Prediction of rock failures in mines, with application to the Näsliden mine in northern Sweden

PREDICTION OF ROCK FAILURES IN MINES

WITH APPLICATION TO THE NÄSLIDEN MINE

IN NORTHERN SWEDEN

by

TORGNY BORG

Division of of Rock Mechanics

JUH

_{HÖGSKOLAN I LULEA}

PREDICTION OF ROCK FAILURES I N MINES, WITH APPLICATION THE NÄSLIDEN MINE I N NORTHERN SWEDEN

av

Torgny Borg

AKADEMISK AVHANDLING

som med vederbörligt tillstånd av T e k n i s k a Fakultetsnämnden v i d Högskolan i Luleå för avläggande av t e k n o l o g i e doktorsexamen

kommer a t t o f f e n t l i g e n försvaras å Högskolan i Luleå, s a l F 341 ( g u l a h u s e t ) f r e d a g e n den 6 maj 1983 k l 10.00

PREDICTION OF ROCK FAILURES I N MINES, WITH APPLICATION THE NÄSLIDEN MINE I N NORTHERN SWEDEN

by

Torgny Borg

D i v i s i o n o f Rock Mechanics Luleå U n i v e r s i t y

I ABSTRACT C o n t i n o u s c u t and f i l l m i n i n g w i l l cause i n c r e a s e d l o a d i n g o f t h e r o o f o f t h e s t o p e s i n m i n i n g o p e r a t i o n s . The aim o f t h e p r e s e n t work i s t o p r e d i c t what m i n i n g l e v e l s w i l l i n -duce c r i t i c a l l y l a r g e l o a d s i n t h e r o o f o f t h e s t o p e s i n c u t and f i l l m i n i n g and i n p a r t i c u l a r t h e Näsliden Mine, i n N o r t h e r n Sweden.

The q u a s i - s t a t i c e l a s t i c response o f r o c k masses i n t h e Näsliden Mine i s p r e d i c t e d i n t e r m s o f t h e s t r e s s e s and

s t r a i n s i n d u c e d by m i n i n g o p e r a t i o n s , by u s i n g t h e F i n i t e Element Method. C r i t i c a l s t a g e s i n t h e development o f f a i -l u r e s a r e i d e n t i f i e d and d e f i n e d f r o m o b s e r v a t i o n s i n t h e mine. I n - s i t u measurements a r e a l s o t a k e n f r o m a s e l e c t e d r e f e r e n c e s t o p e i n t h e mine, a t c e r t a i n s e l e c t e d s t a g e s i n a sequence o f m i n i n g o p e r a t i o n s . The e x t e n s i o n s t r a i n c r i -t e r i o n o f f a i l u r e i s used i n o u r c a l c u l a -t i o n s . From o u r c a l c u l a t i o n s , and t h e o b s e r v a t i o n s f r o m t h e r e f e r e n c e s t o p e , c r i t i c a l l e v e l s o f m i n i n g , f o r a l l s t o p e s i n t h e mine, a r e i d e n t i f i e d . The p r e d i c t e d r e s p o n s e o f t h e r o c k mass f r o m c a l c u l a t i o n s based on a l i n e a r l y e l a s t i c model i s compared w i t h p r e d i c -t i o n s f r o m c a l c u l a -t i o n s based on a j o i n -t e l e m e n -t model. The j o i n t e l e m e n t s a r e used t o s i m u l a t e weak c o n t a c t a d j a -c e n t t o t h e o r e body. B o t h models a r e e v a l u a t e d and a d j u s t e d by comparisons w i t h i n - s i t u measurements o f d i s p l a c e m e n t s and s t r e s s e s . A s e n s i t i v i t y s t u d y i s used t o i n v e s t i g a t e t h e e f f e c t o f v a r i a t i o n i n r o c k p r o p e r t i e s and c a l c u l a t i o n e r r o r s due t o n u m e r i c a l d i s c r e t i z a t i o n s . F u l l s c a l e s t r e n g t h d a t a a r e compared w i t h d a t a f r o m u n i a x i a l c o m p r e s s i o n t e s t s c o n d u c t e d i n t h e l a b o r a t o r y .

The r e s u l t s o b t a i n e d w i t h t h e e l a s t i c model and t h e e x t e n -s i o n -s t r a i n c r i t e r i o n -show -s a t i -s f a c t o r y agreement w i t h d a t a o b t a i n e d f r o m t h e mine. A l m o s t i d e n t i c a l r e s u l t s , as f a r as

t h e p r e d i c t i o n o f r o o f f a i l u r e o f stopes i n t h e mine i s c o n c e r n e d , a r e o b t a i n e d when t h e j o i n t e l e m e n t model and t h e Coulomb f a i l u r e c r i t e r i a i s used i n o u r c a l c u l a t i o n s . Our q u a n t i t a t i v e p r e d i c t i o n o f r o c k f a i l u r e can be used t o d e t e r m i n e when r o c k s u p p o r t i s needed o r when changes i n m i n i n g methods s h o u l d be a d o p t e d . Our c o m p u t a t i o n a l model c o u l d a l s o be used t o i n v e s t i g a t e a l t e r n a t i v e mine l a y o u t s .

Key words

Cut and f i l l m i n i n g , r o c k mechanics p r e d i c t i o n o f r o c k f a i -l u r e , b r i t t -l e f r a c t u r e o f r o c k , e x t e n s i o n s t r a i n f a i -l u r e c r i t e r i o n , Coulomb f a i l u r e c r i t e r i o n , n u m e r i c a l p r o c e d u r e s , j o i n t element.

I I I

CONTENTS

Page

NOTATIONS AND SYMBOLS I X

1 INTRODUCTION 1 1.1 Aim o f t h e work 1

1.2 Background 2 1.3 Scope o f t h e work 3

1.4 Method o f a n a l y s i s 4

2 MINING METHOD AND GEOLOGY I N THE NÄSLIDEN MINE 6

2.1 P r i n c i p l e s o f c u t and f i l l m i n i n g 6 2.2 M i n i n g method i n t h e Näsliden Mine 7 2.3 Geology and s t r u c t u r e s i n t h e Näsliden Mine 8

2.4 V i r g i n s t r e s s e s i n t h e Näsliden Mine 12

3 OBSERVATIONS OF FAILURES I N THE NÄSLIDEN MINE 15 3.1 Types and o c c u r e n c e o f f a i l u r e s around

t h e s t o p e s 15 3.2 Stages o f f a i l u r e s 18

4 CONSTITUTIVE RELATIONS FOR THE ROCK MASS 24 4.1 L i n e a r e l a s t i c m a t e r i a l b e h a v i o u r 24 4.2 F a i l u r e c r i t e r i a 25 4.2.1 The e x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n (ESFC) 26 4.2.2 The Coulomb f a i l u r e c r i t e r i o n (CFC) 28 4.3 J o i n t e l e m e n t s 29

Page

5 NUMERICAL PROCEDURE FOR FINITE ELEMENT

CALCULATIONS 33

5. 1 F i n i t e e l e m e n t f o r m u l a t i o n 3 5

5. 1 . 1 The i s o t r o p i c l i n e a r e l a s t i c model 35

5.1.2 The j o i n t e l e m e n t model 38

6 EVALUATION OF THE NÄSLIDEN MINE MODELS 4 0

6. 1 F i n i t e e l e m e n t models o f t h e Näsliden Mine 4 0

6. 1 . 1 F i r s t e l a s t i c model f o r comparisons w i t h i n - s i t u measurements o f d i s p l a c e m e n t s and s t r e s s e s 40 6.1.2 F i r s t j o i n t e l e m e n t model f o r com-p a r i s o n s w i t h i n - s i t u measurements 46 6.1.3 A d j u s t e d e l a s t i c model f o r com-p a r i s o n s w i t h i n - s i t u measurements 4 6 6.1.4 Second j o i n t element model f o r

com-p a r i s o n s w i t h i n - s i t u measurements 48 6.2 Comparisons o f r e s u l t s f r o m FEM c a l c u l a t i o n s w i t h i n - s i t u measurements i n t h e back f i l l e d e x c a v a t i o n s 48 6.2.1 H o r i z o n t a l f i l l p r e s s u r e a c r o s s t h e e x c a v a t i o n s 49 6.2.2 Convergence a c r o s s t h e back f i l l e d e x c a v a t i o n s 52 6.3 Comparisons o f r e s u l t s f r o m FEM c a l c u l a t i o n s w i t h i n - s i t u measurements i n t h e o r e body 55 6.3.1 S t r e s s e s i n t h e o r e body above s t o p e 3 5 5 6.3.2 Convergence o f o r e body and

v Page 6.4 Comparisons o f r e s u l t s f r o m FEM c a l c u -l a t i o n s w i t h i n - s i t u measurements i n t h e open s t o p e s 61 6.4.1 D i s p l a c e m e n t o f t h e r o o f and t h e h a n g i n g w a l l 61 6.5 P r e d i c t i o n o f r o o f f a i l u r e s based on a b r i t t l e f r a c t u r e c r i t e r i o n f o r r o c k 65

FINITE ELEMENT CALCULATIONS FOR PREDICTION OF

FUTURE MINING CONDITIONS 68 7.1 F i n i t e e l e m e n t model o f t h e Näsliden Mine 68

7.1.1 M i n i n g s t a g e s 68 7.1.2 Rock mass p r o p e r t i e s and v i r g i n

r o c k s t r e s s e s 69 7.1.3 Boundary l o c a t i o n s and f i n i t e e l e m e n t mesh 70 7.2 R e s u l t s o f t h e f i n i t e element c a l c u l a t i o n s 73 7.2.1 D i s p l a c e m e n t s i n t h e r o c k 73 7.2.2 P r i n c i p a l s t r e s s e s i n t h e r o c k 76

APPLICATION OF ROCK FAILURE CRITERIA TO THE

NÄSLIDEN MINE 8 0 8.1 Q u a n t i f i c a t i o n o f t h e f a i l u r e c r i t e r i a 82 8.1.1 D e t e r m i n a t i o n o f t h e p a r a m e t e r f o r t h e e x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n (ESFC) 82 8.1.2 D e t e r m i n a t i o n o f t h e p a r a m e t e r s f o r t h e Coulomb f a i l u r e c r i t e r i o n (CFC) 87 8.2 Check o f t h e s t r e n g t h d a t a by i n - s i t u s t r e s s measurements 91

Page

8.3 Comparisons o f s t r e n g t h d a t a f r o m l a b o r a

-t o r y -t e s -t i n g and i n - s i -t u e s -t i m a -t i o n s 94

8.4 Summary 95

9 DEVELOPMENT OF POTENTIAL FAILURE ZONES I N THE

NÄSLIDEN MINE 9 6 9.1 P o t e n t i a l f a i l u r e zones o b t a i n e d w i t h t h e e l a s t i c model 96 9.2 P o t e n t i a l f a i l u r e zones o b t a i n e d w i t h t h e j o i n t element model 104 9.3 D i s c u s s i o n o f t h e p o t e n t i a l f a i l u r e zones

o b t a i n e d w i t h t h e Näsliden Mine models 107 9.3.1 F a i l u r e zones i n t h e r o o f o f

t h e s t o p e s 107 9.3.2 F a i l u r e zones i n t h e s i l l p i l l a r s 107

10 PREDICTION OF FAILURES I N THE NÄSLIDEN MINE 109 10.1 P r e d i c t i o n o f r o o f f a i l u r e s i n t h e s t o p e s o f t h e Näsliden Mine 109 10.2 Comparison o f t h e p r e d i c t i o n and i n - s i t u o b s e r v a t i o n s i n t h e Näsliden Mine 112 10.3 D i s c u s s i o n o f t h e p r e d i c t i o n o f f a i l u r e s i n t h e Näsliden Mine 114 10.4 F u r t h e r a p p l i c a t i o n s o f t h e p r e d i c t i o n model 115 11 CONCLUSIONS 119 11.1 T h e o r e t i c a l c o n c e p t 119 11.2 P r a c t i c a l i m p l e m e n t a t i o n s 120

V I I

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12 FUTURE WORK 121 12.1 Rock mechanics p r o g r a m i n t h e Näsliden Mine 121

12.2 Rock mechanics programs i n mines where f r a c t u r e s and f a i l u r e s a r e f r e q u e n t l y e n c o u n t e r e d 122 12.3 F u r t h e r d e v e l o p m e n t o f t h e p r e d i c t i o n model 122 ACKNOWLEDGEMENTS 124 REFERENCES 126 APPENDIX I : L i n e a r e l a s t i c f i n i t e e l e m e n t f o r m u l a t i o n o f t h e Näsliden Mine 1.1 D e r i v a t i o n o f t h e e l e m e n t s t i f f n e s s m a t r i x 1.2 Nodal f o r c e s 1.3 Flow c h a r t f o r t h e f i n i t e e l e m e n t c a l c u l a t i o n o f a m i n i n g sequence APPENDIX I I : P e r t u r b a t i o n a n a l y s i s and s t a t i s t i -c a l methods t e s t e d f o r t h e f i n i t e e l e m e n t models o f t h e Näsliden Mine 11.1 P e r t u r b a t i o n a n a l y s i s o f e l a s t i c models 11.2 E x p e c t a t i o n and v a r i a n c e o f FEM s o l u t i o n v a r i a b l e s APPENDIX I I I : S e n s i t i v i t y s t u d y o f t h e e l a s t i c f i n i t e e l e m e n t models o f t h e Näsliden Mine A1 3 I I I . 1 E l a s t i c p r o p e r t i e s o f t h e r o c k s i n t h e Näsliden Mine A1 3 A1 A1 A6 A7 A9 A9 A1 2

Page

111.2 V i r g i n s t r e s s e s i n t h e Näsliden

Mine A16 111.3 Boundary c o n d i t i o n s A18

111.4 E x c a v a t i o n t e c h n i q u e s A20 111.5 F i n i t e element mesh d e s i g n A23

111.6 Geometry A24 APPENDIX I V : C o n d i t i o n s f o r i n s t a b i l i t i e s i n t h e

r o o f o f t h e s t o p e s A27

I V . 1 G e n e r a l A27 IV.2 Roof i n s t a b i l i t y A29

I V . 3 I n s t a b i l i t y o f s i l l p i l l a r s A30

APPENDIX V: U n i a x i a l compression t e s t s o f o r e

samples f r o m t h e Näsliden Mine A31 V. 1 Choice o f specimen and t e s t

p r o c e d u r e A31 V.2 R e s u l t s A32

IX

NOTATIONS AND SYMBOLS

The n o t a t i o n s and symbols a r e e x p l a i n e d i n t h e t e x t when t h e y f i r s t o c c u r . I n a d d i t i o n t h e n o t a t i o n s and symbols most f r e q u e n t l y used a r e l i s t e d below.

Roman l e t t e r s

c c o h e s i o n

d d i s t a n c e between j o i n t s

E Young's modulus o f i n t a c t r o c k

E e f f e c t i v e Young's modulus o f t h e r o c k mass rm ^ F o u t e r boundary t r a c t i o n v e c t o r ~s JRC j o i n t roughness c o e f f i c i e n t JCS j o i n t c o m p r e s s i v e s t r e n g t h kg t a n g e n t i a l j o i n t s t i f f n e s s i n shear k n o r m a l j o i n t s t i f f n e s s n J L c r i t i c a l i o i n t l e n g t h c 3 S^ boundary o f open s t o p e S^ boundary o f back f i l l e d e x c a v a t i o n S° o u t e r boundary o f t h e model Tg j o i n t t e n s i l e s t r e n g t h uv v e c t o r o f d i s p l a c e m e n t s i n d u c e d by v i r g i n s t r e s s e s u v e c t o r o f d i s p l a c e m e n t s i n d u c e d by e x c a v a t i o n s u v e c t o r o f t o t a l d i s p l a c e m e n t s u v e c t o r o f o u t e r boundary d i s p l a c e m e n t s s t r a i n energy i n t h e back f i l l m a t e r i a l Ur s t r a i n energy i n t h e r o c k U t o t a l p o t e n t i a l energy i n a r o c k volume w i t h e x c a v a t i o n s

p o t e n t i a l energy o f t h e o u t e r boundary t r a c t i o n s ( x , y, z) C a r t e s i a n c o o r d i n a t e s Greek l e t t e r s ß a n g l e o f f a i l u r e p l a n e e c r i t i c a l e x t e n s i o n s t r a i n v a l u e c i n i t i a l s t r a i n t e n s o r i n f i l l m a t e r i a l ev v i r g i n s t r a i n t e n s o r i n r o c k c s t r a i n t e n s o r due t o e x c a v a t x o n s e t o t a l s t r a i n t e n s o r , e2, £ 3 p r i n c i p a l s t r a i n s AL h o r i z o n t a l convergences o f o r e , o r a c r o s s back f i l l e d e x c a v a t i o n , due t o m i n i n g Ah v e r t i c a l d i s p l a c e m e n t o f t h e r o o f due t o m i n i n g o f one s l i c e v P o i s s o n ' s r a t i o o f r o c k P o i s s o n ' s r a t i o o f f i l l m a t e r i a l d e n s i t y o f f i l l m a t e r i a l c^"^ i n i t i a l s t r e s s t e n s o r o f f i l l m a t e r i a l v a v i r g i n s t r e s s t e n s o r o f r o c k Q o s t r e s s t e n s o r due t o e x c a v a t i o n s a t o t a l s t r e s s t e n s o r a^t a2' p r i n c i p a l s t r e s s e s an s t r e s s component normal t o a j o i n t o r f r a c t u r e ac u n i a x i a l c o m p r e s s i v e s t r e n g t h T shear s t r e s s p a r a l l e l t o a j o i n t o r f r a c t u r e T shear s t r e n g t h IT <f> a n g l e o f f r i c t i o n

X I S i g n c o n v e n t i o n T h r o u g h o u t t h i s work we a d o p t t h e c o n v e n t i o n t h a t compre s i v e s t r e s s e s and s t r a i n s a r e t a k e n t o be p o s i t i v e . A b b r e v a t i o n s ESFC E x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n CFC Coulomb f a i l u r e c r i t e r i o n ys The n o t a t i o n [ y s ] i s f r e q u e n t l y used i n o u r t a b l e s and diagrams when r e f e r r i n g t o s t r a i n s and [ y s ] i n such cases d e n o t e s a s t r a i n o f 10

1 INTRODUCTION

1.1 Aim o f t h e work

E f f e c t i v e and s a f e m i n i n g o f an o r e body depends upon t h e e x t e n t t o w h i c h t h e f a i l u r e p r o c e s s e s o f t h e r o c k around t h e s t o p e s can be c o n t r o l l e d . Two t y p e s o f f a i l u r e s can be d i s t i n g u i s h e d w i t h r e g a r d t o t h e i r l o c a t i o n s i n c u t and f i l l m i n i n g o f s t e e p d i p p i n g o r e b o d i e s .

The f i r s t t y p e o f f a i l u r e o c c u r s i n t h e r o o f and b o t t o m o f t h e s t o p e s , F i g u r e 1.1. P r o g r e s s i v e m i n i n g w i l l cause an i n c r e a s e d l o a d i n g o f t h e o r e body. When t h e l o a d i n g has r e a c h e d a c r i t i c a l l e v e l , f r a c t u r e s o c c u r i n t h e r o o f o f t h e s t o p e s . A t an advanced s t a g e o f m i n i n g t h e m a g n i t u d e s o f l o a d s can r e a c h a c r i t i c a l l e v e l i n l a r g e p a r t s o f t h e s i l l p i l l a r between t h e s t o p e s . A t t h i s s t a g e s i g n i f i c a n t r e d i s t r i b u t i o n s o f l o a d s can o c c u r . F i g . 1.1 I n c r e a s e i n l o a d i n g o f r o c k due t o p r o g r e s s i v e c u t and f i l l m i n i n g .

2 A second t y p e o f f a i l u r e o c c u r s i n t h e f o o t and h a n g i n g w a l l . Of s p e c i a l i m p o r t a n c e a r e l a r g e s c a l e f a i l u r e s i n t h e h a n g i n g o r f o o t w a l l , w h i c h may o c c u r a t advanced s t a g e s o f m i n i n g when t h e s i l l p i l l a r s between t h e s t o p e s a r e mined o u t . T h i s work i s m a i n l y r e s t r i c t e d t o a n a l y s i s o f f a i l u r e p r o -cesses i n t h e r o o f and b o t t o m o f t h e s t o p e s due t o l a r g e m a g n i t u d e s o f l o a d s . These f a i l u r e s a r e o f g r e a t i m p o r t a n c e

as t h e y can l e a d t o an i n t e r r u p t i o n o r a change i n t h e m i n i n g p r o c e s s , i n p a r t i c u l a r o f m i n i n g a t l a r g e d e p t h s .

The a i m o f t h e p r e s e n t work i s t o p r e d i c t what m i n i n g l e v e l s w i l l i n d u c e c r i t i c a l l y l a r g e l o a d s i n t h e s t o p e s o f t h e Näsliden mine, i n N o r t h e r n Sweden.

1.2 Background

The d e s i g n o f s t o p e s and t h e c h o i c e o f s u p p o r t methods i n mines have t o d a t e been based on p r a c t i c a l e x p e r e i n c e and

t h e o r e t i c a l a n a l y s i s . A n a l y t i c a l s o l u t i o n s a r e s t i l l v e r y i m p o r t a n t b u t a r e r e s t r i c t e d t o s i m p l e g e o m e t r i e s o f t h e a c t u a l s t o p e s , e.g. c i r c u l a r o r e l e p t i c a l openings and e l a s t i c r o c k b e h a v i o u r . The development o f computer a i d e d n u m e r i c a l methods, l i k e t h e F i n i t e Element Method - FEM, and t h e Boundary Element Method - BEM, now makes i t pos-s i b l e t o d e t e r m i n e t h e d i pos-s p l a c e m e n t pos-s and pos-s t r e pos-s pos-s e pos-s around t h e s t o p e s f o r t h e a c t u a l mine geometry. F u r t h e r m o r e , t h e f a i l u r e p r o c e s s e s o f t h e r o c k can now a l s o be t a k e n i n t o a c c o u n t .

P a r i s e a u and K e a l y (1972) c o n d u c t e d a s t u d y o f c u t and f i l l m i n i n g by u s i n g FEM. These a n a l y s e s were made f e a s i b l e due

t o t h e e x t e n s i v e i n - s i t u measurements p e r f o r m e d i n t h e Coeur d'Alene m i n i n g d i s t r i c t o f t h e U n i t e d S t a t e s . The aim o f t h e f i r s t a n a l y s i s was t o e v a l u a t e t h e s u p p o r t p e r

-formance o f h y d r a u l i c back f i l l i n t h e s t o p e s . The o r i g i n a l work was e x t e n d e d t o i n c l u d e a n a l y s e s o f s t r e s s e s i n t h e s t o p e p i l l a r , n o n - l i n e a r i t y o f t h e f i l l m a t e r i a l and t h e h i g h h o r i z o n t a l v i r g i n s t r e s s e s i n t h e r o c k , P a r i s e a u e t a l (1976) . Brady (1977) a p p l i e d BEM f o r t h e d e s i g n o f p i l l a r s i n a t r i a l s t o p e a t t h e Mount I s a Mine, Queensland, A u s t r a l i a . I n - s i t u measurements o f t h e v i r g i n s t r e s s e s i n t h e r o c k and o b s e r v a t i o n s o f r o c k f a i l u r e s i n t h e s t o p e were used t o e s t a b l i s h t h e BEM model used f o r t h e d e s i g n p u r p o s e .

1 .3 Scope o f t h e work

The p r e s e n t work i s a c o n t i n u a t i o n o f t h e Näsliden P r o j e c t i n Sweden. I t aims t o e v a l u a t e t h e s u i t a b i l i t y and r e l i a b i l i t y o f f i n i t e e l e m e n t models as a r o c k mechanics p l a n n i n g i n s t r u m e n t i n m i n i n g . I n t h e p r o j e c t a l i n e a r l y e l a s -t i c model and a model i n w h i c h j o i n -t e l e m e n -t s were i n c l u d e d , s i m u l a t i n g t h e weak c o n t a c t s a d j a c e n t t o t h e o r e body o f t h e Näsliden Mine were d e v e l o p e d . The agreement w i t h t h e a c t u a l mine b e h a v i o u r was e s t a b l i s h e d by c o m p a r i s o n w i t h r o c k mechanics o b s e r v a t i o n s and measurements c a r r i e d o u t f r o m t h e s t a r t o f m i n i n g i n 1970 o f t h e Näsliden Mine. A p r e s e n t a t i o n o f t h e Näsliden P r o j e c t has been g i v e n a t t h e C o n f e r e n c e on t h e A p p l i c a t i o n o f Rock Mechanics t o Cut and F i l l M i n i n g , Luleå, June 1980, Stephansson and Jones ( 1 9 8 1 ) .

For t h e purpose o f t h e p r e s e n t s t u d y t h e main o b j e c t i v e s o f t h e Näsliden P r o j e c t were:

- The development o f a l i n e a r l y e l a s t i c model w i t h a known degree o f agreement between model

and mine b e h a v i o u r .

4

i n f o r m a t i o n on t h e r o c k b e h a v i o u r i n t h e imme-d i a t e v i c i n i t y o f an e x c a v a t i o n when t h e r o c k mass approaches a s t a t e o f f a i l u r e . However, t h e i n t r o d u c t i o n o f j o i n t e l e m e n t s i n c r e a s e d q u a n t i t a t i v e u n c e r t a i n t y . T h i s was due t o d i f f i -c u l t i e s i n d e t e r m i n i n g t h e n e -c e s s a r y m e -c h a n i -c a l p a r a m e t e r s o f j o i n t s . C o m p l e x i t y and c o s t o f model w o r k i n c r e a s e d s i g n i f i c a n t l y . The Näsliden P r o j e c t c o v e r e d t h e e a r l y s t a g e s o f m i n i n g . Only one s t o p e was a p p r o a c h i n g t h e f i n a l s t a g e o f excava-t i o n and e x p e r i e n c i n g r o o f f a l l p r o b l e m s . From a s excava-t r i c excava-t l y t h e o r e t i c a l p o i n t o f v i e w , a p r e d i c t i o n o f r o c k f a i l u r e s a t v a r i o u s s t a g e s o f m i n i n g r e q u i r e s d e t a i l e d s i m u l a t i o n o f t h e f a i l u r e b e h a v i o u r , and o f t h e s u p p o r t methods i n t h e mine. T h i s s i m u l a t i o n i s n o t f e a s i b l e y e t , as t h e p h y s i c a l p r o p e r t i e s o f t h e r o c k mass i n t h e f a i l u r e regime c a n n o t be d e t e r m i n e d q u a n t i t a t i v e l y . Mine measurements and o b s e r v a t i o n s i n d i c a t e t h a t t h e r o c k i n t h e Näsliden Mine behave i n a b r i t t l e manner. I t i s known f r o m c o m p r e s s i o n a l t e s t s t h a t a b r i t t l e r o c k behaves a l m o s t l i n e a r l y e l a s t i -c a l l y up t o t h e f a i l u r e s t r e n g t h . We t h e r e f o r e -c o n s i d e r e d we a r e j u s t i f i e d i n u s i n g a l i n e a r l y e l a s t i c model f o r t h e p r e l i m i n a r y p r e d i c t i o n o f f a i l u r e s i n t h e mine, even i f i n advanced s t a g e s o f m i n i n g we m i g h t be u s i n g t h e model be-yond i t s t h e o r e t i c a l l i m i t , g i v e n by t h e s t r e n g t h o f t h e r o c k .

1 . 4 Method o f a n a l y s i s

The approach used f o r p r e d i c t i o n o f t h e f a i l u r e s i n t h e Näsliden Mine i s as f o l l o w s :

a) C a l c u l a t e t h e e l a s t i c response o f t h e r o c k mass t o m i n i n g i n terms o f s t r e s s e s and

b) Determine a f a i l u r e c r i t e r i o n f o r t h e r o c k mass. c) D e f i n e t h e c r i t i c a l s t a g e s i n t h e development o f f a i l u r e as d e t e r m i n e d f r o m o b s e r v a t i o n s i n t h e mine: 1) B r i t t l e f r a c t u r e , o f t e n v i o l e n t . 2) Onset o f r o c k f a l l ; m i n i n g can s t i l l be c o n t i n u e d . 3) U l t i m a t e r o o f f a i l u r e ; m i n i n g has t o be s t o p p e d o r changed. The c r i t e r i o n f o r t h e s e s t a g e s a r e q u a n t i f i e d by comparing o b s e r v a t i o n s f r o m s t o p e 3 i n t h e mine w i t h c a l c u l a t e d v a l u e s o f s t r e s s e s / s t r a i n s f o r t h e same s t a g e o f m i n i n g . d) Compare e l a s t i c s t r e s s e s / s t r a i n s w i t h t h e s t r e n g t h d a t a o f t h e r o c k . The l o c i o f p o i n t s where t h e s t r e s s e s / s t r a i n s e q u a l t h e s t r e n g t h o f t h e r o c k d e f i n e s t h e boundary o f a p o t e n -t i a l f a i l u r e zone. e) P r e d i c t i o n o f t h e c r i t i c a l l e v e l s f o r a l l mine s t o p e s up t o t h e end o f m i n i n g . V a l i d i t y o f t h i s p r e d i c t i o n i s checked by comparing p r e -d i c t e -d c r i t i c a l l e v e l s f o r t h e s t o p e s w i t h p r a c t i c a l e x p e r i e n c e f r o m t h e mine.

Two f a i l u r e c r i t e r i a have been t e s t e d i n o u r w o r k , t h e e x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n - ESFC, proposed by Stacey ( 1 9 8 1 ) , and t h e Coulomb f a i l u r e c r i t e r i o n - CFC.

6

2 MINING METHOD AND GEOLOGY I N THE NÄSLIDEN MINE

2.1 P r i n c i p l e s o f c u t and f i l l m i n i n g

I n c u t and f i l l m i n i n g w i t h h o r i z o n t a l d r i l l i n g , t h e o r e body i s mined i n s l i c e s 3-7 m i n h e i g h t , F i g u r e 2 . 1 . The open s t o p e w h i c h w i l l o c c u r i s back f i l l e d w i t h n a t u r a l sand o r t a i l i n g f r o m t h e d r e s s i n g p l a n t o r r o c k f r o m d r i f t i n g . A f t e r t h e back f i l l i n g i s c o m p l e t e d i n a s t o p e , m i n -i n g o f t h e n e x t s l -i c e s t a r t s w -i t h e n t r -i e s f r o m a ramp -i n t h e f o o t w a l l . The back f i l l m a t e r i a l i s p l a c e d h y d r a u l i -c a l l y i n t o t h e s t o p e s . F i g . 2.1 Cut and f i l l m i n i n g w i t h h o r i z o n t a l d r i l l i n g .

I n most cases t h e c u t and f i l l m i n i n g method i s used i n s t e e p l y d i p p i n g t a b u l a r o r e b o d i e s , w i t h w i d t h v a r y i n g f r o m a few m e t e r s t o t e n s o f m e t e r s . F i v e d i f f e r e n t c u t and f i l l methods a r e a p p l i e d i n Sweden, F i g u r e 2.2. The c h o i c e o f method depends on t h e geometry o f t h e o r e body and t h e s t r e n g t h o f t h e o r e body and t h e w a l l r o c k . De-t a i l s o f De-t h e developmenDe-t o f De-t h e c u De-t and f i l l meDe-thod i n

t h e Swedish m i n i n g i n d u s t r y i s p r e s e n t e d by Almgren ( 1 9 8 1 ) .

Mined and back f i l l e d Unmined

A d e t a i l e d d e s c r i p t i o n o f t h e m i n i n g method w i t h h o r i z o n t a l d r i l l i n g and t h e equipment used i n Näsliden i s p r e s e n t e d by Stephansson and Jones ( 1 9 8 1 ) . M i n i n g i s a t p r e s e n t c a r r i e d o u t i n f o u r s t o p e s and ramps a r e d r i v e n f o r o p e n i n g o f s t o p e 5 and s t o p e 6 i n t h e y e a r s 1983-84, F i g u r e 2.3. The b o t t o m o f t h e s t o p e s a r e s i t u a t e d 5-10 m above t h e d r i f t s a t t h e 160, 260, 360 and 460 m l e v e l s . The s t r i k e o f t h e r o c k

s t r a t a i s N2 0°W. The main o r e body i s known a t t o e x t e n d down t o a d e p t h o f a t l e a s t 640 m. The h o r i z o n t a l c r o s s

2 s e c t i o n o f t h e o r e body d i m i n i s h e s w i t h d e p t h , f r o m 3450 m

2

a t t h e 40 m l e v e l t o 610 m a t t h e 610 m l e v e l . An open p i t was mined t o t h e 40 m:s l e v e l d u r i n g 1972-73. The a n n u a l p r o d u c t i o n i n 1975 was 250 000 tonnes o f o r e and i n 1981 200 000 t o n n e s .

8

F i g . 2.3 V e r t i c a l s e c t i o n o f t h e Näsliden Mine, a) c r o s s s e c t i o n , b) s e c t i o n a l o n g t h e s t r i k e , c) m i n i n g sequences.

2.3 Geology and s t r u c t u r e s i n t h e Näsliden Mine

The Näsliden d e p o s i t c o n s i s t s o f one l a r g e s u l p h i d e o r e body and a s m a l l o r e l e n s , F i g u r e 2.4. The l o w e r p a r t o f t h e s t r a t i g r a p h i c sequence i s made up o f a c i d v o l c a n i c s , such as p y r o c l a s t i c s and q u a r t s p o r p h y r y . The sequence o v e r l y i n g t h e o r e body c o n s i s t s o f p h y l l i t e s and s h a l e s i n t e r l a y e r e d w i t h t h i n h o r i z o n s o f f e l s i t i c t u f f . L i m e s t o n e l e n s e s a r e c o n t a i n e d i n t h e s h a l e s . A l t e r e d c o n t a c t s e x i s t between t h e o r e and t h e w a l l r o c k .

==9 S H A L E , P H Y L L I T E fl I I F E L S I T I C T U F F [ E S L I M E S T O N E [ • • ORE r QUARTZ PORPHYRE AGGLOMERATE Q U A R T Z I T E BRECCIA QUARTZITE 0 5 0 m -O BOREHOLE F i g . 2.4 H o r i z o n t a l s e c t i o n s o f main o r e body a t t h e Näsliden Mine. Mapped by C. T r e p k a - B l o c k .

1 o

The g e o l o g i c a l s t u r e t u r e s i n t h e Näsliden Mine a r e d e s c r i b e d by Stephansson ( 1 9 8 1 ) , and t h e most dominant s t r u c -t u r e s a r e i l l u s -t r a -t e d i n F i g u r e 2.5, The a l -t e r a -t i o n zone on t h e h a n g i n g w a l l s i d e i s t h e sequence o f t h e mine w i t h t h e h i g h e s t j o i n t f r e q u e n c y . Exposures o f t h e s i d e w a l l s i n t h e s t o p e s o f t e n show l a r g e , s l i g h t l y u n d u l a t i n g p l a n e s o f s c h i s t o s i t y , o f t e n c o v e r e d w i t h c a l c i t e o r c h l o r i t e , F i g u r e 2.6. Minimum j o i n t i n g i s f o u n d i n t h e compact s u l p -h i d e o r e . T-he main s t r u c t u r e s can be summarized as f o l l o w s :

- F a u l t s s t r i k i n g p a r a l l e l w i t h t h e o r e i n t h e upper p a r t o f t h e o r e body. - S t r i k e and d i p o f t h e o r e body i s p a r a l l e l w i t h t h e r e g i o n a l s c h i s o s i t y , - Three s e t s o f j o i n t s f o r m an o r t h o g o n a l j o i n t system. The d i r e c t i o n s o f t h e j o i n t s e t s a r e : 1) N-S/70°W 2) A p p r o x i m a t e l y p e r p e n d i c u l a r t o t h e d i p o f t h e o r e 3) E - W / v e r t i c a l

n = 98

N

Faults i n stope 1 , 2 and 3 Faults i n stope 4 , level -300 to -318 m

Faults i n stope 4 , level -321 to -354 m

= 64 h = 44

f a u l t s

5 a l S t e r e o g r a p h i c p r o j e c t i o n o f f a u l t s i n t h e Näsliden Mine. Data t a k e n f r o m m i n i n g s u r v e y and mapping. A ) " h a u l a g e s a t a l l l e v e l s , B) s t o p e 1 , C) s t o p e 2, D) s t o p e 3. A f t e r Stephansson, 1981. b) S t e r e o g r a p h i c p r o j e c t i o n o f p e r c e n -t a g e o f j o i n -t s , -t , e x c l u s i v e o f shear zones, a l o n g s e c t i o n 35-160 m i n haulage o f 460 m l e v e l , Näs-l i d e n Mine. A f t e r Stephansson, 1 9 8 1 .

12 F i g . 2.6 Example o f a p l a n a r s i d e w a l l i n s t o p e 3, Näsliden. A f t e r B a r t o n , 1 9 8 1 . 2.4 V i r g i n s t r e s s e s i n t h e Näsliden Mine The m a j o r i t y o f r e s u l t s f r o m s t r e s s measurements i n c o n t i -n e -n t a l s h i e l d a r e a s show t h a t t h e h o r i z o -n t a l s t r e s s e s

ex-ceeds t h e v e r t i c a l s t r e s s , Hoek and Brown ( 1 9 8 0 ) .

I n t h e Näsliden Mine, r e s u l t s o f s t r e s s measurements, w i t h t h e Leeman t r i - a x i a l o v e r c o r i n g method, were s u c c e s f u l l y o b t a i n e d a t f o u r l o c a t i o n s , L e i j o n , C a r l s s o n and Myrvang

( 1 9 8 1 ) . The mean v a l u e s , f r o m each t e s t s i t e were used t o

a s s i g n t h e s t a t e o f v i r g i n s t r e s s as a l i n e a r f u n c t i o n w i t h d e p t h , F i g u r e 2.7;

v _{- 0.026 x [MPa] (2,1)} a

x

v _{6.1 - 0,045 x [MPa] (2.2)}

Y

where a and o a r e t h e v e r t i c a l and h o r i z o n t a l v i r g i n

x y 3

s t r e s s components r e s p e c t i v e l y , and x i s t h e d e p t h i n m e t e r s . These components a p p r o x i m a t e l y c o i n c i d e w i t h t h e d i r e c t i o n o f t h e p r i n c i p a l s t r e s s e s and a^. Througout t h i s work we a d o p t t h e c o n v e n t i o n t h a t compressive s t r e s -ses and s t r a i n s a r e t a k e n t o be p o s i t i v e .

14 N S (a) STRESS [MPa] 10 20 30 40 I © © (b) F i g . 2.7 a) O r i e n t a t i o n s o f measured p r i n c i p a l s t r e s s e s shown i n s t e r e o g r a p h i c p r o j e c t i o n ( W u l f f s n e t , l o w e r h e m i s p h e r e ) . Numbers r e f e r t o l o c a t i o n s o f measurements. A f t e r L e i j o n e t a l , 1 9 8 1 . b) Measured v i r g i n s t r e s s e s as a f u n c t i o n o f d e p t h . L i n e s show s t r e s s d i s t r i b u t i o n s w h i c h w i l l be used as boundary s t r e s s e s i n o u r f i n i t e element models: (1) v e r t i c a l s t r e s s , a^, (2) h o r i z o n t a l s t r e s s , a^, f o r v e r t i c a l s e c t i o n p e r p e n d i c u l a r t o t h e s t r i k e o f t h e o r e . A f t e r L e i j o n e t a l , 1981.

3 OBSERVATIONS OF FAILURES IN THE NÄSLIDEN MINE

An i m p o r t a n t p a r t o f t h e o b s e r v a t i o n and f i e l d measurement program i n t h e Näsliden Mine i s t h e r e c o r d i n g o f f a i l u r e phenomena. These d a t a were p r e s e n t e d by N i l s s o n and K r a u l a n d ( 1 9 8 1 ) . The r e s u l t s o f t h i s program a r e used t o c l a s s i -f y t h e d i -f -f e r e n t t y p e s o -f -f a i l u r e s and d e t e r m i n e c r i t i c a l s t a g e s i n t h e development o f t h o s e t y p e s o f f a i l u r e s t h a t a r e due t o l a r g e m a g n i t u d e s i n t h e l o a d s .

3.1 Types and o c c u r e n c e o f f a i l u r e s a r o u n d t h e s t o p e s

F i g u r e 3.1 shows t h e common t y p e s o f f a i l u r e i n c u t and f i l l m i n i n g as o b s e r v e d i n Näsliden and o t h e r mines o f t h e B o l i d e n M i n e r a l AB. F i g u r e 3.1 A shows a wedge f a i l u r e w i t h a s l i g h t l y i n c l i n e d f a i l u r e s u r f a c e . T h i s t y p e o f f a i l u r e o f t e n o c c u r s t o g e t h e r w i t h f r a c t u r e s i n t h e r o o f and i n t h e f o o t and h a n g i n g w a l l , F i g u r e 3.1 D, E and G. F i g u r e 3.1 B shows a t y p e o f f a i l u r e w i t h a s t e e p l y i n c l i n e d f a i l u r e s u r f a c e . T h i s t y p e o f f a i l u r e o f t e n o c c u r s t o g e t h e r w i t h r o c k f a l l i n t h e h a n g i n g w a l l , F i g u r e 3.1 F, and i s t h o u g h t t o be due t o t h e r e g i o n a l s c h i s t o s i t y p l a n e s i n t h e c o n t a c t between t h e o r e and w a l l r o c k . F i g u r e 3.1 C shows a t y p e o f f a i l u r e w h i c h o f t e n o c c u r s a t t h e e a r l i e r s t a g e s o f m i n i n g , i f adverse s t r u c t u r a l f e a -t u r e s a r e p r e s e n -t .

1 6 D E F G F i g . 3 . 1 Types o f f a i l u r e s . A) wedge f o r m a t i o n i n r o o f , o f t e n v i o l e n t , B) r o o f f a i l u r e when h a n g i n g w a l l i s v e r y weak, C) r o o f f a i l u r e due t o s t r u c -t u r a l f e a -t u r e s i n -t h e r o o f , D ) , E) f o o -t w a l l f r a c t u r e s , F ) , G) h a n g i n g w a l l f a i l u r e s . A f t e r N i l s s o n e t a l , 1 9 8 1 .

The most commonly f o u n d t y p e s o f f a i l u r e i n v a r i o u s s t o p e s

o f t h e Näsliden Mine a r e summarized i n Table 3 . 1 . I t i s

f o u n d t h a t f a i l u r e o f t y p e C i n F i g u r e 3 . 1 i s o b s e r v e d o n l y

i n s t o p e 1 and a t t h e e a r l i e r s t a g e s o f m i n i n g . The f a i l u r e

o f t y p e B i n F i g u r e 3 . 1 i s n o t common i n t h e Näsliden Mine.

The f a i l u r e s o f t y p e A i n F i g u r e 3 . 1 o c c u r , however, f r e

-q u e n t l y . A l o n g w i t h t h e r o o f f a i l u r e s wedge f a i l u r e s o c c u r

i n t h e f a c e o f t h e m i n i n g , T a b l e 3 . 1 , a t l a t e r s t a g e s o f

m i n i n g . These t y p e s o f f a i l u r e were common d u r i n g m i n i n g o f

t h e l a s t two s l i c e s i n s t o p e 3 b e f o r e m i n i n g had t o be t e r

18

There i s , o f c o u r s e , a v a r i a t i o n i n g e o l o g i c a l c o n d i t i o n s and hence i n m e c h a n i c a l p r o p e r t i e s o f t h e r o c k mass, r e s u l -t i n g i n -t h e v a r i a -t i o n s o f -t y p e s o f f a i l u r e s o b s e r v e d . One s i g n i f i c a n t d i f f e r e n c e i s t h e c o m p o s i t i o n o f t h e f o o t w a l l a l t e r a t i o n zone. I n t h e s o u t h e r n p a r t o f s t o p e 3 i t con-s i con-s t con-s o f h a r d con-s e r i c i t i c q u a r t z i t e . I n t h i con-s a r e a o f t h e mine v i o l e n t f a i l u r e o f t h e r o o f o c c u r r e d , whereas t h e f o o t w a l l remained i n t a c t . I n t h e n o r t h e r n p a r t o f t h e o r e body t h e f o o t w a l l a l t e r a t i o n zone c o n s i s t s o f s o f t c h l o r i t i c q u a r t -z i t e w i t h low s t r e n g t h and a low modulus o f e l a s t i c i t y . Here t h e r o o f i n t h e o r e body remained i n t a c t , whereas t h e f o o t w a l l was f r a c t u r e d .

3.2 Stages o f f a i l u r e s

The f o l l o w i n g c r i t i c a l s t a g e s i n t h e development o f f a i l u r e can be d i s t i n g u i s h e d i n s t o p e 3. Only f a i l u r e t h a t a r e due t o l a r g e m a g n i t u d e s i n t h e l o a d s a r e c o n s i d e r e d . Stage 1: B r i t t l e f r a c t u r e o f t h e r o o f o c c u r s , o f t e n v i o -l e n t . N e a r -l y h o r i z o n t a -l f r a c t u r e s u r f a c e s a r e t h e n c r e a t e d . S t a b i l i t y o f t h e s t o p e i s m a i n -t a i n e d by f a i r l y s p a r s e r o c k s u p p o r -t , m a i n l y r o c k b o l t i n g . Stage 2: F a i l u r e o f t h e f a c e c o n s i s t i n g o f subsidence o f l a r g e wedges and r o o f f a i l u r e s . M i n i n g can s t i l l be c o n t i n u e d , b u t f r e q u e n t s c a l i n g and i n c r e a s e d amount o f s u p p o r t a r e n e c e s s a r y . Stage 3: I n t e n s i v e r o o f f a i l u r e s i n c l u d i n g c r u s h i n g , a l s o a t a g r e a t d i s t a n c e b e h i n d t h e f a c e . Roof f a i -l u r e s cannot be a -l -l e v i a t e d by s c a -l i n g and r o o f b o l t i n g . T e r m i n a t i o n o f m i n i n g o r change over t o a n o t h e r m i n i n g o r s u p p o r t method.

A g r a p h i c a l p r e s e n t a t i o n o f t h e f i e l d o b s e r v a t i o n s d e f i n i n g f a i l u r e s t a g e s 1-3 i n s t o p e 3 i s g i v e n i n F i g u r e 3 . 2 . The c r i t i c a l l e v e l s f o r t h e r o o f o f s t o p e 3 a r e shown i n T a b l e 3 . 2 . £ 100 X Stope number Failure stages 1 2 -100 -300 -500 Depth below surface , x [m]

LEGEND

1 Roof level at onset of b r i t t l e f r a c t u r e

2 Roof level at onset of rock f a l l

3 Roof level at u l t i m a t e roof f a i l u r e • Termination of cut and f i l l mining

Failures occurring between f a i l u r e stages 2 and 3

|;-;| Failures occurring between f a i l u r e stages 1_ and 2

20 T a b l e 3.2 Stages o f r o o f f a i l u r e s as o b s e r v e d i n s t o p e 3 o f t h e Näsliden Mine. Stage o f r o o f f a i l u r e Year H e i g h t o f back f i l l e d e x c a v a t i o n H e i g h t o f o r e between s t o p e 2 and 3 E l e v a t i o n o f r o o f H e [m] H r tm] [m] 1 1973 34. 66. -321 . 2 1979 70. 30. -285. 3 1 981 89. 11 . -266 . A summary o f f a i l u r e o b s e r v a t i o n s o f a l l s t o p e s i n Näsliden up t o 1982 i s g i v e n i n F i g u r e 3.3. The f i r s t f r a c t u r e s and f a i l u r e s a r e f o u n d t o o c c u r a t e a r l i e r s t a g e s o f m i n i n g i n t h e more d e e p l y l o c a t e d s t o p e s . F o o t w a l l d r i f t s a l o n g t h e o r e body e x i s t a t t h e -160 m, -260 m and -360 m l e v e l s . They a r e s i t u a t e d m a i n l y i n t h e a l t e r a t i o n zone a t d i s t a n c e s o f 0 t o 4 m f r o m t h e o r e body. E x t e n s i v e damage t o t h e s e d r i f t s i s o b s e r v e d as m i n i n g

approaches f r o m below and as t h e h e i g h t o f t h o r e body between t h e s t o p e s above and below t h e d r i f t d e c r e a s e s . The r o o f l e v e l s a t t h e o n s e t o f f a i l u r e i n t h e d r i f t s a r e shown i n F i g u r e 3.3.

S u r f a c e H- i • 1 — • — • — i — • — • — i — • — • — i — — i • 'i 0 - 1 0 0 - 3 0 0 - 5 0 0 Depth below s u r f a c e [ m ] LEGEND

1_ Roof level at onset of b r i t t l e f r a c t u r e 2 Roof level a t onset of rock f a l l

3 Roof level a t u l t i m a t e roof f a i l u r e I—I Present roof level

n Roof level at onset of f a i l u r e s i n f o o t wall d r i f t s • Termination of cut and f i l l mining

%M Failures occurring between f a i l u r e stages 2 and 3

|:;| Failures occurring between f a i l u r e stages 1 and 2

F i g . 3.3 O c c u r r e n c e o f f a i l u r e s i n t h e Näsliden Mine up t o 1 9 8 2 .

22

The f a i l u r e s i n s t o p e 3 when t h e m i n i n g was t e r m i n a t e d a r e i l l u s t r a t e d i n F i g u r e 3.4. The most c r i t i c a l f a i l u r e o c c u r r e d i n t h e r o o f beyond t h e f a c e o f t h e m i n i n g , a f t e r s c a l -i n g and b o l t -i n g . A t t h e same t -i m e o b s e r v a t -i o n s o f f r a c t u r e s were c o n d u c t e d f r o m t h e d r i f t on -260 m l e v e l and f r o m an

e x i s t i n g r a i s e i n t h e o r e body between s t o p e 3 and t h e -260 m l e v e l , F i g u r e 3.5. F a i l u r e s t h a t have o c c u r r e d i n s t o p e 4 d u r i n g m i n i n g o f s l i c e s 9, 10, 1 1 , and 12 a r e shown i n F i g u r e 3.6. F a i l u r e s o f t h e h a n g i n g w a l l and i n t h e f r o n t o f t h e m i n i n g f a c e a r e b e e i n g e x p e r i e n c e d . I - I Sections I I - I I n F i g . 3.4 I l l u s t r a t i o n o f t h e o b s e r v e d f a i l u r e s i n s t o p e 3 d u r i n g m i n i n g o f t h e l a s t s l i c e , No. 24, p l a n v i e w and c r o s s s e c t i o n I - I and I I - I I . Mapped by R. Tjärnlund.

Foot wall d r i f t / / / F r a c t u r e d rock Stope 3 F r a c t u r e s i n t h e d r i f t -260 m e t e r s below s u r f a c e , S e c t i o n a c r o s s t h e s t o p e . Mapped by R. Tjärnlund, LT - n S e c t i o n s Sequence o f f a i l u r e s d u r i n g m i n i n g o f s l i c e s 9-12 i n s t o p e 4. P l a n v i e w s and s e c t i o n s I - I t o V-V. Mapped by R. Tjärnlund.

24

4 CONSTITUTIVE RELATIONS FOR THE ROCK MASS

E v i d e n c e o f b r i t t l e f r a c t u r e i n t h e Näsliden Mine, has been

d e m o n s t r a t e d i n c h a p t e r 3 . I t i s known t h a t r o c k s , e x h i b i

t i n g b r i t t l e f r a c t u r e b e h a v i o u r , shows a l m o s t l i n e a r e l a s -t i c b e h a v i o u r up -t o s -t r e n g -t h l i m i -t s i n -t h e d i r e c -t i o n o f -t h e maximum p r i n c i p a l s t r e s s . These f i n d i n g s l e a d t o t h e app-r o a c h a d o p t e d i n t h i s woapp-rk. The c o n s t i t u t i v e app-r e l a t i o n s h i p s f o r t h e models are l i m i t e d t o t h e f o l l o w i n g i t e m s : - E l a s t i c c o n s t a n t s f o r t h e model m a t e r i a l s t o be used i n t h e f i n i t e element a n a l y s i s , a r e g i v e n i n s e c t i o n 4.1. - F a i l u r e c r i t e r i a t o be used f o r d e t e r m i n a t i o n o f s t r e n g t h l i m i t s f o r t h e r o c k , a r e g i v e n i n s e c t i o n 4.2.

I n a d d i t i o n t o t h i s a p p r o a c h , j o i n t elements have been t e s t e d i n t h e Näsliden P r o j e c t , t o s i m u l a t e weak c o n t a c t s between t h e o r e body and t h e a d j a c e n t a l t e r a t i o n zones. The c o n s t i t u t i v e r e l a t i o n s h i p s f o r t h e s e c o n t a c t s a r e d i s -cussed i n s e c t i o n 4.3.

4.1 L i n e a r e l a s t i c m a t e r i a l b e h a v i o u r

The r o c k mass i s d i v i d e d i n t o g r o u p s , depending on t h e m e c h a n i c a l p r o p e r t i e s o f t h e v a r i o u s r o c k t y p e s and t h e

s t r u c t u r e s o c c u r r i n g w i t h i n them, Stephansson ( 1 9 8 1 ) . Each group i s assumed t o be i s o t r o p i c i n i t s m e c h a n i c a l p r o p e r -t i e s . The r o c k i n each g r o u p i s c o n s i d e r e d as an e l a s -t i c c o m p o s i t e m a t e r i a l , composed o f t h e i n t a c t r o c k and t h e j o i n t s , s e c t i o n 2.3. The e q u a t i o n f o r 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 , d e s c r i b e d by Goodman ( 1 9 7 6 ) , i s a -p -p l i e d : E 1 (4.1 ) rm 1 k • d n

where E i s t h e e f f e c t i v e Young's modulus o f t h e r o c k rm ^ mass and E^ i s Young's modulus o f t h e i n t a c t r o c k f r o m t e s t s o f c o r e s , b o t h measured i n GPa, and k i s t h e n o r

-' n

mal s t i f f n e s s f o r t h e j o i n t s , GPa/m, and d i s t h e d i s t a n c e between t h e j o i n t s i n m e t e r s . 4.2 F a i l u r e c r i t e r i a A x i a l c l e a v a g e f r a c t u r i n g i s f o u n d t o be a p r e d o m i n a n t e l e -ment i n a l l p r o c e s s e s o f t h e f r a c t u r e o f b r i t t l e r o c k , Gramberg ( 1 9 6 5 ) . A t low c o n f i n i n g s t r e s s e s a x i a l c l e a v a g e f r a c t u r i n g can l e a d t o l o n g i t u d i n a l s p l i t t i n g o f a s p e c i -men, F i g u r e 4.1. However, a t moderate c o n f i n i n g s t r e s s e s t h e a x i a l c l e a v a g e f r a c t u r e s w i l l n o t l e a d t o f i n a l f a i l u r e , John ( 1 9 7 2 ) . I n s t e a d shear f r a c t u r e s o c c u r as i l l u -s t r a t e d i n F i g u r e 4.1. l a ) l b ) k ) F i g . 4.1 Types o f f r a c t u r e , a) l o n g i t u d i n a l s p l i t t i n g i n u n i a x i a l c o m p r e s s i o n , b) shear f r a c t u r e , c)

m u l t i p l e shear f r a c t u r e s . A f t e r Jaeger and Cook, 1979.

26

Many o b s e r v a t i o n s o f a x i a l c l e a v a g e f r a c t u r i n g were made i n t h e Näsliden Mine. An e x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n , ESFC, s u g g e s t e d by Stacey ( 1 9 8 1 ) , i s s u i t e d f o r e v a l u a t i o n o f t h e s e f r a c t u r e s . As d e s c r i b e d , t h i s t y p e o f f r a c t u r i n g can o n l y l e a d t o f a i l u r e s i n p a r t s where t h e c o n f i n i n g s t r e s s e s a r e low, F i g u r e 4.1. S p l i t t i n g t y p e s o f f a i l u r e s can t h e r e f o r e most l i k e l y d e v e l o p i n n a r r o w zones s u r r o u n d i n g t h e r o o f and i n t h e m i n i n g f a c e . I n s p e c t i o n s o f f a i -l u r e s u r f a c e s i n t h e r o o f and s i d e w a -l -l s have a -l s o shown e v i d e n c e o f shear f r a c t u r e s and o f f - s h e a r e d a s p e r i t i e s and r o c k b o l t s a r e f o u n d a l o n g t h e f a i l u r e s u r f a c e . The Coulomb f a i l u r e c r i t e r i o n i s s u i t e d f o r e v a l u a t i o n o f t h e s e f r a c t u r e s .

I n summary i t can be s a i d t h a t t h e mechanisms o f f a i l u r e s a r o u n d t h e s t o p e s a r e complex and dependent on t h e e x i s t e n c e o f j o i n t s and s c h i s t o s i t y p l a n e s . C r u s h i n g , s p l i t t i n g and shear f r a c t u r e s a r e , however, f o u n d t o be i m p o r -t a n -t componen-ts i n -t h e f r a c -t u r e p r o c e s s . F a i l u r e s o f -t h e r o o f due t o s p l i t t i n g and shear f r a c t u r e s a r e t h e r e f o r e c o n s i d e r e d t o be o f m a j o r i m p o r t a n c e . T h e r e f o r e b o t h t h e e x t e n s i o n s t r a i n c r i t e r i o n and Coulomb f a i l u r e c r i t e r i o n a r e examined i n o u r work.

4.2.1 The e x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n (ESFC)

The ESFC i s d e f i n e d as f o l l o w s by Stacey ( 1 9 8 1 ) , F i g u r e 4.2: F r a c t u r e o f b r i t t l e r o c k w i l l i n i t i a t e when t h e t o t a l e x t e n -s i o n -s t r a i n i n t h e r o c k exceed-s a c r i t i c a l v a l u e , w h i c h i -s c h a r a c t e r i s t i c o f t h a t r o c k t y p e . F r a c t u r e i n i t i a t e s when 0 > £c > e (4.2) where i s t h e c r i t i c a l v a l u e o f t h e e x t e n s i o n s t r a i n . As m e n t i o n e d a l r e a d y i n s e c t i o n 2.4 we t h r o u g h o u t t h i s work a d o p t t h e c o n v e n t i o n t h a t c o m p r e s s i v e s t r e s s e s and s t r a i n s

a r e t a k e n t o be p o s i t i v e . The f r a c t u r e s w i l l f o r m i n p l a n e s normal t o t h e d i r e c t i o n o f t h e e x t e n s i o n s t r a i n , w h i c h c o r r e s p o n d s w i t h t h e d i r e c t i o n o f t h e minimum p r i n c i p a l s t r a i n . For a m a t e r i a l w h i c h shows i d e a l l i n e a r e l a s t i c d e f o r m a t i o n b e h a v i o u r , t h e s t r a i n i n t h i s d i r e c t i o n i s r e -l a t e d t o t h e t h r e e p r i n c i p a -l s t r e s s e s by t h e f o -l -l o w i n g e q u a t i o n : e3 =

i

( a3 " V( 01 + 02 )] (4°3) rm assuming p l a n e s t r a i n c o n d i t i o n s .

H • i

I e

i i

r i

I

"i

i

L J L

J

d , ( b ) (c) F i g . 4.2 The e x t e n s i o n s t r a i n f a i l u r e c r i t e r i o n , ESFC. a) u n l o a d e d sample, b) e x t e n s i o n s t r a i n Co i n d u c e d by a x i a l and l a t e r a l c o m p r e s s i o n s t r e s s e s , c) e x t e n s i o n s t r a i n f r a c t u r e s . The c o n d i t i o n f o r e x t e n s i o n s t r a i n s i s t h u s g i v e n by: v ( a1 + a2) > a2 (4.4) T h i s i l l u s t r a t e s t h a t e x t e n s i o n f r a c t u r e s can f o r m when a l l t h r e e p r i n c i p a l s t r e s s components a r e c o m p r e s s i v e and t h u s a l s o i n p l a n e s a c r o s s , w h i c h t h e n e t macro s t r e s s i s compres-s i v e . Moreover, a f t e r t h e f r a c t u r e hacompres-s formed t h e n e t

28

4.2.2 The Coulomb f a i l u r e c r i t e r i o n (CFC)

The shear s t r e n g t h w i t h t h e Coulomb f a i l u r e c r i t e r i o n i s g i v e n by: Tp = c + an t a n <f> (4.5) where c i s t h e c o h e s i o n o f t h e m a t e r i a l , cj) i s t h e i n t e r n a l a n g l e o f f r i c t i o n , and i s t h e normal s t r e s s a c r o s s t h e p l a n e o f f a i l u r e , F i g u r e 4.3. M a c r o s c o p i c f a i l u r e i n i t i a t e s i f : T > TP (4.6) where, T = (ø1 - a3) s i n 2ß (4.7) assuming p l a n e s t r a i n c o n d i t o n s . The i n c l i n a t i o n o f t h e f a i l u r e p l a n e s , 0, a r e g i v e n by: ß = ± (45° - <}>/2) (4.8) The e x p r e s s i o n f o r t h e shear s t r e n g t h i n e q u a t i o n (4.5) i s a l i n e a r a p p r o x i m a t i o n o f t h e r e a l n o n - l i n e a r shear f a i l u r e envelope e x h i b i t e d f o r most h a r d r o c k t y p e s . The r o o f f a i -l u r e s i n c u t and f i -l -l s t o p e s o c c u r i n areas where t h e magni-tudes o f c o n f i n i n g p r e s s u r e , a-, have s m a l l v a l u e s . The l i n e a r a p p r o x i m a t i o n i s t h e r e f o r e c o n s i d e r e d a c c e p t a b l e f o r t h e s e c o n d i t o n s .

F i g . 4.3 The Coulomb f a i l u r e c r i t e r i o n , CFC. a) s k e t c h o f t h e l i n e a r s t r e n g t h c r i t e r i o n , b) i n c l i n a t i o n o f t h e p l a n e o f f a i l u r e .

4.3 J o i n t e l e m e n t s

I n a d d i t i o n t o t h e e l a s t i c model, a j o i n t e l e m e n t model o f t h e Näsliden Mine has been t e s t e d . The j o i n t e l e m e n t s a r e used f o r s i m u l a t i o n o f t h e weak c o n t a c t s between t h e o r e body and t h e w a l l r o c k . These c o n t a c t s a r e f o u n d t o be i m p o r t a n t f o r f a i l u r e s o c c u r r i n g i n t h e s t o p e s , F i g u r e 3.1 A and 3.1 B. The r e s u l t s o f t h e j o i n t e l e m e n t model w i l l be compared w i t h t h e e l a s t i c model. We f i r s t d i s c u s s t h e con-s t i t u t i v e r e l a t i o n con-s h i p f o r t h e j o i n t e l e m e n t con-s .

A d e t a i l e d d e s c r i p t i o n o f j o i n t e l e m e n t s a r e g i v e n by G r o t h ( 1 9 8 1 ) . The laws f o r shear b e h a v i o u r a r e shown i n F i g u r e 4.4, B a r t o n e t a l ( 1 9 7 7 ) . The j o i n t s i n o u r c a l c u l a t i o n s a r e assumed t o be n o n - d i l a t i v e . The peak v a l u e o f t h e shear s t r e n g t h i s e v a l u a t e d by use o f B a r t o n ' s f o r m u l a :

T~ ~ , v = °" t a n (JRC l o g (JCS/a ) + cj> ) (4.9)

30 f o r a > a > 0 c n where JRC i s t h e j o i n t roughness c o e f f i c i e n t . JCS i s t h e j o i n t c o m p r e s s i v e s t r e n g t h , and <f> i s t h e r e s i d u a l f r i c -t i o n a n g l e . The r e s i d u a l v a l u e , x , i s assumed -t o be ^ r e s r e a c h e d a f t e r a d i s p l a c e m e n t o f Au = 10 • A u , „ , has c r e s peak

d e v e l o p e d , where Au i s t h e shear d i s p l a c e m e n t . When A ur e s

i s r e a c h e d , JRC = 0, w h i c h g i v e s : T = a t a n $ (4.10) r e s n Tr e s There i s a l s o an u l t i m a t e shear s t r e n g t h v a l u e , T j . , w h i c h i s a f u n c t i o n o f t h e c o m p r e s s i v e s t r e n g t h , a , o f t h e j o i n t w a l l s , w h i c h i s s e t e q u a l t o t h e j o i n t c o m p r e s s i v e s t r e n g t h , t h u s : T , , — JCS t a n <b (4.11) u l t Yr e s B a r t o n e t a l (1977) f o u n d , i n g e n e r a l t h a t T a^ i s f o u n d t o be a t t a i n e d a f t e r a shear d i s p l a c e m e n t o f 1 % o f Lc a l o n g a j o i n t w i t h a c r i t i c a l j o i n t l e n g t h v a l u e , L . A c c o r d i n g l y t h e t a n g e n t s t i f f n e s s v a l u e , k , i s g i v e n by: k = 100 T . /L (4.12' s peak c

Shear d i s p l a c e m e n t s along t h e j o i n t , A u (c)

F i g . 4.4 R e l a t i o n s h i p between t h e shear r e s i s t a n c e and t h e shear d i s p l a c e m e n t i n a j o i n t e l e m e n t , a) j o i n t i n r o c k , b) f i n i t e e l e m e n t j o i n t d i s c r e t i z a t i o n , c) shear s t r e s s v e r s u s shear d i s p l a c e -ment o f t h e j o i n t , f o r a g i v e n a v a l u e . A f t e r

G r o t h , 1981. n

Compression t e s t s on j o i n t s show t h e normal s t r e s s - normal d i s p l a c e m e n t b e h a v i o u r can be a p p r o x i m a t e d by a h y p e r b o l a , i n F i g u r e 4. 5 . The j o i n t s can be g i v e n a t e n s i l e s t r e n g t h , T , a l s o shown i n F i g u r e 4. 5 . The f o l l o w i n g i n p u t v a l u e s a r e r e q u i r e d t o c a l c u l a t e t h e normal s t r e s s - d i s p l a c e m e n t b e h a v i o u r ; Tg , j o i n t t e n s i l e s t r e n g t h \ , s e a t i n g l o a d v , maximum normal c l o s u r e o f j o i n t s f r o m mc J s e a t i n g l o a d p o s i t i o n n , s p e c i f i e d normal l o a d d i s p l a c e m e n t a c h i e v e d a t l o a d i n g f r o m E, t o

32 ^ 9 ^ 9 I " ! T ^ 1 s -Normal displacement, Av r vtc

/

_c 1 / "D . / _<n 1 W / cu t—

j

1 / _"5 1

f

£ o 1 z (c) F i g . 4.5 R e l a t i o n s h i p between n o r m a l s t r e s s and n o r m a l d i s p l a c e m e n t o f a j o i n t e l e m e n t , a) j o i n t i n r o c k , b) f i n i t e element j o i n t d i s c r e t i z a t i o n , c) normal s t r e s s v e r s u s normal d i s p l a c e m e n t o f t h e j o i n t . A f t e r G r o t h , 1981.

5 NUMERICAL PROCEDURE FOR FINITE ELEMENT CALCULATIONS

The FEM f o r m u l a t i o n s a r e based on t h e c o n s t i t u t i v e r e l a -t i o n s p r e s e n -t e d i n c h a p -t e r 4, and -t h e geome-try o f -t h e mine, c u t and f i l l e x c a v a t i o n t e c h n i q u e , as p r e s e n t e d i n c h a p t e r 2. The f o r m u l a t i o n o f t h e i s o t r o p i c l i n e a r e l a s t i c model i s g i v e n i n more d e t a i l . We p r e s e n t o n l y a s h o r t d i s c u s s i o n o f t h e j o i n t e l e m e n t f o r m u l a t i o n . C o n s i d e r a t w o - d i m e n s i o n a l v e r t i c a l s e c t i o n o f t h e o r e body, F i g u r e 5.1. T w o - d i m e n s i o n a l p l a n e s t r a i n and s t a t i c c o n d i t i o n s a r e assumed. The p l a n e s t r a i n a s s u m p t i o n i s a d o p t e d s i n c e t h e l e n g t h t o w i d t h r a t i o o f t h e o r e body i s o f t h e o r d e r o f 5-10, F i g u r e 2.3. The d e f i n i t i o n s o f t h e i n t e r n a l and o u t e r b o u n d a r i e s , a r e s p e c i f i e d by t h e c a r t e s i a n c o o r d i n a t e system and t h e d i s p l a c e m e n t v e c t o r s a l s o shown i n F i g u r e 5.1. The d i r e c -t i o n s o f -t h e p r i n c i p a l v i r g i n s -t a -t e o f s -t r e s s componen-ts ( a ^ , a^, o"^) a r e f o u n d t o a p p r o x i m a t i v e l y c o i n c i d e w i t h t h e y-, Z-, x-axes r e s p e c t i v e l y .

The model i s assumed t o be i n e q u i l i b r i u m b e f o r e e x c a v a t i o n has s t a r t e d . E x c a v a t i o n s i n d u c e a d i s t u r b a n c e o f e q u i l i b r i u m and r e d i s t r i b u t i o n o f s t r a i n s and s t r e s s e s . I f s t a b l e con-d i t i o n s e x i s t , a new e q u i l i b r i u m c o n f i g u r a t i o n w i l l be e s t a b l i s h e d . T h i s e q u i l i b r i u m c o n f i g u r a t i o n i s used t o e e c a l c u l a t e t h e i n d u c e d d i s p l a c e m e n t s , u , s t r e s s e s , a , and s t r a i n s , e , f r o m m i n i n g t h e s t o p e s . The t o t a l s t r e s s e s and s t r a i n s i n t h e s t r u c t u r e a f t e r e x c a v a t i o n a r e t h e n g i v e n by: v , e o = a + a (5.1 ) £ = v e £ + £ ( 5 . 2 )

34

F i g . 5. 1 FEM m o d e l l i n g o f t h e Näsliden Mine. D e f i n i t i o n s

o f t h e c o o r d i n a t e system ( x , y, z ) , t h e d i s p l a c e

-ment v e c t o r (u , u , uz) , t h e b o u n d a r i e s ( S1 ,

s£, S°). The p r i n c i p a l v i r g i n s t r e s s components (aY, o-v, o p a r e a l s o i l l u s t r a t e d , a) c r o s s sec-t i o n o f sec-thS mine, b) d e sec-t a i l o f an e x c a v a sec-t i o n , c) i d e a l i z e d c r o s s s e c t i o n o f t h e mine w i t h o u t e r FEM b o u n d a r i e s , d) i d e a l i z e d d e t a i l o f an e x c a v a t i o n .

5.1 F i n i t e e l e m e n t f o r m u l a t i o n 5.1.1 The i s o t r o p i c l i n e a r e l a s t i c model The p o t e n t i a l e n e r g y p r i o r t o m i n i n g , UV, i s g i v e n by: Uv = 1 foV: eV dV + fF - uV dS (5.3) 2 f ~ ~ j ~s ~s V so where av i s t h e v i r g i n s t a t e o f s t r e s s , and eV i s t h e v i r g i n s t a t e o f s t r a i n f o r a g i v e n volume V. F and uV ~s ~s a r e t h e boundary t r a c t i o n s and d i s p l a c e m e n t s . The p r o c e d u r e o f e x c a v a t i o n can be s i m u l a t e d w i t h a d i r e c t o r i n c r e m e n t a l " e x c a v a t i o n t e c h n i q u e " , F i g u r e 5.2. F i g . 5.2 E x c a v a t i o n t e c h n i q u e s f o r m o d e l l i n g a mine. a) e x c a v a t i o n i n t h e mine, b) d i r e c t e x c a v a t i o n t e c h n i q u e , c) i n c r e m e n t a l e x c a v a t i o n t e c h n i q u e .

3 6 The d i r e c t e x c a v a t i o n t e c h n i q u e i s a p p l i e d f o r t h e l i n e a r e l a s t i c model. I n t h e d i r e c t e x c a v a t i o n t e c h n i q u e , F i g u r e 5,2 b, t h e s t a t e o f s t r e s s i n t h e e x c a v a t e d volume i s i n -s t a n -s t a n i o u -s l y changed f r o m t h e v i r g i n -s t r e -s -s e -s o f t h e r o c k t o t h e i n i t i a l s t r e s s e s i n t h e f i l l m a t e r i a l , ai f. The m i n i n g i s s i m u l a t e d by o n l y one u n l o a d i n g i n c r e m e n t by r e -l a x a t i o n o f t h e f o r c e s a -l o n g t h e s u r f a c e o f t h e e x c a v a t i o n , S , F i g u r e 5.1 d. Each c a l c u l a t e d m i n i n g c o n f i g u r a t i o n i s e n t i t l e d a sequence. The a c t u a l m i n i n g o f many s l i c e s i s s i m u l a t e d by o n l y one sequence. T h i s a p p r o a c h i s p o s s i b l e s i n c e t h e a s s u m p t i o n o f l i n e a r e l a s t i c i t y g i v e s r e s u l t s w h i c h w i l l be i n d e p e n d e n t o f t h e s t r e s s p a t h f o l l o w e d . The i n i t i a l s t a t e o f s t r e s s i n t h e f i l l m a t e r i a l i s g i v e n by: o^f = - xf pf g (5.4) i f v f i f /r- . °y = ox (5.5) where x^ i s t h e v e r t i c a l d e p t h b e n e a t h t h e r o o f o f t h e

e x c a v a t i o n , and pf and vf a r e t h e d e n s i t y and P o i s s o n ' s

r a t i o o f t h e f i l l m a t e r i a l r e s p e c t i v e l y . The p o t e n t i a l e n e r g y , U , a f t e r a sequence o f e x c a v a t i o n i s g i v e n by: US = Uf + Ur + Ub (5.6) where U_ i s t h e s t o r e d energy i n t h e f i l l m a t e r i a l , U i s f r t h e s t o r e d energy i n t h e r o c k , and i s t h e p o t e n t i a l o f t h e t r a c t i o n s a t t h e o u t e r boundary o f t h e model. I t i s assumed t h a t t h e m a t e r i a l i s l i n e a r l y e l a s t i c . T h i s i m p l i e s t h a t t h e change o f energy due t o t h e e x c a v a t i o n i s e q u a l t o t h e sum o f t h e i n d u c e d s t r a i n e n e r g y i n t h e r o c k

and back f i l l and t h e work a t t h e o u t e r boundary. No energy i s d i s s i p a t e d due t o f r a c t u r i n g , see F i g u r e 5.1,

The i n d i v i d u a l terms i n e q u a t i o n (5.6) a r e g i v e n by:

V = j fa : e dV (5.7) V f ~ U =

i

fa : £ dV (5.8) r 2 •* ~ ^ V ü b = / E s ' ä s d S ( 5'9 : S° where a, e d e n o t e s t h e t o t a l s t r e s s e s and s t r a i n s i n t h e

back f i l l and r o c k a f t e r t h e e x c a v a t i o n , and Vf and Vr

d e n o t e t h e volumes o f t h e back f i l l e d e x c a v a t i o n and t h e r o c k mass r e s p e c t i v e l y . F , u denotes t h e t r a c t i o n s and

f 2 ~s ~s d i s p l a c e m e n t s a t t h e o u t e r b o u n d a r i e s . The s t r e s s e s and s t r a i n s a f t e r t h e e x c a v a t i o n s a r e g i v e n by: a = aV + øe , I n Vr (5.10) a - al f + oe , I n V, (5.11) ~ ~ ~ r e = eV + ee , I n Vr (5.12) £ = £l f + £e , I n V. (5.13) Using e q u a t i o n s (5.10) — (5.13) , e q u a t i o n s (5.7) and (5.8) can be w r i t t e n i n t h e f o l l o w i n g f o r m s :

38 = 1 fae:ze dV + faif:ee dV + \ / a if : ei f dV 'f 2 V, V, v, (5,14) 1 /* e e U__r 2 = — la : £ V / e e ,.T r v e ,T I , 1 T v v ,T T a i £ dV + / a :£ dV + r J a : E dV V V (5.15) U s i n g e q u a t i o n s ( 5 . 6 ) , ( 5 . 9 ) , (5.14) and ( 5 . 1 5 ) , t h e t o t a l p o t e n t i a l e n e r g y , Ue, a f t e r t h e e x c a v a t i o n and back f i l l i n g i s g i v e n by: (5.16)

The p o t e n t i a l energy i s s t a t i o n a r y when t h e e q u i l i b r i u m i s o b t a i n e d . I n t h i s case, we have:

6 Ue = 0 (5.17)

E q u a t i o n s (5.16) and (5.17) a r e used f o r t h e f i n i t e element f o r m u l a t i o n i n t h e l i n e a r l y e l a s t i c model o f t h e Näsliden Mine p r e s e n t e d i n Appendix I .

5.1.2 The j o i n t element model

The c o n t a c t s between t h e o r e body and t h e a l t e r a t i o n zones a r e s i m u l a t e d w i t h j o i n t e l e m e n t s . The two s u r f a c e s o f a j o i n t element a r e f r e e t o move i n t h e normal d i r e c t i o n , t h e shear d i r e c t i o n and i n r o t a t i o n , F i g u r e 5.3.

F i g . 5.3 J o i n t e l e m e n t , a) j o i n t i n r o c k , b) f i n i t e e l e -ment j o i n t i d e a l i z a t i o n , c) d i s p l a c e m e n t and r o t a t i o n o f j o i n t e l e m e n t . A f t e r Jonasson, 1981.

The j o i n t e l e m e n t model g i v e s r e s u l t s w h i c h a r e s t r e s s p a t h dependent w h i c h means t h a t t h e a c t u a l m i n i n g e x c a v a t i o n sequence t o be used i n p r a c t i c e needs t o be r e p r o d u c e d . The i n c r e m e n t a l e x c a v a t i o n t e c h n i q u e must t h e r e f o r e be a p p l i e d , F i g u r e 5.2 c.

The s t r e s s e s i n t h e elements w i t h i n t h e e x c a v a t e d p a r t s a r e r e l e a s e d i n s m a l l s t e p s . For each s m a l l s t e p e q u i l i b r i u m i t e r a t i o n s a r e p e r f o r m e d . The convergence can be a c c e l e -r a t e d by s h i f t i n g t h e s t i f f n e s s e s between t h e l o a d s t e p s . D e c i s i o n s on t h e number o f l o a d s t e p s and i t e r a t i o n s must be based on p r e v i o u s e x p e r i e n c e , s i n c e no convergence c r i -t e r i a a r e a v a i l a b l e . For a d e -t a i l e d d e s c r i p -t i o n o f -t h e j o i n t e l e m e n t f o r m u l a t i o n t h e r e a d e r i s r e f e r r e d t o work by Goodman (1976) and G r o t h ( 1 9 8 1 ) .