Current Pillar Design

Chapter 16

A CRITICAL REVIEW OF THE CURRENT STATE-OF-THE-ART DESIGN OF MINE PILLARS

by Charles V. Logie

and Gordon M. Matheson

DAMES 6 MOORE 1626 Cole Blvd

Golden, Colorado 80401

ABSTRACT

The c u r r e n t s t a t e - o f - t h e - a r t d e s i g n of mine p i l l a r s c a n be grouped i n t o empirical and numerical design techniques. Numeri- c a l design techniques a r e based on soph is t i ca ted computational t echn iques which ana lyze t h e s t r e s s e s and s t r a i n s t h r o u g h t h e i n t e r s t i c e s of a p i l l a r . Empirical techniques a r e l e s s e x a c t , but have been widely used, p a r t i c u l a r l y f o r coa l mine p i l l a r des ign, because of t h e i r ease of computation.

Over the years numerous empir ical equat ions have been proposed t o calcula te the ul t imate s t reng th of coa l mine p i l l a r s . This paper presents a review of the var ious equat ions which a r e most f requen t ly u t i l i z e d and examines the fundamental assumptions, l i m i t a t i o n s and a p p l i c a b i l i t y of the design formulae. The review in tends t o provide the pract ic ing mining engineer wi th an understanding of t h e assumptions and mathematical bas i s of each equation t o a i d i n t h e se lect ion of the c r i t e r i a most app l icab le t o h i s s p e c i f i c design problems. Recommendations a r e made of t h e empir ical equat ions t h a t a re present ly most acceptable f o r design.

INTRODUCTION

The d e s i g n of mine p i l l a r s i s a s u b j e c t which h a s r e c e i v e d considerable a t t e n t i o n over the past s e v e r a l decades. Three b a s i c methods a r e ava i l ab le t o design mine p i l l a r s . These methods a r e : 1) loca l experience o r precedent; 2) empir ical design equations; and 3) numerical design u t i l i z i n g f i n i t e element, f i n i t e d i f f e r e n c e , o r boundary i n t e g r a l formulations.

360 STABILITY IN UNDERGROUND MINING

The numerical des ign methods a r e by f a r t h e most p r e c i s e from a mathematical viewpoint . However, t h e s e l e c t i o n of app rop r i a t e input parameters t o a c c u r a t e l y model t h e s t r e n g t h and f a i l u r e mechanism of mine p i l l a r s normally cannot adequate ly be c a l i b r a t e d t o f i e l d cond i t i ons w i thou t f u l l - s c a l e t e s t i n g of mine p i l l a r s . I n add i t i on , t he development of t h e numerical models and t h e i r computations is gene ra l ly q u i t e expens ive and must be repeated i f t h e s i z e , shape, he igh t , o r g e o l o g i c c o n d i t i o n s , ( i . e . , roof and f l o o r s t r a t a , e t c . 1, of t h e mine p i l l a r s change s i g n i f i c a n t l y .

A s a r e s u l t of t h e s e problems, p r a c t i c a l mine p i l l a r design s t i l l r e l i e s p r i m a r i l y on t h e f i r s t two design methods. Over t h e pas t y e a r s , numerous p i l l a r des ign equa t ions have been proposed. A number of t h e s e e q u a t i o n s have been recommended f o r use by repre- s e n t a t i v e s o f a g e n c i e s l i k e t h e U n i t e d S t a t e s Bureau of Mines (Babcock et a l . , 1981) o r publ i shed a s ca se h i s t o r i e s of recommended p i l l a r des ign (Djahangu i r i and Abel , 1977 and Djahanguiri , 1979).

The purpose of t h i s paper i s t o eva lua t e t h e most common o r most r e c e n t l y proposed p i l l a r des ign equat ions and c r i t i c a l l y d i scus s each of t h e i r advantages and d isadvantages so t h a t t h e p r a c t i c a l mining eng inee r can make a r a t i o n a l choice of which equa t ion ( s ) t o u t i l i z e f o r des ign . Most equa t ions have been developed f o r coa l mine a p p l i c a t i o n , however, r e c e n t work has a l s o been performed t o expand t h e s e equa t ions i n t o o i l s h a l e mine design.

FUNDAMENTALS OF PILLAR DESIGN EQUATIONS

Most p i l l a r des ign equa t ions , al though d i f f e r i n g s l i g h t l y i n form, have two b a s i c components i n common. These a r e a s i z e - s t r e n g t h e f f e c t and a s h a p e - s t r e n g t h e f f e c t component t o e a c h equat ion . These components a r e u t i l i z e d t o i n t e r p o l a t e from t h e s t r e n g t h of l a b o r a t o r y o r l a rge - sca l e i n s i t u rock t e s t samples t o mine p i l l a r s wh ich h a v e v a r i o u s h e i g h t s , w i d t h s , and l e n g t h s .

Rocks have a n a t u r a l s t r e n g t h an i so t ropy which i s predominantly due t o t h e presence of d i s c o n t i n u i t i e s , ( i . e . , j o i n t s , c l e a t s , o r b l a s t f r a c t u r e s ) , but can a l s o be a t t r i b u t e d t o v a r i a t i o n s i n rock f a b r i c , ( i . e . , f o l i a t i o n , bedding p lanes , e t c . ) , and mineralogy. I t i s w e l l documented (Evans and Pomeroy, 1958 , Gaddy, 1956, Bieniawski, e t a l . 1975, P r a t t e t a l . , 1972 and o t h e r s ) t h a t , given a cons t an t shape , a s t h e s i z e of rock samples i n c r e a s e from say a two-inch cube t o a three- foot cube t h a t s u b s t a n t i a l s t r e n g t h l o s s occurs. F igu re 1 shows a t y p i c a l s i ze - s t r eng th r e l a t i o n s h i p found f o r coa l . The c r i t i c a l q u e s t i o n s about t h e s i ze - s t r eng th e f f e c t a r e then:

1. What i s t h e form of t h e s i ze - s t r eng th r e l a t i o n s h i p ? 2. Is t h e r e a l i m i t i n g s i z e above which t h e "rock mass" s t r e n g t h

i s c o n s t a n t ?

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3. Do rocks of d i f f e r e n t t ypes show d i f f e r e n t s ize-s t rength r e l a t i o n s h i p s ?

4. What type of t e s t i n g program i s requi red t o evalua te t h e s i ze - s t r eng th r e l a t i o n s h i p of var ious rocks?

Each empi r i ca l p i l l a r des ign equa t ion uses a s l i g h t l y d i f f e r e n t approach t o s o l v e t h i s problem. These approaches w i l l be discussed i n d e t a i l a s each p i l l a r des ign equa t ion i s reviewed.

The shape-strength e f f e c t i s a l s o a well-documented phenomena (Baushinger, 1878, Holland, 1964 and many o t h e r s ) . The most widely known of t h e equa t ions which d e s c r i b e s t h i s phenomena is:

Where: u i s t h e compressive s t r e n g t h of a prism; P a, i s t h e compressive s t r e n g t h of a prism f o r which W = H ; W i s t h e minimum l a t e r a l dimension; H i s t h e prism h e i g h t .

Th i s equa t ion was o r i g i n a l l y developed by Baushinger i n 1878 but has b e e n w i d e l y a c c r e d i t e d t o o t h e r i n d i v i d u a l s o r o r g a n i z a t i o n s . Seve ra l p i l l a r des ign equa t ions r e l y on t h i s r e l a t i o n s h i p o r use models which c l o s e l y approximate i t a s demonstrated by Hust ru l id (1981). Also, a s poin ted ou t by Hus t ru l id , many i n v e s t i g a t o r s have no t been c o n s i s t e n t i n t h e i r s e l e c t i o n of t h e method of t e s t i n g f o r a rock shape-s t rength e f f e c t . Some i n v e s t i g a t o r s vary t h e sample width while hold ing t h e sample h e i g h t cons tant while o t h e r s do the converse. Unfor tunate ly , t h e shape e f f e c t der ived from two t e s t procedures a r e o f t e n d i f f e r e n t . I n gene ra l , i t is recommended t h a t if shape-strength t e s t i n g i s t o be conducted the sample he ight should be he ld cons t an t wh i l e t h e width is var ied . The shape- s t r e n g t h r e l a t i o n s h i p u t i l i z e d i n each empir ica l mine p i l l a r design equa t ion w i l l be d i scus sed i n d e t a i l a s each equat ion i s reviewed.

REVIEW OF MINE PILLAR DESIGN EQUATIONS

Many mine p i l l a r des ign equa t ions have been proposed throughout t h e pas t s e v e r a l decades. Based on our exper ience we have chosen e i g h t of t h e s e equa t ions t o review and d i s c u s s t h e assumptions behind t h e i r s i z e and shape e f f e c t r e l a t i o n s h i p s and t h e i r r e l a t i v e advantages and d i sadvan tages w i t h r e spec t t o p r a c t i c a l p i l l a r design problems. The des ign methods we have s e l e c t e d a r e a s fol lows:

1. Obert and Duvall and Modified Obert and Duvall Equation

Obert and Duvall (1946, 1967) were one of t h e f i r s t i n v e s t i g a t o r s t o look a t t h e c o a l mine p i l l a r des ign problem and t r y t o develop

REVIEW OF MINE PILLAR DESIGN 363

prac t i ca l design methods. Based on laboratory t e s t i n g they v e r i f i e d tha t equation (1) was v a l i d f o r cubical specimens of coa l , and many o t h e r rock types . A s a r e s u l t t h e y recommended t h e f o l l o w i n g equation be u t i l i z e d f o r design of mine p i l l a r s :

Where: Up i s t h e p i l l a r s t r e n g t h i n pounds per square inch ; 01 is t h e un iax ia l compressive s t r e n g t h of a cub ica l

specimen (w = h) i n pounds per square inch; w and h a r e t h e width and height of p i l l a r s i n inches .

A f ac to r of sa fe ty of 2 t o 4 was recommended by the Bureau of Mines f o r t h i s equation.

A s inferred previously, equation (2) i s a d i r e c t u t i l i z a t i o n of the shape e f f e c t equation (1). However, t h e o r i g i n a l recommended usage p rov ides f o r t h e u t i l i z a t i o n of l a b o r a t o r y compress ive strengths. A s a r e s u l t no s i z e e f f e c t r e l a t i o n s h i p was incorporated i n t o the o r ig ina l equation. This r e s u l t e d i n possible over e s t i - mating the s t rength of mine p i l l a r s . For tunate ly , high s a f e t y fac to r s (greater than 3) have normally been used with t h i s equat ion by designers.

Although not recognized i n t h e l i t e r a t u r e , a modified ve rs ion of t h i s equation has been used which u t i l i z e s an estimated i n s i t u rock mass s t reng th based on any one of severa l s i z e s t r e n g t h s c a l i n g re la t ionships . Usually t h e s t r e n g t h assumed f o r el is the es t imated o r measured rock s t reng th f o r a minimum 5 f o o t (150 cm) s i z e sample. This modified approach f o r t h e ca lcu la t ion of mine p i l l a r s t r e n g t h s has general ly proven t o be comparable t o most o ther design methods. A sa fe ty f a c t o r of 1.5 t o 2.0 i s recommended f o r p i l l a r design using t h i s modified approach. Figure 2 shows a graph of t h e modified Obert and Duvall equation f o r a s p e c i f i c coal.

2. Holland - Gaddy Equation

Holland (1964) reported t h e r e s u l t s of t e s t s on t h i n prisms of coal. The following equation was developed from the experimental evidence representing t h e s t r e n g t h of coa l p i l l a r s having width t o height r a t i o s ly ing between u n i t y and 9 o r 10:

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REVIEW OF MINE PILLAR DESIGN

There: ap i s t h e mean u l t ima te s t r e n g t h of t h e coal p i l l a r i n pounds pe r square inch;

W i s t h e l e a s t width of p i l l a r i n inches ;

H i s t h e height of p i l l a r i n inches K i s t h e c o e f f i c i e n t depending on

coal.

It was suggested t h a t K be determined by t e s t i n g 2 t o 4 inch cubes i n compression. Gaddy found t h a t t h e equation r e l a t i n g t h e constant "K" t o t h e s i z e of a cube was:

Where: K i s t h e c o e f f i c i e n t required f o r equation (3) ;

a, i s t h e mean u l t ima te compressive s t r e n g t h of c o a l cubes t e s t e d i n pounds pe r square inch ;

il i s t h e edge dimension of cube i n inches .

For p i l l a r design a p p l i c a t i o n s , t h e Holland-Gaddy formula c a l l s f o r a f a c t o r of s a f e t y between 1.8 t o 2.2.

An attempt t o v e r i f y t h e formula was made by Holland comparing the f a c t o r of s a f e t y c a l c u l a t e d f o r some 8 d i f f e r e n t mine p i l l a r s with t h e i r observed behavior. I n one of t h e 8 cases t h e p i l l a r was reported t o have f a i l e d . The ca lcu la ted f a c t o r of s a f e t y f o r t h i s case was estimated t o l i e between 1.2 and 2.4. The range i n f a c t o r of s a f e t y was a s a consequence of u n c e r t a i n i t y regarding t h e a c t u a l p i l l a r s t r e s s a t f a i l u r e . This p i l l a r des ign equat ion has been u t i l i z e d with success by des igners f o r t h e pas t f i f t e e n years . Although not wel l documented i t appears t h a t most p i l l a r s which have been designed u t i l i z i n g t h i s method have performed s a t i s f a c t o r i l y .

3. Bieniawski's Equation

Bieniawski, (1967, 1969) repor ted t h e r e s u l t s of t e s t s performed on cubical coal samples ranging i n s i z e from 2 inches t o 79 inches on a side. This was performed a t Witbank Colliery i n South Afr ica . On the bas i s of the t e s t s Bieniawski concluded t h a t f o r samples l e s s than 60 i n c h e s on a s i d e e q u a t i o n ( 5 ) a p p l i e s and f o r s a m p l e s g rea te r than 60 inches on a s i d e equat ion (6) appl ies .

STABILITY IN UNDERGROUND MINING

Where Up i s t h e s t r e n g t h of t h e coal p i l l a r i n pounds per square inch;

W1 i s t h e width of t h e p i l l a r i n f e e t ; H1 is t h e h e i g h t o f t h e p i l l a r i n f e e t .

Bieniawski concluded based on t h e s e t e s t r e s u l t s t h a t t h e s t r e n g t h of coa l decreases wi th i n c r e a s i n g s i z e of cubic samples tending t o an asymptot ic va lue a t 60 inches. Based on t h i s he recommended t h a t t h e mass s t r e n g t h of t h e c o a l be determined on minimum 60 inch c u b i c a l samples. Examining t h e d a t a r evea l s a reduct ion i n s t r e n g t h from 24 i nch t o 79 i n c h cubes of on ly about 23 percent. From 36 i n c h t o 79 i nch cubes t h e d i f f e r e n c e i s only about 9 percent. It appears t h a t f o r p r a c t i c a l purposes 36 inch cubes is s u f f i c i e n t l y l a r g e t o e s t i m a t e mass c o a l s t r e n g t h .

Normalizing equat ion (6) us ing a 60 inch cube s t r e n g t h g ives :

T h i s e q u a t i o n i s s i m i l a r i n fo rm t o e q u a t i o n (1) deve loped by Baushinger (1876).

Bieniawski (1981) r epor t ed t h e r e s u l t s of a Pennsylvania S t a t e Un ive r s i ty survey of room and p i l l a r dimensions and design p r a c t i c e i n U.S. c o a l mines aimed a t improving design p rac t i ces . Based on t h i s survey h i s e a r l i e r formula equa t ion (6) was revised to:

Where: a i s a c o n s t a n t ; up is t h e p i l l a r s t r e n g t h i n pounds pe r inch

square ; U, i s t h e c rush ing s t r e n g t h of a 60 inch cube o r

l a r g e r ; W and H a r e t h e p i l l a r width and height

r e spec t ive ly .

Based on r e s e a r c h a t Pennsylvania S t a t e Univers i ty t h e following va lues a r e recommended f o r a.

a = 1.4 f o r !,! >5 H


This a l t e r a t i o n i n t h e e a r l i e r formula i s apparent ly t o account f o r the e f f e c t s of high width t o height r a t i o s . The equation i s compared t o others i n Figure 2. The equation r e s u l t s i n a s i g n i f i c a n t s t e p s t r e n g t h i n c r e a s e a t a wid th t o h e i g h t r a t i o o f 5. A s a consequence of t h i s s t e p uncer ta inty e x i s t s of which curve t o use a s the width t o height r a t i o of 5 i s approached.

Bieniawski has not presented any f i e l d p i l l a r performance d a t a t o substant ia te h i s p i l l a r s t r e n g t h equations. Sowry (1967) presented d a t a f o r t h e same c o l l i e r y i n which B i e n i a w s k i ' s s t u d i e s were conducted. Based on these da ta , t h e p i l l a r crushing s t reng ths were e s t i m a t e d f o r f i v e - d i s t r e s s e d p i l l a r s from e q u a t i o n ( 7 ) using the estimated overburden s t r e s s and the mining dimensions given. These s t reng ths a r e shown i n Table I. These s t reng ths a r e compared t o the coal cube s t reng th f o r a 60 inch cube a t which s i z e Bienawski indicates i t i s independent of s i d e length. The r e s u l t s of t h i s comparison i s i n extremely good agreement, however, t h e comparison is only v a l i d f o r the case a = 1. No da ta was a v a i l a b l e f o r a = 1.4.

4. Salamon and Munro Eouation f o r Coal and Modified Salamon and Munro Equation For O i l Shale

Salamon and Munro (1967) compiled a l i s t of 98 s t a b l e and 27 collapsed coal mine p i l l a r h i s t o r i e s . These d a t a were s t a t i s t i c a l l y analysed and design mine p i l l a r equat ions developed. The most widely accepted p i l l a r des ign equation developed by Salamon and Ffunro is a s follows:

Where: up i s the p i l l a r s t r e n g t h i n pounds pe r square inch;

Uc i s the un iax ia l compressive s t r e n g t h of one foo t cube of coal i n pounds per square inch;

W i s the p i l l a r l e a s t width i n inches; H i s the p i l l a r height i n inches.

Where s t reng th da ta is not ava i l ab le f o r a one foo t cub ica l sample of coal the following modification of the formula has been recommended :

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si)

A

fte

r B

ien

iaw

ski

o =

50

3

C 40

0 <

oc

< 6

60

uc

=

572

a =

36

0

Ass

umed

H

eig

ht

(f

t)

60

60

60

6

0

60"

Cu

be

Str

en

gth

62

0

62

0

62

0

62

0

Ass

umed

Ov

i (p

si)

100

10

0

100

100

wid

th

to

he

igh

t r

ati

o

1

2 3 4

Ass

umed

Ovp

(p

si)

200

200

200

200

F1

.13

.39

.53

.66

~a

lc

ga

te

d*

Oo

(ps

i)

2875

25

73

24

11

23

03

F2

.19

.37

.44

.48

Cal

cu_

late

d

Oh

(ps

i)

3 2 7 6

97

11

4

Ass

umed

Oh i

(ps

i)

100

100

100

100

Ca

lcu

late

d

pil

lar

s

tsn

gth

Ov

(ps

i)

3105

28

16

27

21

26

68


Where: Us i s the uniaxial compressive strength o f a cubic sample of coal of width Ws and height Hs i n inches;

W p and H p are the width and height o f the pillar i n inches.

These equations speci f ica l ly relate t o coal. Hardy and Agapito (1975) have suggested a modification t o t h i s equation for o i l shale. This modified equation i s as follows:

The exponents i n both o f these equations have been developed based on a s ta t i s t ica l analysis o f observed laboratory and f i e ld strengths o f coal and o i l shale respectively.

A problem ex is ts with both o f these equations i n that they are dimensionally incorrect i f the pi l lar width does not equal the pillar height. In addition, at f i r s t glance, one would assume that the equations suggest that a one foot cube o f coal represents the sample size above which the strength o f coal i s constant. However, th is i s not the case since through the i r s ta t i s t i ca l analysis the size e f f e c t has been included i n both the one foot cubic sample size and the exponents i n the equation. This can be remedied by factoring the exponents as follows:

Salamon - Munro (Coal)

Hardy Modification ( O i l Shale)

OC .597

= 3% H [i]

I f one compares these equations t o the Holland - Gaddy equation:

it i s apparent that they are generally similar. In a l l three cases the shape e f f e c t i s represented by t h e second term w i t h only slight variations in the exponent. The s i ze e f f e c t represented by the f i r s t term, however, appears t o be s igni f icant ly d i f f e r e n t . Figure 1 presents a comparison o f the s i ze e f f e c t suggested by the Salamon and Munro and Holland-Gaddy coal equations. As i l l u s t ra ted


i n t h i s f i g u r e t h e e q u a t i o n s va ry s u b s t a n t i a l l y f o r smal l s i z e samples but converge and a r e e s s e n t i a l l y i d e n t i c a l f o r sample s i z e s between 36 and 60 i n c h e s , t h e l i m i t i n g s i z e range which has been s p e c i f i e d a s a p p l i c a b l e t o c o a l by va r ious i n v e s t i g a t o r s . This shows t h a t a l t hough bo th equa t ions have been der ived independently from d i f f e r e n t d a t a s o u r c e s t h a t t hey provide very s i m i l a r r e s u l t s and provide conf idence t h a t , a t l e a s t w i th t h e type coa l s they r e p r e s e n t , t h e s i z e and s t r e n g t h e f f e c t r e l a t i o n s h i p s a r e s imi l a r . For des ign purposes i n c o a l a f a c t o r of s a f e t y of 1.6 has been recommended by Salamon. F igure 2 p r e sen t s a comparison of t he Salamon and Munro e q u a t i o n t o t h e o t h e r des ign equat ions .

5. Wardell Method

I n 1976, R. Wardell and P a r t n e r s of Newcastle, England i s sued a P h a s e I1 r e p o r t u n d e r a U.S. Bureau o f Mines c o n t r a c t which p re sen t ed a sugges ted p i l l a r de s ign method t h a t i s r a d i c a l l y d i f f e r - e n t t o t hose developed by o t h e r i n v e s t i g a t o r s . This p a r t i c u l a r method would n o t be d i s c u s s e d e x c e p t f o r t h e f a c t t h a t i t was recommended f o r u s e by r e p r e s e n t a t i v e s of t h e U.S. Bureau of Mines (Babcock e t . a l . , 1981).

The r e l a t i o n s h i p developed by Wardell cannot be s epa ra t ed i n t o s e p a r a t e s i z e - s t r e n g t h and shape-s t rength e f f e c t f a c t o r s . The b a s i s f o r t h i s equa t ion i s a n appa ren t hand drawn l i n e between observed " f a i l e d " and "non-fa i led" mine p i l l a r s . Unfor tuna te ly t h e d a t a t o which t h e l i n e was drawn d i d n o t t a k e i n t o cons ide ra t i on t h a t coa l from d i f f e r e n t l o c a l i t i e s o r seams have d i f f e r e n t s t r eng ths . I n o t h e r words t h e y e s s e n t i a l l y assumed a l l c o a l has t h e same s t r e n g t h r e g a r d l e s s of i t s o r i g i n , g rade , f r a c t u r i n g and o t h e r p e r t i n e n t g e o l o g i c f a c t o r s . Th i s i s a n i n c o r r e c t assumption which should warn e n g i n e e r s t o be c a u t i o u s o f t h i s d e s i g n method. Al though n o t s p e c i f i c a l l y s t a t e d i n t h e r e p o r t Wardell appa ren t ly f i t t e d a curve t o t h e hand drawn "ske tch" of " f a i l e d " and "non-fai led" p i l l a r s and d e v e l o p e d a f a m i l y o f c u r v e s f o r v a r i o u s p i l l a r h e i g h t s . The equa t ion sugges ted by Wardell i s a s fo l lows:

Where: S i s s t r e n g t h of p i l l a r i n pounds p e r squa re i n c h ; W is Width o f p i l l a r i n f e e t ; H i s H e i g h t of p i l l a r i n f e e t ; a and b a r e c o n s t a n t s .

The a and b c o n s t a n t s a r e a s s igned va lues of 1000 and 20 respec- t i v e l y based on "judgement". A f a c t o r of s a f e t y of 1.5 has been imp l i ed f o r u t i l i z a t i o n w i t h t h i s equa t ion by Babcock e t a l . Figure 2 p r e s e n t s a comparison o f t h e Wardel l method t o t h e o the r des ign equa t ions .


6 . Induced Horizontal Stress Method

Agapito and Hardy, 1982 have presented an empir ical des ign method for o i l shale pillars which i s based on a combination o f previous empirical relationships and f i n i t e element s tress analysis. This method represents a design method developed a f t e r considerable work at the Colony O i l Shale Pilot Mine near Parachute, Colorado.

The u t i l i za t ion o f t h i s method i s based on the following three eauations:

- u b v - = - l + B p j (15)

Where: Uv i s the pi l lar strength i n pounds per square inch - 0, i s the pillar strength a t zero horizontal s tress ( i . e . ,

rock mass unconfined compressive strength) i n pounds per

- square inch; i s the average horizontal pi l lar s tress i n pounds per square inch ;

B i s a constant o f 3.2 for o i l shale; b i s a constaLt o f 0.82 fo r o i l shale.

- Where: a, i s the i n s i t u unconfined compressive strength i n pounds

per square inch; 01 i s the laboratory unconfined compressive strength i n

pounds per square inch; Vi i s the i n s i t u volume o f the rock mass; V1 i s the volume o f laboratory specimen; a i s a volume reduction c o e f f i c i e n t equal t o 0.08 for o i l

shale and 0.12 for coal; and

Where: a, i s the average horizontal s t ress at mid-pillar height i n pounds per square inch;

' h i i s the i n i t i a l premining horizontal s tress i n pounds per square inch;

0 i s the average ver t ica l s t ress a t mid-pillar height i n pounds per square i n c h ;

uvi i s the i n i t i a l or premining ver t ica l s tress i n pounds per square inch;

F 1 and F 2 are factors dependent on pi l lar shape.


The F1 and F2 f a c t o r s a r e s t a t e d t o depend on t h e p i l l a r width t o height r a t i o incorpora t ing s i t e s p e c i f i c geologic parameters. Also, u t i l i z a t i o n of t h i s method requires t h e es t imat ion of t h e i n s i t u v e r t i c a l s t r e s s f i e l d , both premining and wi thin the p i l l a r . The i n i t i a l v e r t i c a l s t r e s s i s obtained by assuming a geos ta t i c v e r t i c a l s t r e s s condi t ion ( i . e . , weight of one cubic foo t of rock t imes t h e overburden depth i n f e e t ) . The i n s i t u average v e r t i c a l p i l l a r s t r e s s i s obta ined by assuming a " t r ibu ta ry area" load on each p i l l a r . In a d d i t i o n , an es t imat ion of the premining i n s i t u hor izon ta l s t r e s s f i e l d i s required. No method of obtaining t h i s parameter has been presented, however, the authors assume t h a t t h i s i s estimated by e i t h e r measurement o r engineering judgement.

I n o rde r t o u t i l i z e t h i s method a f i n i t e element analys is must be performed and a graph developed which r e l a t e s the F1 and F2 f a c t o r s t o t h e width t o he igh t r a t i o of t h e p i l l a r s . An example of such a graph i s shown i n Figure 4. Agapito s t a t e d t h a t f o r "very squat" v p i l l a r s t h a t F1 w i l l tend t o 1 and F2 w i l l tend t o F5 where v i s t h e Poisson's r a t i o . No d e f i n i t i o n f o r "very squat" p i l l a r s has been presented but based on t h e graph i n Figure 3 i t i s assumed t o be f o r a p i l l a r wi th a width t o height r a t i o g r e a t e r than 5.0. This condi t ion is i n t e r e s t i n g l y enough s a t i s f i e d by most coal mine p i l l a r s . Once F1 and F2 a r e o b t a i n e d t h e a v e r a g e h o r i z o n t a l s t r e s s a t mid p i l l a r he igh t can be calcula ted by equation (17). A f t e r t h i s e q u a t i o n ( 1 6 ) i s u t i l i z e d t o c a l c u l a t e t h e i n s i t u unconfined compressive s t r eng th . These two values a r e then u t i l i z e d i n equat ion (15) t o c a l c u l a t e the p i l l a r s t rength . This method i s s t a t e d by Agapito t o be app l i cab le t o both square and rectangular p i l l a r s . I n t h e case of r ec tangu la r p i l l a r s the minimum dimension should be u t i l i z e d .

Although t h i s method i s s l i g h t l y more complex i t fundamentally can be separated i n t o a s ize-s t rength and shape-strength e f f e c t portion. The s i z e e f f e c t i s represented by equation (16). This equation appears d i f f e r e n t t o t h e s i z e e f f e c t equation developed by o the r i n v e s t i g a t o r s . A g raph ica l comparison wi th the s t r eng th reduct ion u t i l i z e d f o r c o a l , however shows t h a t i t i s very s i m i l a r and f o r p r a t i c a l purposes t h e same a s t h a t proposed by others. This i s i l l u s t r a t e d i n Figure 1. Agapito has s t a t e d t h a t a maximum s i z e e x i s t s above which t h e i n s i t u unconfined s t r eng th is constant, however, no g u i d a n c e i s p r o v i d e d a s t o what t h i s s i z e may be.

The shape-strength e f f e c t i s e s s e n t i a l l y represented by the r e s t of t h e des ign equat ions . The shape e f f e c t i n t h i s case i s d i f f i c u l t t o compare t o o t h e r des ign methods except by example where the s i ze - s t r eng th reduct ion i s held constant. I f one u t i l i z e s the parameters g iven i n t h e c i t e d re fe rence i t appears t h a t t h i s method impl ies a decreas ing rock mass s t r e n g t h per u n i t a rea of p i l l a r with inc reas ing width t o he igh t r a t i o . This i s shown i n Table 11. This observat ion i s i n t e r e s t i n g s i n c e i t i s contrary t o a l l o ther p i l l a r des ign equa t ions which have been proposed.


3 0

4000

- 2 5 In J a 0 In

5 2 0 3000 (3 W - I

.- - .) a - I I-, 15

I

(3 I-

z w

W 2 0 0 0 8 a E I- I- In U)

LL lo PILLAR HEIGHT - 5Ot t . E

4 DEPTH = 1000 t t J Q, J - HYDROSTATIC STRESS F I E L D J

a LABORATORY UNIAXIAL COMPRESSIVE 1 0 0 0 ~

5

FRICTION = 2E0

0 0

P ILLAR WIDTH(M)

COMPARISON OF PILLAR STRENGTH DESIGN METHODS AFTER AGAPITO

FIGURE 3


Other than the apparent contradictory strength calculation the primary difficulty with this method is that it requires the input of design parameters F1 and F2 which must be obtained by finite element analysis. In addition, an evaluation of the in situ premining stress field is required. Although these are not especially difficult procedures or parameters to quantify they are probably beyond the in house capability of most mining companies.

7. Hustrulid and Swanson Method

Hustrulid and Swanson (1981) have submitted a yet unpublished report to the U.S. Bureau of Mines which examines the size-strength and shape-strength relationships which have been proposed by numerous investigators. In this report an analysis is presented of the strengths and weaknesses of each formula and a coal pillar design equation is suggested utilizing the best documented aspects of each design method. The equation proposed by these investigations is as follows:

Where: up is the strength of pillar in pounds per square inch; K is a constant calculated for each coal based on field or laboratory strength data in pounds per square inch ;

H is the pillar height in inches (constant at 36 inches above coal seam thickness of 3 feet);

W is the minimum pillar width in inches.

The K constant turns out to be the theoretical strength of a one inch cube and is calculated by the following equation:

Where: Uc is the compressive strength of a field or laboratory cubical sample in pounds per square inch;

S is the side dimension of the sample for which uc was found in inches.

The size effect relationship which is utilized in this equation is represented by the first term of the equation. This relationship was suggested by Hustrulid (1976) as being a reasonable approxi- mation of the size-strength relationship for coal. Hustrulid stipulates that above a coal test specimen side length "S" of 36 inches the strength is relatively constant. In addition, he suggests that the strength of a 6-inch sample of coal can be utilized to sufficiently accurately estimate the strength of larger samples of coal for mine pillar design.


The shape strength relationship portion o f t h i s method i s represented by the second term o f the equation. This term i s direct ly taken from the work o f Holland-Gaddy and Salamon-Munro as discussed previously, and appears t o be a valid shape e f f e c t scaling equation that has been substantiated for both South African and Eastern United States coals.

8. Confined Core Pil lar Design Equation

This concept recognizes t h a t a "y i e ld" or " f rac tured zone" develops around the periphery o f a pillar that confines a central e l a s t i c core. Because o f t h i s confinement t he in s ide core i s subject t o t r iax ia l s tress conditions. This phenomena has been recognized by several researchers and formulations developed (Labasse 1949, Walker 1957 and Wilson, 1972) t o describe the mechanics. The most widely publicized work i s that of Wilson.

The basis o f the confined core concept i s that the strength o f coal can be represented by the following equation:

a = a o + a tan ,!3 1 3

Where 0 1 i s the fa i lure s t ress; a0 i s the unconfined compression strength; a3 i s the confining pressure; l+sin@

Tan 6 i s the t r iax ia l s tress coe f f ic ien t where @ i s the angle o f internal f r i c t i on o f the coal.

From Wilson's (1972) tes t ing o f Bri t ish coals tan B i s approximated as 4 based on a f r i c t i o n angle o f 37". This may not be true for coals i n other geographic areas.

In Wilson's 1972 work he proposed an equation for de termi i t ion o f the depth o f the yield zone i n rectangular rooms. This equation was based on the yield zone being comprised o f discrete blocks o f rock and the f r ic t ional forces acting between the individual blocks t o achieve equilibrium. This equation i s as follows:

Where: y i s the depth o f yield zone from the ribside i n fee t ; A h i s the seam height i n f ee t ; a i s the maximum pil lar s tress i n pounds per square inch

at the yield zone/confined core interface; a , i s the unconfined compressive strength i n pounds per

square inch.


The v e r t i c a l s t r e s s on t h e p i l l a r is considere! t o vary l i n e a r l y from zero a t the r i b s i d e t o a maximum value u a t t h e innermost extremity of t h e y ie ld zone where t h e value becomes:

A

u = u tan 8 (assuming u approaches zero) . H (22)

This assumed l i n e a r v a r i a t i o n between t h e minimum s t r e s s a t t h e r i b s i d e t o t h e maximum s t r e s s a t t h e y i e l d z o n e l c o n f i n e d c o r e in te r face i s f o r computational convenience. I n p r a c t i c e i t i s probable t h a t t h i s v a r i a t i o n r a t h e r follows some exponent ia l law. The form t h a t t h i s l aw a c t u a l l y t a k e s w i l l have a s i g n i f i c a n t influence on how close t h i s equat ion approximates i n s i t u cond i t ions and t h e r e s u l t i n g p i l l a r s t r eng th . The l i m i t of t h e average core stress i s reasoned t o b e e q u a l t o t h e peak abutment s t r e s s . Based on t h i s assumption p i l l a r s t r e n g t h s can be calcula ted. The determination of p i l l a r s t r e n g t h is a l s o very s e n s i t i v e t o t h e ca lcu la t ion of t h e depth of t h e y i e l d zone. I n add i t ion , t h e depth of the y ie ld zone i s very dependent on t h e value se lec ted f o r t h e i n s i t u rock mass compressive s t r eng th . Since changes i n t h e depth of the y ie ld zone change t h e p i l l a r s t r e n g t h by t h e square , cau t ion must be exercised i n t h e s e l e c t i o n o f - these var ious parameters.

A comparison of the confined core method wi th o the r p i l l a r des ign equations i s shown i n Figure 2 and 3. This method i s genera l ly i n good agreement a t small p i l l a r width t o he igh t r a t i o s . However a t l a r g e p i l l a r w i d t h t o h e i g h t r a t i o s t h e method r e s u l t s i n conservative s t rengths . This i s because a s t h e width of t h e p i l l a r increases i n Wilsons equat ion t h e e f f e c t of t h e mined he igh t on t h e y ie ld zone becomes n e g l i g i b l e and t h e p i l l a r s t r e n g t h t ends toward a constant value.

Wilson i n 1977 and 1981 modified h i s equation f o r c a l c u l a t i n g t h e depth o f t h e y i e l d zone i n t o p o l a r c o o r d i n a t e s . I n t h i s work roadways of c i r c u l a r c r o s s s e c t i o n a r e considered. The equa t ion f o r ca lcu la t ing t h e depth of t h e y i e l d zone i n t h i s case is:

Where: r b i s the r a d i u s from t h e cen te r of roadway; ro i s the r a d i u s of t h e roadway; Uo is the unconfined compressive s t r e n g t h of coa l ; p i s t h e support r e s i s t a n c e plus s t r e n g t h of broken

mater ia l ; q i s t h e s t r e s s f i e l d remote from t h e excavation; k i s t h e t r i a x i a l s t r e s s f a c t o r -, where 4 i s t h e

angle of i n t e r n a l f r i c t i o n . l-sin@


CONCLUSIONS

It i s apparent t h a t a v a r i e t y of mine p i l l a r design equat ions a r e a v a i l a b l e t o t h e mining eng inee r . I n add i t i on depending on what method i s u t i l i z e d a wide s c a t t e r of mine p i l l a r s t r e n g t h s can be obta ined . Based on t h e comparisons made we have d iv ided t h e reviewed equa t ions i n t o t h e fo l lowing four ca t ego r i e s .

1. Acceptable f o r des ign 2. Use wi th c a u t i o n 3. Unacceptable f o r des ign 4. No expe r i ence

Table 111 p r e s e n t s our recommendations f o r t h e s e equat ions . The "Limited Experience" ca t ego ry i n d i c a t e s equat ions w i th which t h e mining i n d u s t r y has had i n s u f f i c i e n t experience t o show t h e method t o be accep tab l e under p roduc t ion opera t ions . The "Unacceptable For Design" equa t ions a r e t hose which have a f a t a l f law o r no apparent r a t i o n a l bas i s . The ca t ego ry "Use wi th Caution" equat ions a r e t h o s e which h a v e been shown t o b e v e r y s e n s i t i v e t o h a r d t o q u a n t i f y parameters o r demonst ra te ca l cu l a t ed p i l l a r s t r e n g t h s which a r e p o s s i b l y c o n t r a r y t o observed rock mass behavior. The "Acceptable f o r Design" equa t ions a r e those which appear t o have a r a t i o n a l b a s i s from both a s i ze - s t r eng th and shape-strength e f f e c t s t andpo in t .

A g r a p h i c a l comparison of t h e c o a l mine p i l l a r des ign equat ions i s made i n Figure 2 u s i n g t h e a c t u a l s i ze - s t r eng th d a t a from t h e Witbank coa l . A s shown t h e f o u r c o a l mine p i l l a r equat ions which a r e recommended a s "Acceptable f o r Design" have t h e same gene ra l t r e n d and have numer ica l ly c l o s e u l t i m a t e p i l l a r s t r eng ths . The equa t ions shown which have been ca t ego r i zed a s "Not Acceptable f o r Design" ( i . e . Bienawski and Wardel l ) d iverge markedly from t h e o the r equat ions . Bienawski 's modif ied equat ion , r epo r t ed ly a l t e r e d t o b e t t e r p r e d i c t s t r e n g t h i n t h e c a s e of w i d t h t o h e i g h t r a t i o s g r e a t e r than 5 , causes con fus ion because of t h e s t e p i n t h e s t r e n g t h a t t h i s width t o h e i g h t r a t i o . Also t h i s equat ion needs p r a c t i c a l v e r f i c a t i o n a t width t o h e i g h t r a t i o s g r e a t e r than 5 where t h i s marked d ivergence occurs .

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M

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Ob

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REFERENCE LIST

Agapito, J.F.T. and Hardy, M.P. , "Induced Horizontal S t r e s s Method of P i l l a r Design i n O i l Shale" 1 5 t h O i l Shale Symposium Proceed- ings, Colorado School of Mines, 1982.

Babcock, C., Morgan, T. and Haramy, K., "Review of P i l l a r Design E q u a t i o n s I n c l u d i n g t h e E f f e c t s of C o n s t r a i n t " , 1 s t Annual Conference on Ground Contro l i n Coal , 1981, pp 23-34.

Bauschinger, J., "Mi t t e i l unge r a u s dem mechnaisch - Technischen Laboratorium d e r K." Technischen Hochschule i n Munchen, Vol. 6, 1876.

B i e n i a w s k i , Z.T. and Van H e e r d e n , W.H., "The S i g n i f i c a n c e of I n S i t u T e s t s on Large Rock Specimens" I n t . J. Rock Mech. Min. Sci . and Geo. Abs., Vol. 23, No. 4, 1975 pp 101-104.

Bieniawski, Z.T., "Improved Design of Coal P i l l a r s For U.S. Mining C o n d i t i o n s " 1 s t Annual C o n f e r e n c e on Ground Control i n Coal, 1981, pp 13-22.

Djahangui r i , F. "Rock mechanics f o r a Longwall Mine Design; Carbon County Coal Company", 1 s t Annual Conference on Ground Control i n Coal. 1981.

Djahangui r i , F. and Abel, J.F., Jr., "Systematic P i l l a r Design f o r O i l Shale" 10 th O i l Shale Symposium Proceedings, Colorado School of Mines. 1977.

Evans, I. and Pomeroy, C.D., "The St rength of Coal i n Uniaxial Compres s ion" , M e c h a n i c a l P r o p e r t i e s of Non-me ta l l i c B r i t t l e M a t e r i a l , Walton, W.H., ed. , Bullerworths, London, 1958 pp 8-11 Tables 1 and 2.

Gaddy, F.L., "A Study of t h e Ul t imate St rength of Coal a s Related t o t h e A b s o l u t e S i z e of t h e C u b i c a l Specimens T e s t e d " , E., VPI, Vol. 49, No. 10, 1956 Tab le s 6-10. -

Hardy, M.P. and Agapito, J.F.T., " P i l l a r Design i n Underground O i l Shale Mines" Proc. 16 Symp. on Rock Mech., 1975, pp 325-335.

Holland, C.T. "The S t r eng th of Coal i n Mine P i l l a r s " . Proc. of t h e S i x t h Symposium on Rock Mechanics, Univers i ty of Missouri , Rol la , Missour i , A p r i l 1964, pp 450-456.

Hus t ru l id , W.A. and Swanson, S.R., "F i e ld V e r i f i c a t i o n of Coal P i l l a r S t r e n g t h P r e d i c t i o n Formulas" Bureau of Mines Contract H0242059 A p r i l 25, 1981.

REVIEW OF MINE PILLAR DESIGN 38 1

Labasse, H . "Ground Pressure i n Coal Mines, Part 111, Ground Pressure Around Horizontal Cross Measure Drif ts" . Revue Uni- verselle des Mines - Vol. 5 , March 1949 p p 78-88.

Obert, L. and Duvall, W.I. Rock Mechanics and the Design of Struc- tures i n Rock. John Wiley and Sons, New York, 1967, p p 542-545.

Pratt, H.R.,. e t al. , "The E f f ec t o f Specimen Size on the Mechanical Properties of Unjointed Diorite" Int . J . Rock Mech. Min. Sci. , Vol. 9 , 1972, p p 513-529, Table 3.

Salamon, M.D.G. and Munro, A.H. " A Study o f the Strength o f Coal Pillars" Journal o f South African Inst . Min. Metall., Vol. 68, 1967, p p 55-67.

Sowry, C.G. "Contribution t o the Discussion o f "A Study o f the Strength o f Coal Pillars" and "A Method o f Designing Board and ~ i l l a ; Workings" Journal o f the South African institute o f Mining and Metallurgy, November, 1967 p p 187-192.

Walker, L. "The Theory o f Strata Control" Mine and Quarry Engineer- ing, July 1955.

Wardell, K . and Partners, "Guidel ines f o r Mining Near Sur face Waters" U.S. Bureau o f Mines - Contract No. H0252021, Phase 11, February, 1976, p p 8-18.

Wilson, A.H. "Research i n t o t h e Determination o f P i l l a r S i z e , Part 1 , An Hypothesis concerning Pil lar S tabi l i ty" . The Mining Engineer, Vol. 131, June 1972.

Wilson, A.H. "The E f f e c t o f Yield Zones on the control o f Ground" 6 t h In ternat ional S t r a t a Control Conference, B a n f f , Canada, September, 1977.

Wilson, A.H. "Stress and S tab i l i t y i n Coal Ribsides and Pi l lars" , 1st Annual Conference on Ground Control i n Coal, 1981.


Question - P i l l a r des ign methods d iscussed a r e pr imar i ly f o r coa l type m a t e r i a l which i s s u b s t a n t i a l l y d i f f e r e n t than o i l sha le . Don't you t h i n k t h a t s h a l e p i l l a r s should be designed us ing some o t h e r method?

Answer - The paper examines t h r e e semi-analy t ica l methods p re sen t ly a v a i l a b l e f o r t h e des ign o i l s h a l e p i l l a r s . These a r e :

1. Confined Core 2 . Modified Salamon and Munro 3. Induced Hor i zon ta l S t r e s s

I n wes tern U.S. o i l s h a l e mining because of t h e l a r g e mining th ick- ness and g r e a t e r s t r e n g t h compared t o coa l t h e shape of p i l l a r s a r e e i t h e r c u b i c a l o r more s l e n d e r ( i . e . , he igh t g r e a t e r than width). Consequently f a i l u r e should fundamental ly be d i f f e r e n t t o t h a t i n wide c o a l p i l l a r s . Squat ( l a r g e width t o he ight r a t i o ) des ign of most c o a l p i l l a r s f o r c e s them t o f a i l i n compression while t h e r e l a t i v e l y s l e n d e r des ign of o i l s h a l e p i l l a r s could al low them t o f a i l i n a buckl ing mode. The buckling f a i l u r e mode is a sub jec t which has no t been s tud ied i n d e t a i l and deserves f u r t h e r a t t e n t i o n . Regardless t h e methods examined i n t h e paper have been developed based on o b s e r v a t i o n of f i e l d and l abo ra to ry behavior of e i t h e r coa l o r o i l s h a l e . A comparison of t h e equat ions a v a i l a b l e i n d i c a t e s t h a t t h e f i e l d and l a b o r a t o r y behavior and a s a r e s u l t t h e equat ions f o r t h e s e two m a t e r i a l s a r e s i m i l a r . However, i t must be recognized t h a t o i l s h a l e i s a fundamental ly s t r o n g e r ma te r i a l than coa l having about a n o r d e r of magnitude g r e a t e r s t r eng th . As a r e s u l t , t h e s i z e of t h e p i l l a r s c a l c u l a t e d f o r t h e two m a t e r i a l s w i l l be sub- s t a n t i a l l y d i f f e r e n t .

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