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A study of stress and strain concentration factors in a transversely isotropic medium relevant to the… Smith, Hubert Rodney 1975

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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 (<fO ^-x LOAD CASE 3 (<K) LOAD CASE A (<f>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 <N O ^ "7 STRAIN CONCENTRATION FACTOR •H rH rH tr> £ 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 <a CT> 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 <o II > •> > > 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 <D Cn LO CN • U O t J o 0 A w w m •rH OJ rH rH 3 • % al o II > > 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 <r. • II i n anticl 1 - • • i 1 » 1 I I I 4) s_ \\ V V V 1 1 ; I i « mea i i t i i i I i • r o 1 I i I i • *• A 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 <N. ' ),5 + j 2 2 (N.'),n + j 2 3 (N.'),C c . = j 3 1 (N.'), ? + j 3 2 (N . ' ) , n + j 3 3 ( N . M , C -[ B / J , however must be expressed i n terms of c y l i n d r i c a l coordinate as that i s the coordinate system being used i n t h i s problem. This i s a rather simple transformation and need not 7 8 . be gone into here. S t r e s s - s t r a i n (Constitutive) Relationship The other matrix needed to f i n d the s t i f f n e s s matrix i s D where, ' {a} = [D] {e} (16) and D = E, ( l + v 1 ) ( l - v 1 - Znv 2 ) X 2 2 n(l-ny ), n(v x+nv 2 ), nv^BA^) • • 2 2 n"(v1+nv2 ), n ( l - n v 2 ), nv 2 (Hv.^ ) n v 2 ( l + v 1 ) , (W 2(Hv 1), (l-Vj2) 0 0 0 0 n ( l - v 1 ~ 2 n v 2 y2,0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n(l+v 1)(l-v 1-2mv 2 ) 0 0 0 n ( H v 1 ) ( l - v 1 - zmv2 .... (17) 0 0 0 0 where n = E^/E 2 m = G 2/E 2 and [D] i s within the x, y, and z coordinate system. However, many of the elements are at a d i f f e r e n t o r i e n t a t i o n to the x, y, and z coordinate system, so a coordinate transformation i s necessary. The coordinates (x, y, z) can be transferred to (x, y, z) by, 79, h £m 1 n l r = V n2 V Z m 3 n 3 I* (18) where I., m. and n. are d i r e c t i o n cosines (or cosines x i x of the angle between two vectors i n consideration). S i m i l i a r l y , a transformation of stresses can be found such that, fa ' x V = [X] xy yz xz / where (19) or 2 m i 2 n i 2S,]Lm1 2m 1n 1 2 n1 _ 5 ' i *2 2 m2 2 n2 2&2m2 2 m 2 n 2 2 n 2 A 2 *3 3 m3 3 n 3 2 £ 3 m 3 2 m 3 n 3 2 n 3 £ 3 lll2 m i m 2 n l n 2 (^m +JI m ) (m 1m 2+m 2m 1)(n^^+n^l) %2%Z m2m3 n 2 n 3 (il 2m 3+il 3m 2) (m2n3+m3m2) ( n ^ + n ^ ) l3l2 m l m 3 n 3 n l ''^3 ml+^l m3) ( m 3 n i + m i m 3 ) (n 3 5- 1+n 1^ 3) (20) {a} = [X] {a} (21) 8 0 S i n c e {a> = [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.,<t.36.6J . 3F(«26a 1 ,U(«26£ ),STFAIN(2<5 ) <r,SlR*<2';. c _ * * * * * * * * * * * * * * * * * * * * * * * * * « _ < » * * * * * * * * * * * * * * * * * * * _ C STAFT HTF REACINC Tl-E CONSTANTS Fc*C(5,lC> 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 * * * * * _ * * * « • * * » « » » * * ^ * « * _ * _ * ^ *_<JM_* < » « « » » * < » • » * * * * * * * _ * * * _ . C ELASTIC YATRI >* ,**»' ,****TFAr^SVERSELY ISOTROPIC********** Fc*C(5,50)E12,E13,C12,G23,G21,P12,P12,P22 FEACIE ,6C) SIA.CCA .... FE/C <E,61)r\CAS1,NCCASE FEZC IE ,62 )*CC,F = SICl),NSTCF C CPIGIML ELASTIC VATPI> . ... . . . „ _ . _ 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<E.12J _ _ _ _ C v 1( 2, 2 )=0M I 2 ,3 ) C H I 2, 2) = n.C-F12**2*E12)«E12 D M U t « ) = 1.0/Ca2/CCCINST . . . CVK E,E)=l.C/G23/CCONST CV 1 (6,t )= 1 .0/G21/CCCNST C._._ X CCFCJf\ ALES .JF.AASF.LLP,_AlI.CN_f.ATJUX_ ANCLE,.. .C.TC5C....CEG . „ _ CC 1C2 1=1,6 CC 1C2 J=l,fc 102 C L V U , J)=C.O .. _. .. _ ... CLM 1, 11 = 1.0 CLr(2,2)=CCA**2 CLM2.21«C.LKi.2.,2.) 8 3 . CLM 2, 2 ) = S IA»*2 CLM 2,2)=Clf (2,3) CLP! ii, 4)=CCA C L M t , t ) = CCA CLf(5,5)=CLM(2,2)-CLr<(2,3) JLL_(Ai_fc L=_S_LA I ClrM6,4)=-SIiS ! CLP(=,2)=-C0A*SIA O i C.IM2, J J=-.2.C*CLr-JJ, 2J | CLr*(5,3 J=-CLM5,2) j DLM2,5>=-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,<irE12=,F12.5 ,4HG12=,F 12.5 ,4HG23* ,F12. JL5__LG3J_= ,JL1 2_^_S_l/JAhPl 2=_U___4_PJ3.=_i£ l 2 .5.4HP2 3=,F12 . 5 . / . 24HSIA=,F12.7,4riCCA = ,F12.7,lCX f6rNCAST=,I5,7HNCCASE = , I5, / , 3'rHACC= ,F12 .5,7rRESIDU=,F 1 2 .7,4HNSTGP*,I 5) >PITE<t,17)( (Cf -2U.»J.>. , . 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 )=-<P*SI 1-(R*DR)*SI2) EL(_)=_.(R*CCl- (R«r.P) »CC2» _ Al (4)=-Ai ( 1) e114 »=-BIc l» A1 (2 ) = -( ( P <C R ) *S IL_R*S.I2J 8 1 ( 2 ) = - ( ( R « 0 R ) » C C 1 - R * C C 2 ) Al(3 )=-*l(2) exm.r__ei.L2i • ' A2<1)=CR*SI2 E2( l J=CR*CC2 _A2(2)=-A2( 1) . B2(3)=-e2(l) A2 (2 )=-CR* SI I E212)=-CR»CC1 : A2(4)=-A2(i) e2(A)=-e2(2) _A2( 1 ) = P*< SI1 _SI2.).__ E3(l ) = P*(CC1-CC2) A3 (2')"=.-A3( 1) _ _ _ _ _ 2.12J = - £ 3X1) A3(3)=-(R*CR)*(SI1-SI2) B3(2)=-(R*CK>*(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_)_<IC_Jj;_2J_ I F d S . L T . 7 ) GC TC 205 307 SCSJ=0.5 _2C5._CCNST1=D£S10.B.* (3..Q + S3 (I « ) « S 2 (.IS.).). TEr' = A 2 ( I R ) * A l < I S » + A l ( I R l * Z 2 ( I S )-(2.C*R/QR*1.0)*A2(IR)*A2(IS) ABC(1,1 )=CCNST1*(6.0*TEN* ((3.0*f1(IR)*A1(IS)*A2(IR>*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<JJ+TEMl F I A * 1 >=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<n=TEK*I FCCS<NT)-FCCS<NT-»1>) 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 =<N*12)+N4*N5 _KNi 1_7J_L( M+13)+N.< NMie) = (N*14MN't*N6 KM 19) =(N*MW4)*N2*N3 + N5-»N<3 NN(2C)=(N»MJ-12) •» N 2 • N 2-»N 9 NN (21) = (N + MI-12MN2 + N3 + N6 + NS NN(2 2)=(N+NU-1)+N2*N5*Ne J_N 123 )_(N*NUJ*N2_tN.fi _ NN(2M = (N+M) + l)+N2*N6*Ne NN(25)=(N+Mj*12)+N2«-N4 + N5-»N7 ..NN(26) = (N»NUU2) •N2*N4 + N7_ NN (27) = (N'TMJ + 14) • N2 + N4*N6-*N7 * * 4 44 * * * * 49 * * * 4 * __C_.2 C2 C.J = 1 ..3 __ . C * * * * « N C DISPLACEMENT JINF* * * * • IF(NNSCS.EC.l) GC TO 2155 GC TC (215 1,2152,2153),J _. 2151 IF (NT.EC. l ) GC TC 202C IF (NT.EC12) GC TC 2020 / I V TFINP..E.C.XU_-GC__LC 2020 IF( ICASE.NE.4)C0 TC 2155 IF IJ .EC.DCO TC 2155 2153 IF ( N Z . EC ..13 l_C0__TC_2flZC_ 2155 CCNTINLE JJ=3-J .SUM =0.0 SIJP2 = 0.C OC 2C30 1=1,13 IE (NN ( I ).. L I . a)„ .GC__TD._2Q3C_. II=3*\N(IJ-3 K K = 3 *(15 — I )-JJ •JD.C_2C7jL_lJJ=lt3 11=11+1 SUM=SLf l + SKJKK, II )*U(I I) _2C7.1_CCNT INLE 2030 CCNTINUE I I=3 + N-JJ 0G_2C4 C-I = 1 ,_U-IF(NN( I-H3).LT.O) GO TO 2C4C KK=3*I-3 J<KK=2*NN(.I*12)-3 CC 2C72 JJJ= l ,2 KK = K K*1 _KK.K_=KKJ<.+1 SUP2=SUr_*SK(KK,II KIMKKK ) 2C72 CCNTINLE 2C40 CCNTINLE _ IF(NNSCS.EC.O) GC TO 2055 NC DISPLACEMENT JLMP GC_jrC_.t2Q51.2Cf 2.2C5.3 ).,J IFINT.EC.l ) GC TC 2054 IF(NT.EC.12) GC TC 2054 .2052 _1F(NP.EC.1C) GC TO 2C54 IF(ICASE.NE.4)G0 TC 2055 IF(J.EC.1)GC TC 2055 . 203 3_LE.( N2...NE..12.1_-G C__TC_2G 15 90. ( 2CE4 CCNTINLE j F ( 11 »=F < I I J + SLM + SUM2 ! CC TC 2020 ! 2C55 CONTINLE ! CU=(F(II)-SUfl-SUM2)/SK<J,II) >^ U-U.U=utU-i>-Arx*-rjj _ _ f DLSOS = ABS< L<II)) I IF (CLSCS.LT.C.COGl) GC TC 2C2C 1 CLSl> = 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<V3_X1 - a . - i - ^ 3 1 o- O o - o 887 - 0 - 4 - _ _ \ 5 3 0 - 2 . - q 3 l 4 - 4 - o • O o • o 888 - 0 - 4 ^ b _ X _ - _-i _ o - i 3 > — 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 . - l<BV_ 3 - \3• 3 o- O o o 888 -0-4-4-4-002, - \ _ • X S b O O • O o - o 889 - _'X_ " 55 i — \3- 2.X_S o • O o - o 890 - o - i m _ 5 - 3 * 4 - \ 5 * 0 — \ 3 ' ^ 3 X A - o • O o - o 891 - \ 3 - 0\<32_ o - o o - o 892 o - i n o i i - 3 - 4 - 5 o n - \ a - 8,°! 34- o - o o - o 893 o- o o - o 894 O ' 4 1 4 - O O X. — 3 • \3>1b3 — \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-<o \ o.-2.ob8a. o • o . o- o 888 O - 33qo~tq O.- \fc>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<H \ 2 0 - 2 - 8 ^ 4 - 4 - 0 -'fo -31 _'V_ o - o o - o 894 O- -Ol -x-io84-6 - b - 3 3 3 5 0 o - o o - o 895 o-X4-5fooi 1 - 2 - b X \ 1 2 -6-3 \ I 31 o - o 0 * 0 896 O- V34-1 fob ! - 2 - 5 6 6 0 1 — fo-30\3S o-o O ' O 897 O-O - 2 - ,4-5-13 - b- 29 2-4- o- o NODAL DISPLACEMENTS AND LOADS F o l i a t i o n Angle: 6 0 Load Case: r/2a = 0.1 nodal #' u V w u V 885 O-O - 4 - * 5 8139 -3 -^214 -6 O- 054-4-4-^2 o - o 886 - o - 2 . n u _ — 6 4 - 2 O 4- -3*8H~1I5 o - 0 o - o 887 b i <n - 3 - 8 2 ^ 6 4 - o - o 0 - 0 888 - 0 - 5 S 8 0 3 0 - 3 * O I I 6 4 - o - o 0 - 0 889 - o - 5 2 _ l 5 8 - 5 - 2 8 5 6 5 - 3 - 5 6 8 8 0 o - o 0 - 0 890 - 0 - 2 1 8 1 3 ^ -5-5-1 02°! - 3 - 3 1 3 5 8 0- 0 0 - 0 891 - 3 - \44-Ro 0 - 0 0 • 0 892 0 - ^ - l b 8 8 b ; - 5 - 5 1 b m - 2.-S \238 o - 0 o - 0 893 0 - 6 2 Q 5 8 1 ' - 5 - 3 1 4 - 5 0 - a * 1 n i l o - o 0 - 0 894 - 5 - \ l 5 3 5 - 1 - 5 34-3\ o - o o - o 895 ! - 4 - q 5116 - a - 5 1128 0 - 0 0 - 0 896 O-2,83)56 -4--854-R3 ~ a « 4 - i c l 4 - o o - o 0 - 0 897 O-O - 4 * 821 9 2. 0 * 0 . 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 -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 — \.<V 0 5 9 0 9 895 60 9 - 2 . 4 - 4 - 4 - 8 - - 1 - 3 2 . 4 - 5 9 - 2 . 3 - ^ 9 2 . 1 5 - 1 9 - 9 9 0 8 3 896 75 AO - 4-2.9 2 1 - 8 - 8 ^ 4 9 4- - 21 - O b 3 l b - 2 2 . - 6 6284-897 90 \ o - i 8 4 - i 9 — ^ . 4 . 3 0 4 - 4 - - 2 . 8 - 1 2 U 9 - 2 2 > - 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 -<b - \ b 4 -O- 14>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 

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