2nd Congr. I n t . Soc. Rock Mech., Belgrade, V o l . 3, pp. 175-182. Coates, D.F. and Yu, Y.S. (1970). \"A Note on the St r e s s Con-c e n t r a t i o n s a t the End of a C y l i n d r i c a l Hole,\" I n t . Journal of Rock Mech. and Min. Science, V o l . 7, pp. 583-588. Crouch, S.L. (1969). \"A Note on the Str e s s Concentrations a t the Bottom of a Flat-Ended Borehole,\" J o u r n a l of the South A f r i c a n I n s t i t u t e of Mining and M e t a l l u r g y , V o l . 70, No. 5, pp. 100-102. Crouch, S.L. (1970). \" E r r a t a on A Note on the S t r e s s Concen-t r a t i o n s a t the Bottom of a Flat-Ended Borehole,\" J o u r n a l of the South A f r i c a n I n s t i t u t e of Mining and M e t a l l u r g y , J u l y 1970, p. 386. 5 0 . Cruz, de l a , R.V. and R a l e i g h , C.B. (1972). \"Absolute S t r e s s Measurements at the Rangely A n t i c l i n e , Northwestern Colorado,\" I n t . J o u r n a l of Rock Mech. and Min. Science, V o l . 9, pp. 625-634. Denkhaus, H. (1967). \"The General Reporter, Theme 4, R e s i d u a l Stresses i n Rock Masses,\" Proc. 1 s t Congr. I n t . Soc. Rock Mech., V o l . 3, p. 316. Desai, C.S. and A b e l , J.F. (1972). INTRODUCTION TO THE FINITE ELEMENT METHOD. A NUMERICAL METHOD FOR ENGINEERING ANALYSIS. New York: Van Nostrand Reinhold Co., 477 pp. F a i r h u r s t , C. (1968). \"Methods of Determining I n - S i t u Rock Stresses at Great Depths,\" T e c h n i c a l Report No. 1-68, M i s s o u r i R i v e r D i v i s i o n , U.S. Army Corps of Engineers, Omaha, Nebraska 6 8102. G a l l e , E.M. and Wilhout, J.C. (1962). \"Stresses Around a Wellbore Due to I n t e r n a l Pressure and P r i n c i p a l G e o s t a t i c S t r e s s e s , \" Soc. of Petroleum Engineers J o u r n a l , V o l . 2, pp. 145-155. Gray, W.M. and Toews, N.A. (1967). \" A n a l y s i s of Accuracy i n the Determination of the Ground St r e s s Tensor by Means of Borehold Devices,\" Proc. 9th Rock Mech. Symposium, Golden, Colo., AIME, New York, pp. 45-78. Hawkes, I . (1968). \"Theory of the P h o t o e l a s t i c B i a x i a l S t r a i n Gauge,\" I n t . J o u r n a l of Rock Mech. and Min. Science, V o l . 5, No. 1, pp. 57-64. Hendron, A.J., J r . (1963). \"The Behavior of Sand i n One-dimensional Compression,\" Ph.D. Thesis, Department of C i v i l E n gineering, U n i v e r s i t y of I l l i n o i s , Urbana, 281pp. Hiramatsu, Y. and Oka, Y. (1968). \"Determination o f the St r e s s i n Rock Unaffected by Boreholes o r D r i f t s , From Measured S t r a i n s or Deformations,\" I n t . J o u r n a l of Rock Mech. and Min. Science, V o l . 5, pp. 337-353. Hoskins, J.R. (1966). \"An I n v e s t i g a t i o n of the F l a t j a c k Method of Measuring Rock S t r e s s , \" I n t . J o u r n a l of Rock Mech. and Min. Science, V o l . 3, pp. 249-264. Hoskins, J.R. (1967). \"An I n v e s t i g a t i o n of S t r a i n Rosette R e l i e f Methods of Measuring Rock S t r e s s , \" I n t . J o u r n a l of Rock Mech. and Min. Science, V o l . 4, pp. 155-164. Hoskins, J.R. and H a l l , C.J. (1973). \"A Comparative Study of Sel e c t e d Rock S t r e s s and Property Measurement Instruments,\" presented a t the AIME Annual Meeting Chicago, I l l i n o i s Feb. 25 - Mar. 1, 1973. 51. Jaeger, J.C. and Cook, N.G.W. (1964). STATE OF STRESS IN THE EARTH'S CRUST, ed. W.R. Judd, London: Methune & Co. L t d . , pp. 381-395. Kiyama, H. (1972). \" S t r a i n A n a l y s i s of a Borehole Bottom i n an A n i s o t r o p i c Body.\" Unpublished p o s t - d o c t o r a l research, U n i v e r s i t y of B r i t i s h Columbia. Leeman, E. (1964). \"The Measurements of S t r e s s i n Rock,\" J o u r n a l of the South A f r i c a n I n s t i t u t e of Mining and M e t a l l u r g y , V o l . 65, No. 2, pp. 45-114. Leeman, E. (1967). \"The Determination of the Complete S t a t e of S t r e s s i n a S i n g l e Borehole - Laboratory and Under-ground Measurements,\" I n t . J o u r n a l of Rock Mech. and Min. Science, V o l . 5, pp. 31-56. Lieurance, R.S. (1932). \"Stresses i n Foundations a t Boulder Dam,\" T e c h n i c a l Memo. 346, U.S. Bureau of Reclamation, Denver, Colo., 12 pp. M a r t i n e t t i , S. (1970). \"Lessons Drawn from F i e l d Experience i n Rock S t r e s s Measurements,\" Proc. 2nd Congr. I n t . Soc. Rock Mech., V o l . 4, pp. 390-394. Obert, L. and D u v a l l , W.I. (1967). ROCK MECHANICS AND THE DESIGN OF STRUCTURES IN ROCK, Chapter 9, New York: John Wiley and Sons Inc., pp. 236-274. P a l l i s t e r , G.F. (1967). \"The E f f e c t of a T r i a x i a l S t r e s s F i e l d a t the F l a t End of a Borehole D r i l l e d P a r a l l e l t o one of the P r i n c i p a l S t r e s s e s , \" Transvaal and Orange Free State Chamber of Mines, P r o j e c t No. 107/65, Research Report No. 73/67, Johannesburg (P.O. Box 809). Panek, L.A. (1966). \" C a l c u l a t i o n of the Average Ground S t r e s s Component From Measurements of the Diametral Deformation of a D r i l l Hole,\" U.S. Department of the I n t e r i o r , Bureau of Mines, RI 6732, 41 pp. Roberts, A. (1968). \"The Measurement of S t r a i n and S t r e s s i n Rock Masses, Chapter 6, ROCK MECHANICS IN ENGINEERING PRACTICE, eds. Stagg and Z i e n k i e w i t z , London: John Wiley & Sons. Timoshenko, S.P. and Goodier, J.N. (1970). THEORY OF ELASTI-CITY, 3rd ed. Engineering S o c i e t i e s Monographs, New York: McGraw H i l l , 567 pp. 52. van Heerden, W.L. (1968). \"The E f f e c t of End o f Borehole C o n f i g u r a t i o n and Str e s s L e v e l on St r e s s Measurements u s i n g Doorstoppers,\" Rock Mech. Div., Nat. Mech. Eng. Research I n s t . , C o u n c i l f o r S c i e n t i f i c and I n d u s t r i a l Research Report MEG 626, P r e t o r i a , South A f r i c a . Varga, R.S. (1962). MATRIX ITERATIVE ANALYSIS, Englewood C l i f f s , New Jersey: P r e n t i c e - H a l l Inc. Voig h t , B. (1966). \" I n t e r p r e t a t i o n of I n - S i t u S t r e s s Measurements,\" Proc. 1st Congr. I n t . Soc. Rock Mech., Lisbon, V o l . 3, pp. 332-348. Wilson, E.L. (1965). \" S t r u c t u r a l A n a l y s i s of Axisymmetric S o l i d s , \" American I n s t i t u t e of Aeronautics and A v i a t i o n , V o l . 3, No. 12, pp. 2269-2274. Z i e n k i e w i t z , O.C. and Cheung, Y.K. (1967). THE FINITE-ELEMENT METHOD IN STRUCTURAL AND CONTINUUM MECHANICS, London: McGraw H i l l . Z i e n k i e w i t z , O.C. (1971). THE FINITE-ELEMENT METHOD IN ENGINEERING SCIENCE, London: McGraw H i l l , 521 pp. 5 3 . DEFINITION OF TERMS MID SYMBOLS u - Displacement a t any p o i n t i n the x d i r e c t i o n . v - Displacement a t any p o i n t i n the y d i r e c t i o n . w - Displacement a t any p o i n t i n the z d i r e c t i o n . h - In t h i s r e p o r t , a d i s t a n c e on the f i n i t e - e l e m e n t model i n the z d i r e c t i o n from the o r i g i n on the bottom of the borehole to the top of the c y l i n d e r . U - Forces at any p o i n t i n the x d i r e c t i o n . V - Forces a t any p o i n t i n the y d i r e c t i o n . W - Forces a t any p o i n t i n the z d i r e c t i o n . A, - S t r a i n measured i n the gauge leng t h when one u n i t of s t r e s s i s a p p l i e d i n the x d i r e c t i o n . A_ - S t r a i n measured i n the gauge le n g t h when one u n i t of s t r e s s i s a p p l i e d i n the y d i r e c t i o n . A_ - S t r a i n measured i n the gauge leng t h when one u n i t of s t r e s s i s a p p l i e d i n the z d i r e c t i o n . A^ - S t r a i n measured i n the gauge le n g t h when one u n i t of shear s t r e s s xxy i s a p p l i e d . Ag - S t r a i n measured i n the gauge leng t h when one u n i t o f shear s t r e s s - x y z i s a p p l i e d . A, - S t r a i n measured i n the gauge leng t h when one u n i t o f shear s t r e s s xxz i s a p p l i e d . a - S t r e s s component. T - Shear s t r e s s component. e - S t r a i n component. 54. FIGURE 1 : S T R E S S TENSOR .ACTING ON A CUBE 5 5 . FIGURE 2: I L L U S T R A T I O N OF E L A S T I C CONSTANTS FOR A TRANSVERSELY ISOTROPIC MEDIUM THE DOORSTOPPER 4 Connector Pin s Moulded P l a s t i c Body Rubber C a s t i n g F o i l Rosette S t r a i n Gauge Alignment Notch INSTALLING TOOL -Dummy Gauge attached to p iece of rock E l e c t r i c c a b le -Doorstopper •Spring E l e c t r i c a l Plug I n s t a l l i n g Rod FIGURE 3: THE DOORSTOPPER AND INSTALLING TOOL 5 7 . A standard BX borehole is d r i l l e d £ r £ E £ E £ E £ H 3 ^ = - ^ to the depth in the rock where ^^^-^L^L^i the s t ress determination is ~~~. '•-de s i red . The end of the borehole is ground f l a t , and po l i shed. It is then c leaned, dr ied and inspected. 3. The doorstopper is i n s t a l l e d . 4. I n i t i a l readings are taken of the s t r a i n gauges. 5. I n s t a l l i n g equipment is removed from the borehole. 6. The doorstopper is overcored using a standard BX diamond b i t . a The rock core with the door-stopper attached is removed. 7. The s t r a i n r e l i e f due to s t re s s r e l i e f is measured. FIGURE 4: THE FIELD PROCEDURE FOR THE DOORSTOPPER 5 8 . STRESS 1 > 5 CONCENTRATION FACTOR \"a\" i . o 4 S T R A I N CONCENTRATION FACTOR \" A \" 1 . 5 H 1 . 0 i 0.1 0 .2 0 .3 0 . 4 POISSONS RATIO van Heerdon (1968) — de l a Cruz (1972) -— Crouch (1969) — Hiramatsu and Oka (1968) Coates and Yu (1970) P P a l l i s t e r (1967) 9 G a l l e and Wilhout (1962) — Bonnechere & F a i r h u r s t (1968) FIGURE 5 : VALUES OF STRESS AND S T R A I N CONCENTRATION FACTORS \" a \" AND \" A \" FROM VARIOUS AUTHORS 0.5 STRESS 0.1 0.2 0.3 0.4 0.5 POISSONS RATIO van Heerdon (1968) de l a Cruz & Raleigh (1972) Crouch (1969) Hiramatsu & Oka (1968) Coates and Yu (1970) p P a l l i s t e r (1967) g Galle and Wilhout (1962) Bonnechere S Fair h u r s t (1968) FIGURE 6: VALUES OF STRESS AND STRAIN CONCENTRATION FACTORS \"t>\" AND \" B \" FROM VARIOUS AUTHORS 60. -0.5 4 S T R E S S C O N C E N T R A T I O N F A C T O R \" C \" -1.0 S T R A I N - O . S C O N C E N T R A T I O N F A C T O R \" C \" -1.0 x 0.1 0.2 0.3 P O I S S O N S P %ATIO 0.4 0.5 • van Heerdon (1968) de l a Cruz & R a l e i g h (1972) —- Crouch (1969) Hiramatsu and Oka (1968) Coates and Yu (1970) P P a l l i s t e r (1967) g G a l l e and Wilhout (1962) Bonnechere & F a i r h u r s t (1968) F I G U R E 7 : V A L U E S O F S T R E S S A N D S T R A I N C O N C E N T R A T I O N F A C T O R S \" C \" A N D C F R O M V A R I O U S A U T H O R S 61. 1443 1456 I 10 - nodal number FIGURE 8 : FINITE ELEMENT GRID USED TO MODEL BOREHOLE I n i t i a l i z e [K] and {R> Start with f i r s t element I Compute element matrix .fel\" and vector {Q,} Convert so that degrees of freedom same in loca l and global coordinates I Use d i r e c t s t i f f n e s s procedure. Add k and {Q} to K and {R> NO Add external loads to 1 Continue) FIGURE 9: ITERATIVE LOOP FOR DIRECT STIFFNESS METHOD (AFTER DESAI AND ABEL, 1972) FOLIATION OR PLANE OF ISOTROPY FOLIATION OR PLANE OF ISOTROPY = 30° FOLIATION OR PLANE OF ISOTROPY = 60° FOLIATION OR PLANE OF ISOTROPY = 90' FIGURE 10: FOLIATION ANGLES ELEMENT ANALYSIS CONSIDERED IN THE .64, LOAD CASE 1 C^) LOAD CASE 2 (J FIGURE 11:. THE FOUR LOAD CASES APPLIED TO THE BOUNDARY \\ OF THE MODEL FIGURE 12: STRAIN CONCENTRATION FACTOR \"A^ FOR VARYING FOLIATION ANGLES '\". tr tr FIGURE 13: STRAIN CONCENTRATION FACTOR A 2 FOR VARYING FOLIATION ANGLES 67 . i o O o o c C O 3 U J o O vo o n o l o vo I o o O I CM I I STRAIN CONCENTRATION FACTOR? o U J . U J C O o 03 L U U J cc C D U J (=1 s •rH r-l o n o o cn FIGURE 1/4: STRAIN CONCENTRATION FACTOR\"A 3 ' FOR VARYING FOLIATION ANGLES 6 8a. FIGURE 15: STRAIN CONCENTRATION FOR VARYING FOLIATION ANGLES FACTOR \"A4' I / / . • - . O / clockw -i / / •- d anti 1 i. if 1 l\\ li i\\ - f 1 l i — A i 1 l \\ \\ N - -\\ y-V \\ \\ \\ \\ \\ \\ 1 i 1 i i \\ o o o o o . • • • • «tf £ 3 o II II FIGURE 16: STRAIN CONCENTRATION FACTOR A 5 FOR VARYING FOLIATION ANGLES 69. FIGURE 1 7 : STRAIN CONCENTRATION F A C T O R ' ^ FOR VARYING FOLIATION ANGLES f/ o — > — 1 d i, $1 i [• * 11 <*r 1 » V / / i t 9 t jdock\\ i. it/ * j / t c CO T3 — a — 1 i ft 'Ii measun : i j ! t i i i Mi • 11 ! Ii 1 1 1 ' 1 1 I 1 1 » I • i \\ t I % I i * i \\ * i » i t » % \\ \\ * i \\ \\ • \\\\ • — •'• % \\ * \\x i \\ V V % i \\ , * V M % \\ t1 \\ 4 A \\ 1 'i o O O n o m l o I o o • • CN MH f ^ 0 CO VD Cn in rH in CN ^ CM id • • u O a C O 0 o •o 0 M -d o 0) +J <+H U 4-> XI cu 0) w in cn cu in •H cu rH CD K rH rH 3 • 0 • ,q o i—I c o II rd •> > > P r - .../ o o CN I O I o CTv I S T R A I N - C O N C E N T R A T I O N ^ F A C T O R } o o L U . IXl CC f— C O o cc • to L U L U cc C D L U F I G U R E 1 8 : S T R A I N C O N C E N T R A T I O N F A C T O R A X F O R A N I S O T R O P I C M E D I U M 71a > I 1 4 \\ ' 0) in . \\\\ i % \\ \\ \\ lockw! % \\ * \\ * \\ \\ * \\ %. o C a A \\ v \\ \\ \\ » ».\\ * \\V\\ < \\ \\ - a o i_ 73 l/> JO \\ \\ \\ \\ i A * \\ i V * me ¥ b 1 • Cn * • V it \"/ * / f/ l ' i i 1 * • tig i f j I / * i l * H * »n ' i y / l l t t j * ii ' t ff i * f * i f * i f i -' f i f t # i •II / i ' i i t !!, i l l - -o «W O > r - \"7 o o 10 o n o r o i o I o CN o o o CXl I— o HI . U J I-o DC U L . CO LU . LU CC CD . U i <=> CM I I STRAIN CONCENTRATION FACTOR i FIGURE 19: STRAIN CONCENTRATION FACTOR A 2 FOR AN ISOTROPIC MEDIUM 71b. I 1 t i • o / / * // / • th Vi t i r I I 1 J I CD II VO ockwi s I 1 ( • I Hi I • I 1 % \\ !\\ i i t t i i i \\\\\\ Ml \\ i t i l i i i i i I i j i j ' / ' r i L * ft / 1 J / r i / • / < u ' / ' A -o o d) tn i n C CN ffl • U o •d o cu •p >c to 10 i n •H cu rH rH • •9 al o II > •> o CN o ON O VO O CO o r o I o VO I o o o c n o L U CC-. L U »—« oo o cc V) L U . L U CC C O . L U CN I I STRAIN CONCENTRATION FACTOR • < FIGURE 2 0 : STRAIN CONCENTRATION FACTOR A 3 FOR AM ISOTROPIC MEDIUM TO A, APPENDIX A MATHEMATICAL FORMULATION FOR FINITE ELEMENT TECHNIQUE from H. Kiyama (1972) 7 3 . APPENDIX A THE FINITE ELEMENT FORMULATION As stated i n the text the element was chosen from Zien kiewitz and Cheung and i s a general s i x sided, l i n e a r displacement element. To f i n d the s t i f f n e s s matrix of t h i s element, the Examine the Displacements To ensure continuity between elements, the displacement function was made to vary l i n e a r l y within the element. Generally then, the displacement function stated as: u = 1/8 (1-5) (1-n) ( l - O u , + U+0 ( l - c ) u + U+0 (l+n) (l - e)u + (l-£) (l+n) (l-C)u + {1-0 (l-n) ( ln)u' + (l+0 (l-n) (i+e)u + (l+0 (l+n) (i+?)u + (1-5) (l+n) (i+c)u where 5 = x / a n = y/b following procedure was used. = z/c (1) u = N. ' l u. l S i m i l i a r l y (2) V = N. ' V. x X W = N . ' W. x X 74. where u^, V^, and Vi\\ are displacements of the node i Therefore, displacement within the element where 1 i s a three by three i d e n t i t y matrix and = 1/8 (1-C) ( l - n ) (1-C) etc. e and 6 i s the displacement at the nodal points. Examine the St r a i n Vector Very simply, the s t r a i n vector i s the displacement vector acted on by an operator matrix. So i n general, {e> = [ L l ' { 6 } In t h i s case X \"3/3 x 0 o -y / 0 3/3 y 0 / M 0 0 3/3 z Y xy 3/3 y 3/3 X 0 Y yx 0 3/3 ' z 3 / 3 y V Y z x 3/3 z 0 3/3 x_ (3) Since we know {5} = We can f i n d values such that [ V V B. {6}' (4) 75. Examine the Stress Vector The stress vector i s re l a t e d to the s t r a i n vector by the c o n s t i -tute matrix. In t h i s case, fa \\ x a y {a} = J°Z \\ = [D]{e> .... (5) / T y z \\ \\ T z x J where CD] i s a s i x by s i x matrix i n the x-y-z coordinates Formulation of the Element S t i f f n e s s Matrix By the ap p l i c a t i o n of v i r t u a l displacement to the nodes of the element and some matrix manipulation, the s t i f f n e s s matrix can now be found. In general by conservation of energy, we see that {A } T {F} e = /{Ae} T {a} dV ( 6 ) volume From t h i s we can see that ' {F} e = LK] { 6 } e . . . . (7) where DO = ^[B] T [D] [B] dV (8) Ox] i s known as the s t i f f n e s s matrix and i s the basis for f i n i t e element manipulation of structures. \"BUILDING\" THE ELEMENT STIFFNESS MATRIX 76. As seen i n the previous section, the s t i f f n e s s matrix of an element can be expressed as, Therefore, to obtain the element s t i f f n e s s matrix, we have only to consider the s t r a i n displacement r e l a t i o n s h i p [B] , the s t r e s s -s t r a i n r e l a t i o n s h i p [D] , how they d i f f e r from element to element, the product, and the volume i n t e g r a l . Strain-displacement Relationship a p a r t i a l d e r ivatives with respect to x, y, and z. The d i s p l a c e -ments however, were expressed i n terms of the area coordinates £, n, and £. So a Jacobian matrix i s needed to transform from one system to another, such that [D] [B] dV In the previous section the operator matrix was shown as (9) where [JJ = ~3x/3£; 3 x / 8 n dx/dr, 9 y / 3 £ 3 y / 3 n 3 y / 3 ? 3 z / 3 ? dz/dT) (10) Since x N. 1 x. x 1 y N. ' y. I I (11) z N. * z. 7 7 , where x., y., and z. and p o s i t i o n coordinates of the 1 1 i c node i , we w i l l know the value of the Jacobian matrix f o r each element. From equations ( 3 ) , ( 9 ) and ( 1 3 ) we can e a s i l y see that, {£} = [J] fa/Hi < 3/3TI, . ( 1 2 ) or. •{e} = 1 [JJ ' 1 1 ' 2 1 3 1 1 2 ' 2 2 3 2 ' 1 3 ' 2 3 ' 3 3 3/3-n 3/3? ( 1 3 ) Taking the inverse of the jacobian and operating on i t with the shown vector i n equation ( 1 3 ) w i l l give {e} with respect to {e} e as shown i n equation ( 4 ) . The B ± i n equation ( 4 ) w i l l look l i k e , A. l 0 0 0 B . l 0 0 0 c . 1 B . l A. l 0 0 C i B . l C. L 1 0 A. i J (14) where A ± = J i ; L (N ±') ,5 + j 1 2 (N^ ) ,n +' J 1 3 ( N ^ K C B. = j 2 1 = [5] {e} {a} = [D] {e} (a> = [A] {5} {£} = [x] T {e} .... (22) Then, by the e q u a l i t y o f work. { c } T {5} = { e } T {a} So t h a t , [D] = [X] [Dj [ X ] T So we can see t h a t the e l a s t i c m a t r i x can be found i n any c o o r d i n a t e system. M u l t i p l i c a t i o n and I n t e g r a t i o n We know t h a t , [K] = X K T M K d v and we can f i n d v a l u e s f o r both [B] and [D] . The m a t r i x [K] i s a twenty-four by twenty-four but may be broken up i n t o 64 i n d i v i d u a l t h r e e by t h r e e m a t r i c e s . Each of these t h r e e by t h r e e p i e c e s [KrsJ can be i n t e g r a t e d and handled s e p a r a t e l y , so t h a t , [Krs] = ^ [ B r ] T [D] [ B j abs |j| d ^ d n d? where (Bf] has been g i v e n i n e q u a t i o n (14) . I n t e g r a t i o n of these terms i s l e n g t h y and w i l l n ot be gone i n t o here. APPENDIX B COMPUTER PROGRAM FOR STRAIN ANALYSIS OF THE BOREHOLE BOTTOM IN A TRANSVERSELY ISOTROPIC BODY by H. Kiyama (1972) 82. Q * » * « « * • * • » « * » » * * * * « » * * * * 4 * * » » * * * * » * * * * * * * * * * * * * C STR/IK ANiLYS CF THE RCREHCLE eCTTCC IM ANYSCTROPIC BOCY CIVEr»SICr-: f S I M 1 2 J ,F CGS( 12 J ,Ffl ( S ) ,FCR I <5) ,FQZ I 12 ), S 1( a ) ,S2 (8 ) ,S3 (8) l,NS(26),IV (6,2),CVl(S,6),C k'2<6,6),NM27 ) ,ALm,A2ie) ,A2ie>,OlrM6,6 \\ 2 J.i E 1 (e.),B.2.i.S.L, E.3 (.8.).j.A.e.C.( 3_,JJ_,..__S__t.3].. SKX't2., SIC tJEtL.lt . lU.Uf S IM.U_,I_.1,.1J_ _ _ _ FE*C (5 ,10 ) IFCCSI I) .1 = 1, 13J FEAO IE-, 1C ) (FR ( I ) ,1 =1 ,<5) RE/C15.1GI IFCR(I),1=1,9) . _ FE* C IE .10MFCZ (I ) , 1=1 ,12) FEADI5,2C) IS1C I) ,1=1 ,8) _F.£/C IE ,20.) (S2.l.lL,i=l...e.J : _ J u ~ : c E *C I; , 2C ) ( 5 2( I) , I=1 , 8 ) FSAC(5,2C)ns i I) ,1=1 ,36) PltC (5 ,4C ) (I I M I , J ) , 1=1,6 ), J=1,2L _ _..;._„ 13 FCF i -£T<6F l i .C ) . . 20 FCFV*TItFA.0) 20_E.CFr,AJ.{lB.I.2l ! 40 FCFVATIieiZ) 50 F CF v AT I 2F1Z.0»/»2F12.0»/, 2F12.C) ..: . 6 0 FCFV/TI2F12.0J . . : 61 FCP\"ATt2I?) 62 FC*V*T (2F12.C, 15 ) Q * * * * * _ * * * « • * * » « » » * * ^ * « * _ * _ * ^ *_* ,**»' ,****TFAr^SVERSELY ISOTROPIC********** Fc*C(5,50)E12,E13,C12,G23,G21,P12,P12,P22 FEACIE ,6C) SIA.CCA .... FE/C . ... . . . „ _ . _ 7.L \" ' . [-'J 1C1 1=1, e~~\" . . . . . . CC IC1 J= l , f 101 c n I,J)=0.0 : : CCCNST=l.U/(tl3*(E12-2.C*F12*P22*P12*E12*E13-P12**2*E12*E13 1-F23**2*£ 1 2-F 12**2*E12**2)) .._ C. V 1 ( 1,1) = £ 12 * i i 12 - F 2 3 * * 2 *.E 12 ) ^ CM ( 1,2 ) = E 13* ( F12*E12*F 12*P22*E 12) CV 1( 2, 1)=0M( 1,2) Cl» 1(1,2)=E13*IF12*F22+P13)*E12 r\"i<3,i )=c-ni ,3) C v 1 { 2,2) = E 13*1 l.C-P13**2*E12) CKit2.3) = E.13* ( F23+P12«P 12=-CLf<2,S) O 1 C ELASTIC MATRIX IN X.Y.Z CCCROINATES | CC 1C3 1 = 1,6 DC 1C3 J=I ,6 C | Off=0.0 ' C C 1C4 IP=I,6 DC 1 C 4 IQ=1,6 £JL£=JZ¥J*£V1{ IF I IC )»DLH ( I. IP )»CLM J . IC) 1 0 4 C C I V T I M E Cf2( I. J ) = CCONST*CKr* _IO.3_C.CNT. IN UE CC 1C5 1=2,6 11 = 1-1 _C._.C__J = 1.,JLL DP2< I ,J)=Dr2(Jt l ) 105 CCNTINUE fcJ?.IT.EJ6 ,.16.)t 12 »E.13j.G1.2.f.i:_2.2J.G.2.1jF12j PI3, P23.JS IA_,C0A ,NCASX»N0CA5E_ 1 ,ACC .RESICl.NSTGF 16 FCPw/sT Ur0,<,rE12=, F12.5,PITE. , . I= l . i6 ],.„.*.l,6) _ 1 7 FCPVATI1H ,45r-CM2< I, J ) : ELASTIC KATRIX IN X, V , Z-CCCRC IN ATES , l / ( l h ,fcel5 .5)1 C »»»4».*4»«4'»«44a4»4«'44444444iM44*»»444i<>»4»»«4444.4» C SETTING TFE STIFFNESS fATRIX j C »4*4*4*******4*>4>1>***4*44 4444 4444**4 444 4 4*4,* 4*44*4. C ; __CC 150_.J=lVt3i€ i CC 1 5 0 1 = 1 , 4 2 SK(I , J ) = 0 . C ( _ L 5 . 0 _ C . C N X 1 N . L E ; NNZ = C C C H O C N Z = 1 , 1 2 ..NNP=.C : N-R E = S M J = 1 2 0 1 1 0 3 N U = 9 1 1 1 0 2 N R E = t _ L 1 0 . L _ . C C N T I N I E _ C Z = F C Z ^ Z > D C 1 2 0 C N R = 1 , N F E : ) _ N N _ Z J J V N R _ P = F R ( N R ) C R = F C R ( S R ) -SICRCZ=SIC*0R**2/288.C/CZ-CZSICR=CZ/K4 .C/S IC/DR**2 CPPLCG= 0 . 0 O V LF-LNR.NE.5 ) CRPLCG°ALCG ( 1 . C 4 C P / B ) CC 120C NT = 1,12 C ELEMENT NCD6 NO TC ABSCLLTE LCCATICN NCCE NC N = N+1 NN(1 ) = N NN(2)=N+1 NN (_1=J__ : NN(4)=N*14 NN(5) = N*M NN(6 ) = NNC5 )•!.._ _ . ___ _ NN<7 ) = f>M5 1+13 NN«8 ) = NN( 5)*14 S_L_IS_VLN_) : SI2 = FSIMNT*1 ) CC1=FCCS(NT) CC2=FCCS(NT*1) C A l ( I ) , f 2 ( I ) , * * * * * * * * * ,63 (1 ) A1(1 )=-*(CCi-CC2) A 2 (4 ) = -A3( 2) . e3(4)=-E3(2) IF(NR.NE.9 ) GC TC 401 A 1 < 3 ) = C R / 2 . 0 » ( S I 1 - S I 2 ) : A1(4)=A1(3) A2(2) = A1(2) A2 (4 )=A1C2 ) A2(2)=C.0 A3(4)=C.O E l i 1A =L R /__. 0»(CC1-CQ_) : e i ( U = El (2) B2(2)=E1(3) E2(4 ) = E1(2 ) B2(2)=C.C B3(4)=C.C ____C.CN.T_I,\\L.£_.. : . OC 2C1 1 = 5,b A i d >=M< 1-4) AZ.il .) = »_.< .1-4.) 1 A2(I)=A3(1-4) e i ( I ) = E K I - 4 ) _2_11_.E_L_.4_) 85. C . ; : ! B3(I ) = e3(1 - 4 ) I 201 CONTINUE Q | C fr************************************************ I C STIFFNESS KATFIX I C STEF 1 ***INTEGFATICN PAIRI* AeC(3,2) N S S = I , : _ : >- 00 2C1 IR=1»E \\ SCSI=1.0 O 1 I F (NR. IvE.5 )_GC._T.C_30.2_ ! IF( IP.LT.3) GO TC 202 I I F U E . l T . 5 ) GC TC 203 O ' I F 11 P. LT. 7 ) GC TC 302 2C3 SCSI=0.5 302 CC 3C4 IS=IR,8 __SCSJ = 1.C .. IF(NP.NE.9) GC TC 305 IF (I S.LT. 1 ) GC TC 305 JJL(XJ..Al_5_)_*A3(IS))-(2.0*R J_!_1. . .CJ±3. . .C_TE.„.*C„ TErv = /2 ( IR )*B l ( lS ) + M(IR)*E2(IS)-(2.C+R/CR+l.C)*A2(IR)*B2(IS) A8C( l ,2 )= -CCNST l *U.a *TEP* ( (3 .C*A l ( IF)*ei ( I S )«A3( IP )*B3( IS) )-(2.0* JL.P./.CR_* 1...0.)_2• 0*TCf ).*P.PRLJC.CJ. „ TEME2 (IR)*A1 ( IS J*E1 ( IR )*#_ (IS J-(2.C*R/CR+l.C)*e2lIR)*A2(IS) AEC(2, l ) = - C C N S T l * ( e . O * T E f « ( ( 3 . C * B 1 ( I R ) * A 1 ( I S ) + B 3 ( I R ) * A 3 ( I S ) ) - ( 2 . 0 * 1B/CP+1 .0 ) *3 .0 *TEM»0RRLCG I TEr- = E2(IR)*Bl( IS) + e i ( I R ) * £ 2 ( I S ) - ( 2 . C * R / C R U . C ) * e 2 ( I R ) * e 2 ( I S ) ABC(2,2)=CCNSTl*(6.0*TEf*((3 .0*E 1 (IR )*B1( IS)*E3( IR)*83( IS))-(2 .0»R Jt/.CP • l.C.)*3..0.*TEV..)*CR.RL.CG) ABC( 1, 2) = S2( IS ) /48.0* (3.0*M (IP ) «S1 ( IS)*A2 1 IR )+S2( IS )*A3( IR ) ) 1 * S C S J 1*SCSJ AeC (3, l ) = S3(IP )/48.0*(3.0*A 1(IS )*S1( IR)*A2(IS)*S2(IR)*A3(IS)) 1*SCSI _ • ABC(2,2)=-S3( IR)/4E.0*(2 . C*E1 (IS ) + S 1 (IR)*B2( IS>*S2(IR)*e3(IS)) 1*SCSI 212 ASC( ;.2)=SICRC2*S3(IR)*S3 < I S)*( 3 . 0 * S 2 t I R ) » S 2 ( I S ) ) * t ( 2.0 * R / C E + 1 .0 )* 1 ( 3 . 0 * S 1 ( I R ) * S 1 ( I S ) ) + S 1 ( I R ) + S 1 ( I S ) ) I F 1 N R . N E . 9 ) GC TC 314 _.I.F ( SCS I . E C .0 . 5 ) GO T.C.31.2 : I F ( S C S J . E C . l . O ) GC T C 3 14 213 A8C(2.2)=ABC(3 , 3 ) * S O S I * S C S J * 3 . 0 / ( 3 . C * S 2 ( I R )* S2 ( I S ) ) _1A_CJCJSJ_IN.U.E , STEF 2 *****ELEr'ENT STIFFNESS f ATR I > * * * * * SKERS( I»J) DO 320 1=1,3 CC .22l._J=l.i2 . . TEF=C.C CC 222 ICC=l,fc . . LC^ lM lCC^J i I F (I C. ECO ) GC TC 222 DO 223 IPP=1,6 _I.P = IM IPP.t I) IF( IF.EC.C) GC TC 323 TE^=TEf + ABC(IP.IQ)*0H2(IP F,ICC) O v 323 CONTINUE 222 CCNTINLE SKERS(I,J)=TEr' 221 CCNTINLE 220 CCNTINLE C STEP 2 * * * * * E L E f E N T STIFFNESS MATRIX TO SYSTEM STIFFNESS MATRIX C »»«»»r'ATRI X»»»**SK( NX,NY ) CC 220 1=1,3 NX=3*NN(IP)-2*I „CC 321 J = l , 3 J : : NY=3*NS(NSS)-2»J SK (NY,NX) = SK.EPS(I,J)+ SK(NY,NX) 2 31 CCNTINLE : : 330 CCNTINLE NSS=NSS+1 2CV-CCNTINLE 3C1 CCNTINLE 13C0 CCNTINLE » &NR_=N_I+_3 : : '. : 1200 CCNTINUE NN? = NNZ.*NU _11CQ_.CCM.INLE C ECCNCARY CCNCITICNS CC 8e88 ICASE=NCAST,NCCASE c *****CCr'r'CN«***«l , 2 , 2 . « D C . ICC C 1 . A J 2 . 6 8 : F 11 I =0 .0 U(I)=C.C _1.CCQ_C.CI\\.T.I NX E : . r, ***»»»***»** GC TC, (eGCl,eCC2 ,8C03,800« ) ,ICASE _ C * + *. * * C A S E _1 ._.__ . ecoi F C C = I . C e C l l CCNTINLE N ti z = c , : . . CC 3000 NZ=1,12 CZ=FCZ(NZ) ... NU=130 . - . _ IF(NZ.CE.8 ) NC = 91 TErV=1.25*C2 N_M_. . CC 2100 NT=1,12 N = N+ 1 NN(11 = N _ , ' ., ' , : ' ••' _ _ _ NN(2)=N*1 NN<3)=N+NL N _ _ J = A _ i _ U : \\ TEr*l=TEP*( FSIN(NT+1)-FSIN (NT ) ) T6f'2=TEf*( FCCS (NT«-1)-FCCS CNT J ) CC. 3 200..1=1,A - 1 J=3*NN(I)-2 F(J)=F=F I . H I >-TFiv?*FGC : 2200 CCNTINLE 31C0 CCNTINLE NNZ = NNZ.*NU . . 3C00 CCNTINLE GC TC EC05 r *»*««CA.<;C ? 8CC2 C C N T I N L E FCC=-1.0 GC TC ecu C •••••CASE 3 80 0 3 CCNTIMUE N A E _ X : DC 3C03 NR=1,S P=FR(NR» CR = F CR (NR) . : T E r v=-0.0321*CR«(CR+2.0*P) N = NNF CC 3 1 0 3 NT=1.1Z : N = N*1 F ( 2 ^ N > = F ( 2 < N ) « T E f F ( 3*.N.t.3 ) =F (3*N *3 ).t.TtK _ F( 2*N*29) = F( 2^N*39)+TEK F ( 3 * N + 4 2 ) = F(3^N*421 *TE tV _U_3__CA.T__LE^ : : NNF=NNF +13 3C03 C C N T I N L E GC TC . EC05 C •••••CASE >> eCCA C C N T I N L E NNZ.=.C . . — CC 2004 NZ=1,12 CZ=FCZ(NZ> l\\L=12C _ : _ _-_ I F I N Z - C E . 9 ) NU = 91 TEK=1.25*CZ N = NN z . . : : : CC 3 104 NT=1,12 N = N* 1 N N I 1 ) = N _ - '. : N N ( 2 ) = N * 1 KKr 2r=N•nv KN L 4 J = N.tNL.:»l..._ : TE) DO 3204 1=1,4 J = 2*NM 11 ) _ J _ 3 204 F ( J ) = F U ) » T E M 3104- C C N T I N L E NN2=NNZ»NU 3CCA- C C N T I N L E NNR = C L\\C..„_C14 NR=.1.,.S : : - : F = FR (NR ) DR=FOR(NR) ____!£__= 0 ._Q.2.__7__ D RjLl.C R_2...Q_EUL : : -T£( v = - T E f N = NNR CC .3114 N.T=.l., 12 N = N+1 F( 2 » N - U = F <3^N - l ) *TEr -F 1 2 ^ N * 2 ) = F(3*N-*2 ) 4 T E r . ; F ( 3 * N + 3 S » = F ( 3 ^ N * 2 8 ) * T E V F ( 2 * N + 4 1 ) = F ( 2 + N * 4 1 ) * T E f _3114- _C C NT. 1N Li E _ - — Ni\\F = NNP + 13 3014 C C N T I N L E . C »•••»•••••*»•»»•»•<••••••<•••••••*•••••• ••.»••••• : e\\i v c r r - ^ i r= t m \\ N ; , »7g + ?M» »/N»TN + ( ^T+1M+N) =( S )NN — ^VM+VN* T vJ • (11 «1 >J '• vn = ( 3) MM : » * ^ S N * V N » TMM . I •1M*M> =( _>NN 9N + IN*(I+1M+M)=(9)MN TWTTVWr=TSTMN SMMN* ( I -TM*N) =( V) MN + ? I - 1 N + N ) = l £ ) N N 1 £ N , f T ^ t j E f -1 v• V) = ( Z) MM SN*CNMM*( trt-\" IN*N) =( T ) NN (Z.21NN « { 11 >JV * DN 33DN a i sY lDSa * 3L LZ M *3N 300>J 3 M 1 . 1 3 S 3 : — — . aO\"O0T\"=\"=9V_TTT',33''*\"rVTin 0OD0I-=5N ( T 3 3 ' l N ) d I D = 9M ; ^ ' — ' — — — • 0 = 5 ^ : [ I •M = N' - I * I = i N 3232 DO i _ 3 ^ N r x M D D — | 0O33 I -= iN I S *33*dN>3 I 0033T-=3V ( 3 * 3 D * « M ) d I — —•- — OO00T-=_M~TI*3D'*-aN) 31 ~ i 3902 3J. 33 (_ *3M*2N)d I 3TUI1M03 0832 _ D _ 3 . 0 T _ _ . . V N _ . 0303T = H»N(E *33*ZN )3 I 0802 31 33<3dN'3N * aM)d I 3 033T-=CM\" ( T-33'cJM) 31 1 3=VbM 0 = * 7 N : • — : : : —O=TM ~~i «vj>.*ZMW = M I 3tH« -MM 0T32 33 1032 I I 6 = n.M 2002 1S-=TN £002 \" rD\"02*2002*T3Cr2T3\"=7 W O l 0£1 = riM 3€T-=3N _ — _ or=\"3'aM 0 = dMN 3 0 3 0 I - = 2 M ( S f 3 3 ' Z N ) d I — — — CTOODI -= TM ( i 'Tr^rmn 0=6.M 0 = 3N 0=2M 0=TN n ' T W O J D Z~UTJ— 0 = ZMM I=SDSMM C _31S^*t33* OMI) a I — T O N I O M I 0*0=1M333 0*0 = r,nSf13 6662 : 3T=DN3VT 3 = 3MI 0=S03NN 33NI1M33 5003 * * * * * * * * * * * * * « * * * * * * * * * * * * * * * * * * * * * * * * • * « * * . * * * . * 3 c o r n s * o i i v a a i i * a e u n > n DMIMOS 3 89. O NN(12)= (N-12)»N3*N6 NNU3) =(N-1)+N5 NN(14)=N NN(15)=(N+1)+N« NNU6J =^ U-U.U=utU-i>-Arx*-rjj _ _ f DLSOS = ABS< L = CLS-f*Ae_(C_/U.(I.UJ : ; \\ 2C56 CCLNT=CCCN1*1.C 2020 CONTINUE ! tJ\\£=.hK£j>±3 : i 2C10 CCNTINLE 1 NNZ=NNZ+NU j 2C0Q ..CCNT INUE _ '. '. | DLSLf = CLSLtVCCLNT i IF(INC .NE. INCNC ) GC TC 2CIE i LNCNCJLIKCKCJLLC] _ i 2C57 WRITE(t,15) INC,CUSUM ,L(2 IS 2) ,L<259A ) ,U<2595 ) 15 FCFMAT ( 10X ,4i-IN3 = , 13, £i-TUSl.iv= ,F 12.5 ,2E15.6) I 2C58 CCNTINLE . : ' '. j IF(NNSCS.EC.l) GC TO 9E88 IFCCUSL'P .LT.RESICU) NNSCS=1 i _ G.c_xc__s-s.s . — : : ! fj * * * * * * * * * * * * * * * * * * * * * * ! C CLT PLT * * « * * » * L A'JC F * * * * * * * * L_ C * * + * * *.**.*-« * » * <*.•*•***.*** <.*..*.*.+*.***j*jta.*.*.****.*.**.*.*.* * : | 9B88 CCNTINUE I k>RITE ( £ ,11 ) ICASE, IN3 ,CUSLF ,CCLNT I 11 FCPMATtlH tinx^.PHTHSPLAChfFNTg ANC NCCAL FCPCES\" , / / , i 16HICASE = ,I3,4HNC=,I5,6hCLSUM=,E20. 1C,6HC0UNT=,F10.3,//, 25H I ,5X,lFL,l4X,lFV,14>,l l-k.,19X,2FFX,14X,2FFY,14X,2hFZ) L K R I TE ( £ »12.) t..(iV.ll I 3*.N- 2 ).,UA2 *.N-_1) .,.UX2*N. L,FJ 3* N.-.2 ) ., F(3*_iN_l) *F_3*I_L1». IN=1,1456) 13' F C P K A T f 1 5 , 2 5 1 5 . 6 , 5 / , 3 E 1 5 . £ ) : f- STRAIN f.n FF F I CI FNT : ! E 1=0.0 CO 4C2C JJ=1,1C E i=s l+iooo co.c : . : • ! N=e59 J = l • cc ACCC NR . : . . : . : . 1 = 896 ! R = FR (fiR ) I CC 4CIC .NT = 1,11 _______ : i SI = FSIN(NT ) CC = FCOS(NT ) I F 1 = U t_t.N-2 )/_£_! . : '. : : F2=U(3*N-1)/El F2=L(3*N)/E1 J F4=LM3*.I1/E1 : - - . — Tcf> = (R*CC + F1 )**2*( R*S I»F2 l*«2«( F3-F4 J + *2 T=f=SQRT(TEM) STFA IN(.I1= (TEf-R )/P • : : STPA(J) = (CC*F1*SI*F2 )/R 1 = 1*1 '• J = J*1 ; : j N=N+1 i 4C10 CCNTINLE 1 • AOnn rrNT I N U F ; _ : 91. © : , f W R I T E I f c , 1 2 ) £ l , ( S T R A I M J ) , . ; * l , 3 9 ) , < S T F M J ) , J = l , 3 9 l : 12 FORMAT(lh ,3F£ 1=,F15.3,/,tIH , '© ! 15HF = C.5 ,5X ,7E15.5,/, lOX.fcE15.5,/, ! 25rR=C.3,5X,7E15.5,/ , lCX,6E15.5,/ , 35r-P = C . l ,5X,7E15.5 , / , l C X . t E U . E ) ) o 4C20 CCNT INLE EE83 CCNTINLE O ' C * * * * * * * 4 4 * 4 4 4 * 1 4 * 1 * * * 4 4 + 4 * * * * * 4 + * * * + 9999 CONT INLE STCP C ENC , • G APPENDIX C NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: O* NODAL DISPLACEMENTS AND LOADS Load Case : \\. r/2a =0.1 nodal # u V w u V 885 O • O O - ^ - l ^ - b 8 2 o - o 886 o - o 2 - 6 2 . ^ 8 1 - o - o c i . V - U O - O 0 0 887 O - O 5 0 1 1 0 = 1 - 0 - 0 8 1 S 8 I O O - 4 . - 1 4 - 6 S 0 O - O 0 0 888 0 - 0 ~ 1 \\ 8 \\ 2 - _ — o o ~ A \\ _ 4 - 4 - b O O o - O 889 O - 0 ~ i S 5 8 3 ~ 0 - 0 5 0 8 i ^ - l 0 - 4 \" 1 ^ b O O o - o 890 o - o s _ v n _ - O - O Z.io'b OJ 5 0 0 0 • 0 891 o - i o i 6 o 3 - 0 - 0 O O O 3 1 1 0 - 4 - ^ 4 - 5 ^ 8 o - 0 0 0 892 0 0 0 - 0 893 0 - 0 8 1 W X 0 4 1 4 - 1 0 3 O - O o - o 894 0 - o n \\ 8 b i 4 - 0 - O H 8 1 0 1 o - 0 o - o 895 O O S O 8 I O 4 - 0 0 o - o 896 0 - 0 1 b i S 8 4 - 0 - O S S 1 0 4 4 0 --4-.-14--104- 0 0 o - o 897 O O o - i o \\ _ 1 \\ 0 - 4 - ~ i 4-~l04- ~ O O i O _ 5 ^ > 6 o - o . NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle : 0 ° Load Case: 2 . r/2a = 0.1 nodal # u V w u V 885 O O O-182 O b i - 0 0 0 0 \\ 0 » 2 b - o - o z v u 4-8 O - O 886 0'04-15 2 1 \\ O- 1 1 5 ^ 4 - 0 o - o o o i i ^ - i f o - o o - o 887 CD-OS \\ h^OS 0 - I 5 1 S - 0 0 ooofe 'S _2 O O 0 0 888 o- 12.94-2. _ o- u q ) 3 6 0 - 0 0 \\ l ( o 2 1 o - o 0 - o 889 0 \\ _ S 2 0 ^ 0 • 0 O \\ ^ 0 5 3 o - o 0 • 0 890 0 ' n b l 0 3 O 0 4 - ^ 4 - 3 O O O Z 3 S X X 0 0 o - o 891 0 - \" - 8 2 3 2 . 3 - 0 - 0 0 0 3 b H 0 - O O 2 b 0 2 S 4 0 0 o - o 892 0 - \\ l b 2 O 3 - 0 - 0 4-1^04- 0 0 0 2 - A - ^ c j T 0 0 o - o 893 O- \\ 5 8 X X 5 - 0 - O S X 0 4 - 5 0 O O 2 . O S c _ _ 0 0 o - o 894 ~ o - \\ i c n I O O O O 15 32.4/ o - o o - o 895 o - o o o ^ 6 o - o o - o 896 o 0 4 - 1 5 6 * 1 6 ~ 0 - \\ - i 6 3 5 X 0 - 0 O 0 5 V 1 \\ 0 o - o • o - o 897 O - O - O - *_ 2.4-4- 0 - 0 0 0 4 n \\ - 0 - 0 2 - \\ » 3 4 - 3 o - 0 NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 0 ° Load Case: 3 r/2a = 0.1 nodal # u V w u V 885 O O 0 0 8 X 0 2 8 ( 0 o - o n o o _ / - - o • o 886 - O - O H \\ b O S o - o n ^ 2 . 5 ^ 5 - \\ _ - i 8 S 2. o - o o - o 887 - 0 04-03801 o - o i u i n - i b - i 8 n o - o o - o 888 - o - 0 3 1 9 5 8 2 3 O O S 8 \\ 2 - c _ t - \\ b - i - 9 4 . o • o o - o 889 - O - 0 1 0 ^ 0 - 9 0 - 0 4 - » \\ R _ 2. - \\ ( o * l 8 S 3 o • o o - o 890 0 - O X \\ 4 - b 5 5 - l b - l & S o o - o o o 891 - O - 0 8 l S l b S 0-0002-b0l1 - \\ 6 - l 8 S _ O • o o - o 892 - O - O l S l O ^ O - O - O i O ^ b l © - i b ' i 8 ^ 4 - o - o o - o 893 - 0 - 0 1 0 U 1 8 - 0 * 0 4 - 0 1 6 b 6 - l b - 1 8 ^ 5 o • o o - o 894 - O ' O S l S 12b - o - O S i i 6 & 1 CD • O o • o 895 - 0 - 0 ^ - 0 ^ 5 5 5 - O - o i O b V ^ b - l b - i < 8 c l 4 - o - o o - o 896 -o - o 2 » 2 0 i 6 - o - O l _ S & 5 8 - i b - 1 8 ^ 4 - o • o o - o 897 O • Q - O - 0 8 \\~lh ~l b - 1 6 - 1 8 ^ ^ O - O l l O l l Q O - o . NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 3 0 ° Load Case: i . r/2a = 0.1 nodal # u V w u V 885 o - o i • q 4-5\" \\ ' i - 0 - 0 _ ^ 0 3 b o - o 886 O«o8\\-O~i4- 0 - 8 5 ^ ^ 0 o • o O O 887 o • o o - o 888 O- n n U \\ -. o o m o o • o o - o 889 o- \\bO X R S H l o a / _ o O • D o • o 890 0 - H 1 5 0 4 - \\« V03 \\-©8^><5>o o • o o - o 891 0 ' 0 4 - 4 - ^ O l O \\- 8 5 b ^ o • o o - o 892 \\ • 2 .3 _n S o • o o - o 893 O O o - o 894 - o - o c i o n 3 \\ - \\ 1- ifon 1 _> o - o O O 895 o - o o - o 896 l - o s i ^ s - o • o o - o 897 O-O \\ -cn84-Q \\. 1 6 4- 5 1 -0'0085£,4-73 o • o . NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle : 3 0 ° Load Case : CL . r/2a = 0.1 nodal #' u \" V w u V 885 O • O - O - 4 - 5 R 5 0 1 —O - CMA-b lobb o - o 886 0-0032>3q_j3 - 0 *4 - ~ i S 8 b 4 - . 0-Sbr300\\ O- O O O 887 O-O 1^)^4-0-1 - 0 -53SB01 O • O o - o 888 0-0 4r4- < S>5X\\ -0 - -62. b_4-^ 0 - 8 1 3.0 1 O- O o - o 889 0-0^-2.3^61 — 0-~V ~ - b H _) o • o o • o 890 0 - \\ 5 3 5 2.5 — O- <_xrifo;_ o - o o - o 891 - 0 - o S 5 b 4 - 2 O S O S O - 5 3 o • o o - o 892 0 - _ b 0 3 b b -0-q4>0q~\\6 o - o o • o 893 O-XT4-135 o ^ s s o o i o - o o - o 894 -0 - qb l4 -bb o • o o - o 895 O- \\8< i2.2.5- O - ^ - ^ l b ^ o - o o - o 896 0-\\01 b i b - 0 - ^ 4 - _ U 3 o - o o - (_ 897 o- e -0*0_>0311 \\ O • D • NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 0° Load Case : 4~ r/2a = 0.1 nodal # u V w u V 885 O O - 0 - ^ 1 3 1 1 s O- o 4 - \\ _Sb_ o - o 886 -O- 2 — O * b ° \\ 0 O \\ \\ o - o o - o 889 - 0 - 4 - X l b 8 b - 0 - 4 - E > ^ 3 i l O - O o • o 890 - o - x 4 - 3 4 c * 5 - O - X S ^ ^ S o - O o - o 891 - o - o o o o i n o ^ . - 3 - - - b l b S O • O o - o 892 O • X4-_4--l 5 ~ 3 - (oOXbS 0 - X 4 - S ' _ X l o - o o • o 893 0 - 4 - 2 . 1 - 8 4 - - 3 - 4 - X 4 - 1 0 o - o o - o 894 0 - 4 - ^ 8 1 ^ - 3 - \\ _ . \\_ \\ 0 -6 ftOXS^ o - o o - o 895 0 - 4 - X \\ b O S O - 8 3 4 - 3 1 0 o - o o - o 896 0 -2_4 -33R8 - 1 - l b O l O 0- c12>\\X4-X o - o o o 897 O O -ZL-bS 5 O O 0 - ^ b 4 - 3 1 8 - 0 - 0 4 l c i O l 4 - o - o . NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 2>0° ' Load-Case : o r/2a = 0.1 nodal #' u V w u V 885 O o - l _ - 4 - ^ 2 O- 0 4 - _ 3 \\ 0 _ o- o 886 - 0 - l l b b 5 0 - *-_*3<34-b o- O o • o 887 - o - i n ^ i o ? , - 2 . - l1b3 — \\X- b l X S o- o o • o 895 0 - 3 ^ 0 ^ 8 — \\ X - S ^ b D o - o o - o 896 - a - 5 i i - 4-2 o - o o • o 897 O • O -2 . -8\\X\\Z- o- o . NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: _>0° Load Case: 4 ~ • r/2a = 0.1 nodal #' u V w u V 885 O - O - _ • a S . S l O -02.b4-60 8 o-o 886 _ 0 . 1(34.4.-1:2. - 3 * I B ^ I X o - o o - o 887 - 0 - 1 3 8 0 4 - 0 -3-3-4-e>\\^ O • O o - o 888 -O- _ 3 4 - l 3 0 O • O 0 • 0 889 -0-2-82-A-X\\ - 3 - 0 5 1 3 2 ) O • O o - o 890 - 0 - 1 4 - 0 0 ^ 5 o - o o - o 891 0 - 0 5 1 0 1 0 8 0 • 0 0 • 0 -892 - 2. • S SR3>8 o - o 0 • 0 893 84-4-4- -2. - 3 ^ 1 2 5 o - o 0 • 0 894 0 * 3 ~ l b O _ ? 6 - •2 , • b V I 4-4- - 2 - 1 5 5 3 3 0 - 0 o - o 895 -_-4-8b4-8 o - o o - o 896 O - 1 1 1 X 5 8 i o b b ~ i 0 0 0 0 897 o - o -2.-__ba.oo - a - O S 1 8 b - 0 - 0 ^ 4 - 1 8 S 0 • o-NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: - 0 ° Load Case: \\. r/2a = 0.1 nodal #' u V w u v 885 o-o X-OA-4 -5 \\ - O - O 4 - 3 4 - S S 0 o-o. 886 _ _ X ^ _ L b * ~ l o- o o-o 887 a- __q4-S8 1 o • o o-o 889 O - _o8 _3,1 a • \\ i o n . o- o o - o 890 o- \\ s 4 - \\ 5 a X - S1 b b \\ 2 . ' 0 _ b02_ o -o o - o 891 O-ooi _ \" 8 b 0 , 0 3- oioqo 1 - S 8 0 o • o O • O 892 — O - m bi O 3 • 0__b1 8 I • 9 0 5 0 0 , o • o 893 —o-xi bi 3,q a-qqoxo \\ -84-bbo o-o o o . 894 •>.. - O - 0 « 1 3 S _ - • \\ - 8 \\ 4 - \\ S o-o o • o • 895 -O'X3b333 \\-803Ob o -o o • o • 896 -o- 103 _i~i 2.'- 835^0 o -o o • O • 897 o-o. X-_^_30q . \\ <-xqqb4- o- o . NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 6 0 ° Load Case: \"X. r/2a = 0.1 nodal # u V w u V 885 o-o - \\ - 2 . \\ _ ) l O o - s s n o b i - o - o o a s i 2 » S 2 O - O . 886 - o • o _ o S 5 1 3 0 - 5 ^ 2 5 1 1 o - o O - O 887 -0-04-532.52. O O . o - o 888 - O 02.8*14-65 o - b i ^ n s O O o - o 889.. O- O^ViO-ebbB - \\-631 OS 0 - 6 5 1 0 4 - 1 o - o o - o 890 0 . J34 -S4-1 -1-1 C*32 CV- o - 6 8 _ 6 ? 5 o - o o • o 891 O - 1 5 1 1 5 0 - l - q i 6 9 O o - n s O ^ l O - O o • o 892 0 - 3 5 1 _ b l ~ 1 - S 8 _ 3 2 . o * n o 6 5 3 o - o o - o 893 o - 3 ^ 1 8 ^ 2 - 2 - O I C 5 5 O - _,0»144- o - o o • o 894 0 - 3 b l 8 \\ 8 - 1 - 0 \\ 5 ^ 3 O - 8 . 5 1 4 4 - o - o o - o 895 0 - 1 1 8 1 2 ) 3 - a - o i b m o - 8 H 4 - S 3 o - o o - o 896 o - \\ 4 - 8 1 8 5 - a • 0114-8 O - 3 \\4 -Ob8 O O o - o 897 O O - 1 - O l b ^ S o- 8 . 3 0 2 4 - - o . o 3 ^ 0 B b o - o . ' NODAL DISPLACEMENTS AND LOADS ' F o l i a t i o n Angle: ^ ^ ' Load Case: 3. . r/2a = 0.1 nodal #' . u V w u V 885 o- o 0-031^4 - 0 8 o - o . 886 -0'i4 -~ i 94-b — b - 84-bO 1 o - o o - o 887 - 2 - 4 8 1 8 0 — b-SOSS^ o - o o • o 888 - O _\\ _54/- 5 61*5^2. - ^ , - 1 6 X 1 ^ O'O o • o 889 - 0-0-804-54r -2.-T64-fo2- -_b--ioi2>i o - o o - o 890 -0-15.5 64- O -2-888% -6-624^6 o - o o • o 891 0-O2.A-4-S8S. - a - w o n o - o o - o 892 o- \\ 9 3-2,4- - a - 4 o - \\ - i o _ b - 44 -S \\ b o - o o - o 893 O- 2 < 3 4 8 0 9 2 , . - \\ 8 4 0 q _ 0 - 0 886 o-o\\ fo5l 5 \\ -O- 3~140 S4> 3 - 1 8 5 0 9 O.O 0 - 0 887 0-O_X~i 02_b 3 * 1 8 b 5 9 O- O o - o 888 0-O4-1b2>99 - o - 2 - 9 5 6 4-8 3 - 1 8 8 2 2 . O * O 0 • 0 889 o - o b o ~ i o 4 - 2 - 0 - 2 3 2 5 0 C | 3 - i q o i b o - o o - o 890 O-Ol * O 159 - 0 - 1 5 1 8 5 0 3-194 - 0-1 O - O 0 - 0 891 -O- OnS88\\ I 3 - 1 9 1 9 j O-O 0 - 0 892 0 - 0 \" l 8 0 b 0 5 O-OO813014- 3 - © 0 1 8 * 1 O - O o - o 893 0 - 0 1 2 2 9 1 9 0 - 0 8 1 ^ 6 2 4 - ^ • 8 0 5 4 4 - 0 - 0 0 - 0 894 O ' O b 0 2 < 8 2 S O- 3 - 8 0 8 4 - 1 0 - 0 o - o 895 0-04-2>\\393 3 b 4 - 3 - 8 1 0 1 3 o - o 0 - 0 896 o-o'*- 2-4- 4-63 O - 2- 4-5 1 1 2 3 - 8 I Q . 3 2 . o - o 0 - 0 897 O-O 0 - 2 5 8 0 ) 1 3 - 8 1 3 1 3 -O- 02->0691. 0 * o-NQDAL DISPLACEMENTS AND LOADS ' F o l i a t i o n Angle: 9 0 ° ' ' Load Case: 2.. r/2a = 0.1 nodal #' u w u V 8 8 5 o - o 0-4-84-0 3 3 - l - O X 2 2 . 8 ) - 0 - © 2 . 3-1518 0 • 0 . 8 8 6 o-o5 8 ^ 8 ° 1 0 - 4 - ^ 8 5 0 3 — \\ • O 2 .2_4-3 0 • 0 0 - 0 8 8 7 0 - 4 - 2 3 2 c \\ 5 - » * O 2 . 2 . 4 - 2 0 - 0 0 - 0 8 8 8 0-35 V2._S — \\ - 0 2 . 0 . 3 O 0 - 0 o - 0 8 8 9 O - ^ 3 4 ^ - 6 o - 2 5 6 ^ 5 6 — i - O 2 .2-2 5 o - o o - o 8 9 0 0 « 2 _ l 4 - 4 - 0 8 0 - \\ 4 - b — \\ • 0 2 . 1 5 8 o - o O - o 8 9 1 O - l C L O ^ l 0 - 0 2 . 5 i 0 1 l — 1 * 0 2 3 6 0 0 - 0 0 • 0 8 9 2 O - 2 - ^ 8 1 8 - O - O R S o o o i . — 1- O-a.530 o - o o - o 8 9 3 O - 1 8 ^ 8 0 5 - o - 2 - o i b q o — \\* 02.131 0 • 0 0 • 0 8 9 4 O ' I 'S 8 ~0-304-2\\6 — I- 0 2 9 3 5 0 • 0 0 - 0 8 9 5 o- 1 0 ^ 8 8 2 -o- S 1 8 H 6 — 1 * 0 3 0 S O o - o 0 * 0 8 9 6 o- osfo^^q — o * 4-2.4-14-6 - i - 0 3 \\ 8 6 o - o o - o 8 9 7 o - O - 0 * 4 - 4-08 lO — \\* 03 2 . 2 . 8 -O- OX '_4-bbl 0 • 0 NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: S O ° \" Load Case: 3- r/2a = 0.1 nodal #' u V w u V 885 O-O - s - ^ s i ^ • 0 - 0 . 886 -o- o\\\\l315 O • \\ _ _ ~\\ \\ \\ -<3-^ 4-4-2.^ o- 0 0 * 0 887 -O- OX\"i.55\\4- o-\\v_ c\\38 -5-^4-61 8 0 * 0 0 - 0 888 -O - 0 3 3 5 3 3 4 - O- ^ 0 -5-<3 4 - 8 ^ 8 o- 0 0 - 0 889 -o-Q4-3b5 3 6 O-12.33b4- 0 - 0 0 * 0 890 -0-052.04 -04- o-\\o4-S4-~l - \" 5 - ^ - 1 5 0 - 0 o - o 891 -0-O514-4-15* 0-082.14-1 \\ - 5 * 9 ^ 0 ^ 4 - 0 • 0 0 - 0 892 -0-o_Bvq4-Z O * O B I 8 1 1 1 Q - 0 0 • 0 893 -o- 054.351© o-o^_b5854- - 3-^61 :2.4- . 0 - 0 0 - 0 894 -O-O4-53180 0-OM83b8 o - o 0 - 0 895 - 0 * 0 3 2 3 b H - 0 - O O 5 4 - H 2.0 0 - 0 o - o 896 - 0 - 0 1 ( 9 - 1 1 5 5 - 0 - 0 1 b B 3 l l — 5 , c\\\"\\054- 0 - 0 o - o 897 O- 0 -0-02_\\_ 4-q x - 0 - 0 _qnob9 0 - 0 0 8 014-15 0 - 0 NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 90° Load Case: Ar r/2a - 0.1 nodal # u V w u V 885 O - O - 4 - 8 8 6 6 9 — O - b 5 & 0 2 . 5 0 - 0 4 - S T & 8 0 0 - 0 886 -O - 2 . fc>8 4-4-9 -4--q2,\\ 3 9 -0-b4-2.53b 0 * 0 0 - 0 887 - S * & 2 - b b 9 - o - ^ X < 8 t S 4 - o - o 0 - 0 888 -O- b052.S5 - 5 - 2 0 0 2 - 4 . — 0--4-<=lblbS 0 - 0 0 * 0 889 - 0 - S 6 8 . 1 2 4 - -5 -4-19 3 3 — O - 34-01XO 0 - 0 0 - 0 890 -O- 3 5 D 0 1 5 - s - b l 9 X l — 0 - 1 2 b 3 IX 0 - 0 0 * 0 891 O- 0 0 0 8 8 ^ 2 - 6 - X 3-102_91 O - \\ 2 . b \" 5 i a 0 - 0 o - o 892 o- o s n o s -5-fo\\©5 fo O - 3 1 9 3 0 1 0 - 0 0 - 0 893 o-5b4384- - S* 4 -1814- 0 - B 9 3 > 6 6 8 0 - 0 o - o 894 O - b o b ] 6 3 - 5 - \\ 9 8 b 3 O - 1 4 - 9 0 . 2 . 6 0 - 0 o - o 895 O - 4-8833 \\ - 5 • 0 24-14- 0 - 8 4 5 5 3 3 0 - 0 o - o 896 © • 2.68591 - - 4 - 9 \\ q i 8 , O- 8 9 4 - R M - o - o o - o 897 O • O — 4 - ' 8 8 4 2 2 o-q\\oo8 \\ - 0 - 0 4 - ^ 8 6 K l 0 • 0 . APPENDIX D STRAIN FROM CENTRAL NODE FIRST NODAL RING STRAIN FROM CENTRAL NODE F o l i a t i o n Angle: 0° r / 2 a ~ °* 1 t o nodal # 0 a 2 a 3 835 - 9 0 \\• o \\<5<\\b - \\ - B X O f e ~ l O-h, • RA- I 4 - 3 886 - 7 5 t - o i 5 8 S - 1-5-164-^ 2-fo - a x ^ i b 887 - 6 0 1 -O lS I S - 0 - B 2 . 0 2 5 2\"?>- 3 3 \\ 4-5 888 - 4 5 \\ - o \\ _ 8 l 0»00?.03 - 0 - 8 2 0 2 1 IR - 0 4 - 3 ^ 0 889 - 3 0 \\ o \\ S 8 1 O- \\ °i - O - 8 2 0 O 1 ^ - 4 - b 8 b b 890 - 1 5 1-51^84/ - 0 - 8 \\ 9 8 2 6 ' 9 T \\ b \\ 891 0 \\-Ol 6 0 3 1 - 8 2 3 x 3 - 0-8«q\\b — O- OOO \\ 2-892 15 \\ - o i _ q 5 I - 5 1 8 0 0 - O - 8 1 S 4 0 893 30 l - O i ^ S O 0 - ^ » O 0 3 -13'4-T n 1 894 45 - o - o o \\ 8 \\ — l q • o s 4-0 b 895 60 -<_>• 8 1 8 0 S — _ _ - 3 3 S 4 - 4 896 75 \\ - O I 3 6 8 - f - 5 8 0 3 3 . - 0 - 8 H 8 3 - 2 b • 0 3 \\ b O 897 90 — \\- 824-4-1 ' - o - 8\\~}68 STRAIN FROM CENTRAL NODE F o l i a t i o n Angle: _>0° r/2a = 0.1 to nodal # 9 a l a 2 3 835 - 9 0 - £ - 3 8 2 b 4 a s - \\\\ \\ 0 1 886 - 7 5 - 8 - 0 8 0 4-B 4-- ^ 4 - 3 1 8 OA- - B 4 - 3 4 0 2 . x * 2 » O b 9 5 887 - 6 0 - 1 - 2 . 4 - 4 - 4 6 4-' 1 4 - 5 B 8 2.-X - 2-\\ 3 8 I \\ 9 * 9 3 4 - 8 8 888 - 4 5 - S - ° i O B 2 . 0 4-« \\ u © 3 \\ 8 * \\oc_» 2 . 8 \\6 -2 -4 -4 -OS 889 - 3 0 3. '8\"2>314- \\2. • 3 \\ 9 6 4 - 1 1 - 5 5 ^ 6 1 890 - 1 5 - \\ ' 4 i o o 5 3 - b 0 9 8 2> b - b 9 b l 2. 6 - X 2 . ^ 3 891 0 0 - 4 4 - 3 0 2 2.* \\ 5 1 0 2 O O l 14-ZL 0 - 5 2 0 X 1 892 15 o o nc \\ ' 5 0 - b - 1 3 1 \\ \\ - 5 - 2 . 1 3 4 - 3 893 30 5 ' 4 l ^ l O - 2 - 4 - 2 6 3 5 ~ \\ 3 - 3 3 0 3 2 . - i o - 8 1 5 \" 1 1 894 45 > f a U 5 5 - 5 * 0 3 X 8 1 — \\. - b\\998 STRAIN FROM CENTRAL NODE F o l i a t i o n Angle: b_° r/2a =0.1 to nodal # 6 K l a 3 a 4 885 - 9 0 - 2 . 1 - S ^ - J ^ © 886 - 7 5 l \\ • ° 1 3 ^ 8 \" 1 2.2,- 8 4 - - e ^ _ 4-4.-887 - 6 0 - \\ q . _-3.8-i b 2.0 2-*2.<5n b 3 8 * ^ 4 - O b 888 -45 - 1 5 * t > b l 0 5 l O - \\ b X , 4 - \\ \\ b * 2_bb2.-2_ 3 \\ - IS\" 889 - 3 0 ~ \\ \\ - O V 1 A- \\ U - 3 ^ 4 - 3 ^ 8 4 - ^ 2 - 4 -890 - 1 5 _ s* 1 1 0 0 3 \\ \\- 5 b ^ 3 b 891 0 O - 0 \\ 5 8 b 2 - 5 T 1 5 0 O* 2 .4 -4 -8 )^ 0- S 4 - 8 S 4 -892 15 - ^ 8 4 - 5 ^ - 5 - b 5 8 3 1 893 30 \\a- s^44-n - b - b O b ^ < _ - U - 5 3 . 4 - ^ 4 - - 2 . \\ - \\ns<=\\6 894 45 i©. b \"D4 - -\\ l - \\ \\ - b S - 4 - O l - t b - ^ S O l S - 3 \\ - 5 1 c \\ 5 b 895 60 — l b * O T 4 ^ b b -a \\ - 4-~l \\ 6 b ~ 4 - 0 - 2 . ^ 4 - 4 - b 896 X - i - 0 5 - b b \\ — W 10X3/_ - 2 4 - ' A - 3 1 O \\ - 4 - b - l b 2 . 2 _ 8 897 90 - l O - ^bc) 4 - 8 ' 12 ,8 - 4 - 8 -2.1^ n STRAIN FROM CENTRAL NODE F o l i a t i o n Angle: 90° r / 2 a = °- 1 to nodal # 0 °1 a 2 a3 835 - 9 0 3 -84 '2>09 - 4 - 8 4 / 0 3 - i — l - 5 8 b 4 l 4 - 8 - 8 b b © n 886 -75 3 - 6 5 5 9 9 - 4 ^ 3 * 1 2 1 b 4 - b - 84-2.29 887 - 6 0 3 - \\4-0 4 - 3 - \\ - 4 - 0 3 , 6 0 4 4 * 0 9 3 2 . 5 888 -45 2.- — \\ - x \\ 4 - 1 - l 3 2 ' 4 - 9 1 4 - 1 889 - 3 0 \\ - l o w i - 0-°V° v 2.2.- \\ - l b l 3 890 -15 t - b 9 2 9 4 - - o - n i 3 2 . b \\ \\ . \\ b l l 2 891 0 o - i - i ' b x . A 2.- i o a . q \\ - o - s ^ 4 - 4 - \\ O - 0 0 8 3 * 9 892 15 O ' l l S O S 1- 6 0 0 1 0 - O - 4 4 \"5 34 / - - V \\ - V 4 - 4 - 8 0 893 30 \\ - O b 5 9 3 O - b 0 5 3 O — O - 3 0 2 T 3 - 2 . 2 . - 1 5 9 ^ 8 894 45 \\ - 5 3 9 b 3 - 1 - O S \\ 4-9 — 3 - \\-3-14-5 - 3 2 ' 4 1 3 > ' B 6 895 60 . a * o 4 \" i b i - 2 . - - \\ 2 5 1 1 - O ' 2 0 8 ~ 1 2 — 4 \\ - 0 1 4 . 0 8 896 75 - • 5 - 9 5 5 3 5 - 0 - 2 . 0 b 0 5\" - 4 - b - 8 2 . 0 5 1 897 90 x - 5 8 0 \\ \\ - 4 ' 4 0 8 \\ o - 0 - 2 _ \\ 5 > 4 9 - 4 -8-84 - ' i \\ < 2> APPENDIX E STRAIN CONCENTRATION FACTOR TO FIRST NODAL RING STRAIN CONCENTRATION FACTORS F o l i a t i o n Angle: 0° r / 2 a = 0 • 1 6 A l A2 A3 A4 A5 • A6 -90 - 0 - 4 - 0 3 3 1 * 4 - 1 9 2 . - O - ©191 o- o o • o o - o -75.. - 0 - 2 . 8 I X 1 - 2 . 9 1 1 - 0 - 8 I 9 1 - o - 9 U 3 o - o O - O -60 0 - 0 5 2 3 1 - 0 6 3 6 - 0 - 8 1 9 1 - \\ - 5 1 S 3 o • o o - o -45 O 5 0 1 9 O- 5 D 1 9 - 0 - 8 1 9 1 - 1 - 3 X 3 o - o o - o -30 O - 9 6 3 1 O- 0 5 2 1 - O - 8 \\ ° i \\ - 1 - 5 1 3 3 o - o o • o -15 - O - 2 . 8 1 5 - o - 8 i 9 \\ ~ o - 9 n 3 o - o o - o 0 1-4195 - 0 - 4 - 0 3 1 - 0 - 8 19\\ o - O o - o o - o 15 \\ - 2 9 1 4 - - 0 - Z 8 \\ S — 0 - 8 1 9 1 O - S l \\ b o : o o • o 30 0 - 9 6 3 7 0 - 0 5 2 1 - 0 - S \\ 9 \\ \\- 5 1 8 9 o - o o - o 45 • 0 - 5 0 1 9 o - S O I 9 - o - 8 \\ 9 I \\ - 8 X 3 X o - o o - o 60 0 - O 5 2 3 I• O b 3 b — 0 - B 1 9 \\ 1 - 5 9 8 9 o • o o - o 75 - 0 - 2 . 8 I X 1 - 2 9 1 \" , - O - 8 1 9 1 O - ^V^ b o - o o - o 90 - 0 - 4 - 0 3 3 - 0 - 8 1 9 1 o o o - o o - o STRAIN. CONCENTRATION FACTORS F o l i a t i o n A n g l e : _ 0 ° r/2.a =0.1 e A l A2 '• A 3 A4 A5 A6 -90 \\-8 14-5 - i - >e>2 - 0 - 1 4 - 3 4 - -o - 2-_4-S' O- 5 2 0 2 . I -75 -O -4-1 51 1-6 5 0 -\\- » o 8 - \\ - 2 . 8 9 _ — o - \\ i e > O* 4-14 - 9 -60 -O- l 2 . 1 © H o i 5 - 0 - 8 8 1 - 2 - 1 2 . 5 6 - 0 - 2 , 8 O- 3 4 - 2 . -45 0-2>_4-2 . O- -O- 5 1 4 5 — a - 4 -2 .11 O- I b Z O- \\ b 2 _ -30 - o- \\ i f e 4 -O- 2 . 5 5 5 - 2 - \\ 1 5 b O- 3_4-2. — o- 0^.8 -15 i * n i b - o b i 3>\\ — O - 0 2 O \\ 5 - \\ - 2 8 9 3 O'414-q - O- \\ 1 8 0 ! - 3 0 o 3 » - 0 - 8 5 1 8 O- 014-W9 -0.-V4-34- O- 5 0 2 1 - 0 - 2 . ^ 3 4 - 5 15 \\- in 16 - 0 - b l 3 l - O - o i o \\5 1 - 0 0 3 9 o - 4 - 1 4 - 9 - o - \\ i 3 30 0 - 8 2 1 1 — o - \\ l f o 9 - O -2 5 5 5 v«3 4 - 4 - 1 0 - 3 4 2 . - 0 - 0 2 . 8 45 0-354-2 0 - 4 - 4 1 6 — o- S l < 4 5 2-^^ro.o O- \\ b 2 . O v \\ b 2 _ 60 - o \\ 2 - \\ « 3 \\ - \\ _ l _ - O - S S l \\. 84 - 4-1 — O- _ _ . © 0 - _ _ 4 - 2 _ 75 - o • 4-151 1- b 5 0 — \\- \\ o s - o - \\ 1 8 0 - 4 - l 4 - c \\ 90 - o - 6 1 3 3 \\ - Q 1 4 - 5 \" — \\ • \\ 8 2 . — O- 1 4 - ^ , 4 - — o -2.s -4 -_\" O- 5 2 _ _ _ U STRAIN CONCENTRATION FACTORS F o l i a t i o n Angle: b O ° r/2a = o . l 0 A l A3 A4 A5 A6 -90 - 0 - 5 94- 3 - 4 1 3 - O - 8 3 0 - 0 1 4 6 — \\ - 1 0 2 5 o- 8 ^ 8 5 4 -75 - O - 4 - b l B ' \\ 2 0 - 0 - 1 = 1 4 -2.-3«B4S - I - 0 1 0 5 O- T O I -60 — 0-1\\\\3> 2 . - X 1 8 3 -O- b 2 2 - 5 - 5 8 1 O- 33>5 -45 i- \\ O 8 B -O- 2»5~1 —4- 0 0 b — o-1 b 4 — O-1 b 4 -30 O v © 3 1 b -O- O T 9 I — O- O b b - 3 - 5 8 1 O- 3 3 5 - O- 6 5 0 5 -15 I- 111=1 - o - S 5 b 8 o- \\S~I8 - ^ • 3 , 8 ^ 5 o - 1 0 1 — \\ • 0 7 . 0 b 0 -\\ • 2L8 0 8 o- 2.4-489 - o- 1 4 b o- 8 4 * b 5 4 - 1 - 2 D 2 5 15 1- n a q 0 \\\"51.3 o- 9 \\ O b o- n o n - 1 ' O a oS\" 30 - 0 • 0-19 \\ -O- O b b ZL- \\ 2 9 b o- - o- b 5 0 5 45 0 - 3 b 2 9 •I- 1 0 8 6 -X- S i n 5 -4-60 - O- 1 \\ 3 -O- b l X ZL- \\'2.c\\b — 0 • b s o s 75 - O 4 b 2 3 - - o - i S 4 0 - 9 \\ O b - \\- 0 2 . O S o - n o n 90 -O- S 9 4 - 2>- 4-13 — 0 • 8\"i 0 - O- 1 4 - b — \\ - -2.02.^ 0 - 8 4 B 5 - 4 STRAIN CONCENTRATION FACTORS F o l i a t i o n Angle: 90 r/2a =0.1 6 A l A 2 A 3 A 4 A 5 A 6 - 9 0 — O - 1 0 5 \\ 3 - < 9 \\ 9 a - o - ^ 0 0 4 9 - \\ - X 0 4 - 1 o - 0 O * O - 7 5 — O- 5 b O \\ 3- 6 0 3 9 - O - © b ° i 5 8 - 2 - 9 \\ \\ 9 o- 0 O - O - 6 0 — CD* 1 * 5 9 3 1- 1 5 2 b - o - 8 0 5 - 6 6 -4_. 164-0 o- O o - o - 4 5 O • 3 8 9 4 - 1 - S 9 4r \\ - O - l 1 5 5 b - 4 _ - 6 Z 4 - X O - O 0 - 0 - 3 0 0 - 4 3 8 5 9 0 - 4 - 3 8 5 \\ - O - 6 4 8 8 3 - 4 - \\ b A - 0 o - o o - o - 1 5 I - 3 4 - 0 1 4 — O* 4 - o b O \\ - O - 5-94 -3 - 2 . - 4 \\ 19 0 • 0 0 • 0 0 i - 4 - 3 8 O b - 0 * i b 4 - 3 5 - o - -514.44 — \\ - I 0 4 - I o - o o - o 15 \\ - 3 4 - O l 9 - o - 4 0 b 0 l 5 9 4 - 3 0 - 5 0 0 0 8 o - o o - o 30 O - 9 3 8 5 9 0 * 4 - 3 8 t S l - 0 - ^ 4 - 8 8 3 \\-14-6 8 o - o o- 0 45 0 - 3 3 - 9 4- 4 - - 0 . 1 2 S 5 b 0 0 o- 0 60 -0-1*5-43 a - 1 5 2 b —: 0-<3 0 5 b b 1- 14-6 8 o- 0 0 • 0 75 - O - S b O l 3 - ( 3 0 3 ^ - o - 8 b 9 5 8 0 - 5 0 0 0 8 o- 0 0 • 0 90 - 0 - 1 0 5 I 3- ^ I R X — 0 - 4 0 0 4 4 - M O 4 -9 0 • 0 o- 0 "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0093380"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Mining and Metalurgy"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "A study of stress and strain concentration factors in a transversely isotropic medium relevant to the Leeman doorstopper technique"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/19323"@en .