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The effect of specimen size on compressive strength of coal
Int. J. Rock Mech. Min. Sci. Vol. 5, pp. 325-335.
Pergamon Press 1968. Printed in Great Britain
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE S T R E N G T H O F COAL Z. T. National
BIENIAWSKI
Mechanical Engineering Research Institute, Pretoria, South Africa (Received 10 October 1967)
Abstract--This paper describes the results of extensive underground investigations aimed at establishing the in situ strength of coal. Details are given of the equipment and experimental techniques used for testing cubical coal specimensmeasuring from 0" 75 in.-6.6 ft (2 m) in size. Based on over sixty underground test results, an empirical relationship between the strength of coal and the size of the specimens tested is established and its significance for practical applications is outlined. 1. INTRODUCTION KNOWLEDGEof the in s i t u strength of coal is of vital importance to the coal mining industry in designing safe and economical colliery excavations. In South Africa, where extraction of coal is mainly by the bord-and-pillar method, data on compressive strength of coal is essential for the design of coal pillars. It may seem that the compressive strength of a material is a property easy enough to determine and that this can be done during a simple laboratory test. While this may be so, to a certain extent, in the case of such materials as steel, the compressive strength of coal is one of the most difficult properties to establish experimentally. It has been shown [1] that there are many factors that influence properties of coal, such as anisotropy, behaviour of cracks and fissures, pore pressure, environment, rate of loading, time effects and specimen size and shape. Although the compressive strength of coal has been measured since the beginning of this century* and some research has already been done related to factors influencing the compressive strength of coal [2--4], little reliable data exist for estimating the actual in s i t u strength of coal. This is particularly due to the influence of two important factors, namely: (i) The influence of environment. Coal samples, obtained either in the form of drilled cores or large lumps, weather and crack so rapidly after removal from the mine atmosphere, due to changes in temperature and humidity, that results of laboratory tests on coal can hardly be considered as representing in s i t u strength data. (ii) The effect of specimen size. Since coal is not a continuous solid material but contains various discontinuities such as cracks, cleat and bedding planes, the strength of coal is of necessity a statistical value depending on how many and what types of discontinuities are present. * See Louts H. Trans. Am. Inst. Min. Engrs 28 (1904-05).
325
326
z.T. BIEN1AWSKI These discontinuities give rise to a scatter of results. In addition, however, in a smaller specimen the probability of finding discontinuities is smaller and the strength is thus higher. Consequently, the strength of coal is known to decrease with increasing specimen size.
Therefore, to determine the strength data of coal not only a large number of specimens must be tested, to account for the scatter of results, but also the specimens should be as large as possible to account for the size effect. In addition, all the tests should be carried out underground and not in a laboratory in order to eliminate the influence of environment. It is clear that such underground tests must be very expensive and time-consuming and it is not surprising therefore that, as far as the author is aware, no such investigations have as yet been reported in the literature. With the exception of one series of large-scale tests on coal conducted in situ by GREENWALD et al. [5] in 1939, all the previous investigations were conducted in the laboratory. The tests by Greenwald although pioneering were, however, limited in their number with one specimen only tested for each specimen size and no experimental correlation with smaller specimen sizes was obtained. Thus, up to now no reliable experimental evidence exists for the effect of specimen size on strength of coal under conditions which prevail underground. Since this information is essential for the design of coal pillars efforts to obtain it are warranted. Extensive underground investigations were therefore conducted in South Africa by the Council for Scientific and Industrial Research on behalf of the Coal Mining Research Controlling Council and this paper reports on the findings.* 2. EXPERIMENTAL TECHNIQUES Since specimens of a wide range of sizes were tested, different methods of specimen preparation and loading had to be employed. These will now be briefly discussed for the three classes of cubical coal specimens, namely:
(a)
small-size specimens (b) medium-size specimens (c) large-size specimens
: up to 3 in. : up to 18 in. : up to 6.6 ft (2 m).
It should be emphasized that all specimen preparation and testing, including the small sizes, was done underground using specially developed portable equipment, to eliminate effects of environmental changes on the specimens.
2.1 Specimen preparation The small-size specimens were prepared either from loose coal pieces or from drill cores obtained by means of a heavy-duty hydraulic feed drilling machine. Drilling was done dry with compressed air as the cooling medium. The specimens were sawn to approximate sizes using a special specimen preparation unit fitted onto the drilling machine. This unit, driven by a vee-belt from the main motor of the drilling machine was fitted with a diamond saw for cutting the coal into suitable sizes. Final grinding of the specimens was achieved by means of a sanding machine illustrated in Fig. 1. The medium-size specimens were cut directly to the required size by means of the equipment developed for this purpose. The equipment consisted of a modified forestry chain * The locality for the tests was chosen at Witbank Colliery, Transvaal, South Africa.
FIG. 1. Underground grinding of small cubical specimens of coal.
FIG. 2. Chain saw in underground cutting operation of medium-size coal specimens facing page 326 Rock
FIG. 3. A large coal specimen prepared for in situ testing.
FIG. 4. Experimental set-up for testing small coal specimens underground.
FIG. 5. Loading arrangement for underground testing of medium-size coal specimens.
FIG. 6. A large-size coal specimen with loading system and deformation measuring equipment in place
FIG. 7. Violent failure of a 1-ft cube coal specimen.
FIG. 8. T h e double pyramid failure m o d e of a m e d i u m size coal specimen which had failed in a non-violent manner.
FIG. 9. A large coal specimen after failure.
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
327
saw in which the petrol engine was replaced by an air motor supplied from a compressor. The chain was driven through a flexible shaft and a 5 : 1 reduction gear box. The cutting links of the chain saw were provided with tungsten carbide tipping to prevent their wear and to enable efficient cutting. The chain saw thus modified and shown in Fig. 2 was mounted on a specially designed rig to facilitate underground cutting in both horizontal and vertical directions. The large-size specimens were prepared by means of a Simpson universal coal cutter specially modified to give as smooth a cut as possible. The first step in the preparation of the specimen was to take vertical cuts parallel to existing faces of the coal pillar in order to remove weathered and damaged coal and to expose fresh coal from which the specimen can be cut. It was normally required to remove a thickness of approximately 2 ft. Having thus exposed the corner of the coal specimens, the vertical cuts which separate the coal specimen from the remainder of the coal pillar comprised the next stage in the cutting of the specimen. The specimen faces were then instrumented with extensometer units and while the final horizontal cut was taken measurements of specimen expansion were made. Since a certain amount of unevenness in the cut surface is unavoidable, in order to minimize the creation of point loading at surface irregularities, the loaded surface of the specimen was capped with a cement and sand mix of approximately 3 in. thickness. The cap was then allowed to set for a minimum of two weeks. In order to simulate the effects of roof on the coal pillar a lateral restraint was provided in the form of either steel shuttering fastened around the upper part of the specimen or reinforced concrete capping. A specimen thus prepared is illustrated in Fig. 3. It should be noted that the specimens remain attached to the floor and will thus be tested in situ. 2.2 Loading of specimens All the specimens were loaded in uniaxial compression and in the direction perpendicular to the bedding planes. The small-size specimens were loaded underground in a 100-ton compression machine illustrated in Fig. 4. The load was measured by means of a strain gauged load cell accurately calibrated beforehand in the laboratory. For testing of the medium-size specimens a horizontal cut of 30 in. high and 6 ft deep was made in one of the pillars by means of the universal coal cutter. The specimens were then loaded against the top and bottom surfaces of the cut using hydraulic jacks designed for this purpose as shown in Fig. 5. All the large-size specimens were tested in situ* by the hydraulic jacks placed on top of each specimen which was then loaded against the roof and the floor of the seam as depicted in Fig. 6. The jacks had the following design features: (a) Maximum pressure exerted over the loading area--5000 lb/in 2. (b) Load area of one square foot. (c) Overall height of 11½ in. with a stroke of 5 in.--sufficient to meet practically any testing situation. (d) A spherical seat in order to accommodate slight imperfections on the loaded surfaces. * The cost of these tests was appreciable, amounting to some £2000 per o n e test as determined on the basis of expenses involved during two years of development, equipment manufacture, specimen preparation and testing including salaries of 5-man strong full-time research team. Rocx
514,----D
328
z. T. BIENIAWSKI (e) Total weight limited to approximately 150 lb so that one jack could be handled by two persons. (f) Simple design enabling the jack to be stripped and reassembled under conditions prevailing underground. (g) Overall cost kept to minimum (£225).
The pressure was generated by means of a variable-volume reversible flow pump incorporated into a hydraulic pump station specially designed for compactness and ease of transport. All the connexions between the pump station and the jacks were done using self-sealing quick coupling units which enabled efficient underground operations. Since a large number of jacks was to be used in loading the specimens (36 in the case of 2-m cube specimens), it was necessary to ensure that a uniform load distribution was achieved over the entire surface of the specimen and that possibilities of corner failures were eliminated. For this purpose the jacks were not placed directly on the concrete capped surface of the specimen but on a rigid steel channel arrangement. The channels, 4 in. high and of the length equal the width of the specimens, were welded in pairs to form rectangular 'pipes', all such 'pipes' being held in position by high tensile steel bars. Two sets of so-joined channels, placed on top of each other and mutually perpendicular were used (see Fig. 6). Steel plates were then placed on top of the jacks, the remaining space between the plates and the seam being filled with wooden packings. It should also be noted from Fig. 6 that measurement of deformation was achieved by means of vertical and lateral surface extensometer units together with an axial deformation gauge unit placed in the centre of the specimen, the output of this gauge being read off on a strain indicator. 3. EXPERIMENTAL RESULTS The results of the underground tests are summarized in Table 1. It should be seen from this table that, usually, a number of specimens was tested for the same specimen size in order to cross-check the results. TABLE L RESULTS OF UNDERGROUND TESTS ON CUBICAL COAL SPECIMENS LOADED IN UNIAXIAL COMPRESSION PERPENDICULAR TO THE BEDDING PLANES
Cubic size (in.) 0-75 1 2 2"7 3 6 12 18 24 28 36 48 60
Number off tested
Strength (lb/in2)
10 10 8 5 6 7 4 2 1 1 2 2 2
4260 4760 4880 4575 4070 1850 1158 910 800 774 709 650 643"5
Deviation (lb/in~) 814 700 1070 1250 400 435 115 12 --2 20 20
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
329
Generally, pressure on specimens was increased in steps of about 100 lb/in ~. Constant pressure was maintained after each increment until no further deformation was indicated by the deformation measuring instruments. It may be noted that, for large specimens, the time-interval between readings ranged from 15 minutes to a few hours. On the average each in situ test was completed within 8-10 hr. 3.1
Mode offailure
Failure of all specimens occurred in the direction of the applied compressive load. A characteristic of all small-size specimens and some medium-size specimens was a violent mode of failure, in some cases in a manner tantamount to an explosion as shown in an unusual photograph in Fig. 7. In the case of most medium and all large specimens, however, failure occurred in a distinctly non-violent fashion leading to a double pyramid type of failure as shown in Figs. 8 and 9. In these cases, failure was invariably associated with opening of vertical cleats and spalling from one or more faces, usually near a comer of the specimens, this phenomenon being concentrated in the mid-height of the specimen and gradually spreading throughout the specimen width. It should be observed from Figs. 8 and 9 that although each specimen had failed, a central block usually remained, sometimes fairly solid, even when all the loose pieces were removed as seen in Fig. 9. 3.2
Strength characteristics
The relationship between the strength and size of cubical coal specimens, as determined experimentally from the present tests is graphically depicted in Fig. 10. It may be seen from this figure that the strength decreases with increasing specimen size but that the curve flattens out, eventually approaching an asymptotic value at the specimen size of about 5 ft (1.5 m). It is also interesting to note that the scatter of the results is gradually reduced as the size of the tested specimens increases. Since the present underground investigations have provided what is believed to be the first reliable results obtained for coal for such a wide range of specimen sizes, these experimental results may now be compared with the PROTOOYAKONOVformula [6], given by equation (1) below, and describing the effect of size on strength of rock. The Protodyakonov formula reads as follows:
~d
d ÷ mb d+ b
) (1)
with m = ao/eM where o, is strength of cubical rock specimen with side length d a,~ is strength of the rock mass, i.e. d = a0 is strength of the specimen with d = 0 b is distance between discontinuities in the rock mass. For comparison with the experimental results equation (1) is also plotted in Fig. 10. The constants for this equation were determined as follows. It can be seen from Fig. 10
330
Z.T. BIENIAWSKI
that the value of 0-0 may be taken as 4575 lb/in 2 (321.65 kp/cm 2) and that the strength of coal mass 0-Mis 643.5 lb/in 2 (45.2 kp/cm2). Thus, m = o0/0-M= 7-1. The value of parameter b = 2.6 in. (6.6 cm) was obtained directly from in situ geological measurements.* It will be seen from the experimental results plotted in Fig. 10 that for specimen sizes of less than about 2.5 in. (6.35 cm), the strength becomes constant, i.e. independent of the specimen size. This interesting finding may be explained by the fact that if the size of the specimen is smaller than the least distance between the discontinuities, the strength of such specimens cannot be affected any more by these discontinuities. Consequently in determining parameter b the minimum rather than the average values were used. It is evident from Fig. 10 that a marked discrepancy exists between PROTODYAKONOV'S curve [6] and the experimental data. Calculations also show that Protodyakonov's curve will approach the strength of coal mass, to within 5 per cent only, for the specimen size of about 26 ft (8 m) ! This discrepancy between the experimental data and the values predicted by the Protodyakonov formula [6] stems from the fact that the constants for the formula were determined experimentally, independently of equation (1). However, determination of parameters b, 0-Mand m from equation (1) by solving three simultaneous equations yield the unrealistic values b = 60 in. (1.5 m), 0-M ---- 80 lb/in ~ (5-6 kp/cm 2) and m ---- 13.5. A further comparison between the theoretical predictions and the present experimental results can be made by considering the statistical theory of WEIBULL [7]. Weibull's relationship reads as follows: rn log ~ = log V~ 0" 3 v1
(2)
where al 0-3is the strength of specimens with volume II1 and V2 respectively, and m is a constant representing the slope of a straight line. In Fig. 11, the experimental results are plotted t as log 0- vs. log 1I. It will also be seen from this figure that a straight line, for m = 2.5, can be fitted according to equation (2) to a portion of the experimental data. It is clear, however, that equation (2) does not hold for all the results, the discrepancy being clearly observed at the beginning and at the end of the experimental curve. In both these instances the strengths are constant but the theory indicates a continuous decrease in strength with increasing specimen volume. An important finding of the present investigation is the fact that there appears to be no more change in strength once the specimen size of 5 ft (1.5 m) is reached. Consequently, from this size onwards the strength of specimens will be the same in spite of their increasing dimensions. Thus, a 20 ft cube coal pillar would have the same strength as a 5-ft cube coal specimen which means that results of such in situ tests are directly applicable to the design of full-size coal pillars. It also means that the strength of coal mass is reached by relatively small finite size coal specimens and not specimens of infinitely large dimensions as predicted theoretically. While this experimental finding throws an entirely new light on the strength-size relationship in rock, it is interesting to note that this has actually been anticipated by DENKHAUS * These measurements involved determinations of so-called 'cleat body' sizes by measurements of thicknesses of bed separation layers and distances between the cleat planes within the different cleat systems. The distance between discontinuities is the cubical root of the volume of a cleat body which is formed by two bed separation planes and two cleat planes. ~ For convenience,metric system was used to construct this graph.
331
THE EFFECT OF SFECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
¢
450
6500 -
&25 " 6000 ~00 55OO 375
b"
350 ~ 5000
I
325 " 4~-?s ~;5oo
.
THE EFFECT OF SPECIMEN SIZE ON
300
275
STRENGTH
400O
250 " 3500
OF
COAL
[
8 225 2OO
3OOO l i
1"/5 - 2500
i/
150
EXPERIMENTAL RELATIONSHIP
2000 125 100
/
PROTOOYAKONOVFORMULA 0"d
/
1500
t ~
b = 2.6ins
O. : 6.6cn
m = 0o/0", = 7.1 0'o= 4575 tb/sq.in=321.? kplcmt 0M= 643-51b/sq.in=~5,2 kp/cmt
"/5 " 1000 "~"~
~
--=---...~
~"'
500
2s
] 0
10
5
10
20
15 30
~0.
20 "50
25 60
30 70
35 80
90
~0 100
~5 110
50 120
CUBE SIZE d
FIG. 10. "1"heeffect of specimen size on strength o£ coal.
55 130
1/,0
60 150
65 inch~=s 150
1"/0 c m
332
z . T . BIENIAWSKI
W :::3
z
/ 0
0 u
E u
E .o 0
o
0
/
0
(~ld~)
~
5o3
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
333
[8] even as early as 1962 on the basis of considerations of the influence of specimen size on strength of metallic materials. He pointed out that from the size range of the order of centimeters onwards, the strength of steel remains constant. This is the reason, he suggested, why the design of large structures (boilers, bridges, ships) may be based on strength tests on the well-known standard steel specimens of much smaller sizes. DENKHAUS [8] has then reasoned that a similar situation may exist in the case of rock but it is only now that this has been confirmed experimentally. Since the above reasoning has not been taken into account by either PROTODYAKONOV [6] or WEIBOLL [7] it is not surprising therefore, that the present experimental results do not confirm their respective theories. It is clear from Figs. 10 and 11 that the strength-size relationship for coal (and perhaps for other rock, for that matter) cannot be described by one mathematical expression such as given by equations (1) or (2) but that three different equations are needed for this purpose in order to describe: (a) the initial constant strength relationship; (b) the subsequent strength reduction relationship; and (c) the final constant strength relationship. From the point of view of practical applications only the last two cases are of interest, of which the third case is by far of the greatest importance. Thus, no attention will be given in this study to the first case. As far as the remaining two cases are concerned it must be emphasized that the present paper deals only with the effect of specimen size which must be distinguished from the effect of specimen shape, the knowledge of the influence of which is considered essential for a derivation of a pillar strength formula for practical applications. Space does not permit to consider the shape effect here, but it will suffice to say that this has been done elsewhere [9] and that the following relationships were derived on the basis of 25 large-scale in situ tests.* W0-16
For case (b):
o = 1100 -
For case (c):
o = 400 + 220 w/h
where
h0.55
.
(3)
(4)
~ is the specimen strength in lb/in ~ w is the width in feet h is the height in feet.
Equation (4) above is valid for w/h greater than unity and for specimen sizes equal or greater than five feet. For w/h ratios as well as specimen width dimensions of less than those specified above equation (3) will apply. This is, however, of academic interest only since such ratios and such pillar widths are not used in South African collieries. Equation (4) can now be used for design purposes of full-size square pillars. The opinion is often expressed [8], however, that pillars of rectangular cross section or barriers are not much stronger than square pillars with side length equal to the shortest side of the rectangular pillars. Since no scientific evidence is available to indicate the extent of any possible differences in the strength for such pillar types, the above statement was checked experimentally during the present study. It was found that, for 13 coal specimens tested underground, the average strength of rectangular coal specimens 3- 5 in. × 2 in. cross section and of 2 in. in * See footnote on page 327.
334
z . T . BIENIAWSKI
height was a b o u t 2 2 . 5 p e r cent higher t h a n that o f 2-in. cube specimens. This finding, while n o t fully conclusive, can nevertheless serve as an indication for strength d a t a o f some rect a n g u l a r pillars. I n o r d e r to determine if the results o b t a i n e d at W i t b a n k Colliery, where the present tests were conducted, are generally representative o f South A f r i c a n coal mines, a survey o f coal strength d a t a for other collieries was u n d e r t a k e n . I n Table 2 a s u m m a r y o f test results on 579 1-in. cube coal specimens from all the m a j o r coalfields in S o u t h A f r i c a is given. While these results d o n o t represent the in situ strength o f coal they can, however, be used for c o m p a r i s o n purposes. I n t r o d u c i n g a reference strength index o f 100 for the strength o f coal specimens f r o m the locality in W i t b a n k Colliery where the in situ tests were conducted, strength indices for o t h e r collieries can be determined as given in the last c o l u m n o f Table 2. It m a y be noted f r o m this c o l u m n that, with the exception o f one locality only, all the results are within 4- 12.5 ~ o f the reference strength index. This finding, therefore, justifies the use o f equation (4) for design purposes on South A f r i c a n collieries with accuracies sufficient for practical purposes. TABLE 2.
COMPARISON OF COAL STRENGTH DATA FOR COLLIERIES REPRESENTING ALL MAJOR COALFIELDS IN
SOUTH AFRICA
-o ~_ o O C
Colliery
Coalbrook North
Uniaxial compressive strength (lb/in2)
Coal density (lb/fts)
Mean
94'5
5920
14.1~
54
3
103-9
89"5 Bertha Section Section 5 89-6 Section 40 84-6 84"0 94"7 91-5
6350
16"9~
60
3
111.5
5000 5575 6400 5500 5875
10"0~ 16"7~ 22"6~ 14.3~ 16.9~
52 72 40 39 100
2 3 2 2 5
87-7 97.8 112"4 96-5 103"0
Locality
Second
Number Number of of Strength Standard specimens batches index deviation tested
seam
C
Cornelia
B B A C D
Durban Navigation Durban Navigation Kendal Sigma Springfield
A
Witbank, Wolvekrans
No. 4 Seam
90"7
5700
18-1~
35
6
100.0
A
Witbank, Wolvekrans
93" 1
6200
27.4 ~
78
9
108" 9
A
Witbank, Wolvekrans
No. 2 Seam No. 1 Seam
92- 9
8200
16" 3 ~
49
4
143" 9
Note: (i) All specimens were 1 in. cube in size and were loaded normal to the bedding planes at a constant rate of 100 lb/in2/sec in a standard laboratory testing machine. (ii) Coalfields: A --Witbank-Breyten Coalfield, Transvaal B --Klip River Coalfield, Natal C --Vereeniging Coalfield, Orange Free State D --Balfour Coalfield, Transvaal. 4. C O N C L U S I O N S . Experimental d a t a on the strength o f coal as a function o f the size o f cubical specimens tested u n d e r g r o u n d indicates that f r o m a certain specimen size onwards, a b o u t 5 ft, the strength remains constant.
THE EFFECTOF SPECIMEN SIZE ON COMPRESSIVESTRENGTH OF COAL
335
2. Theoretical relationships by PROTODYAKONOV [6] a n d by WEIBtrLL [7] are f o u n d to be inapplicable since their equations did n o t take into a c c o u n t the possibility of the above finding. 3. Proposed o n the basis of experimental evidence, a new pillar strength f o r m u l a is f o u n d to be generally applicable for design purposes in South African collieries. Acknowledgements The author wishes to thank Dr. H. G. DENKI-~US,Director of the Institute, for his valuable criticism and encouragement and to Messrs. D. MULLIGAN,U.W.O.L. Vogler and W. L. VAN HEEPa~ENof the Rock Mechanics Division for assistance in carrying out the tests. The Coal Mining Research Controlling Council of South Africa, who sponsored the major part of this study, is gratefully acknowledged for permission to publish this work. A part of this paper was presented in German at the Ninth Meeting of the International Bureau for Rock Mechanics, Leipzig, October 1967 under the title "Eine in situ studie des Bruchmechanismus yon Kohle". REFERENCES 1. BIENIAWSKIZ. T. Mechanism of Brittle Fracture of Rock, D.Sc.(Eng.) Thesis, University of Pretoria, August 1967. 2. GADDYF. L. A study of the ultimate strength of coal as related to the absolute size of the cubical specimens tested. VaPolytech. exp. Sta. Tech. Bull. No. 112, 63-76 (1956). 3. EVANSI., POMEROYC. D. and BERENBAUMR. The compressive strength of coal. Colliery Engng 38, 75-80, 123-217, 172-178 (1961). 4. HOLLANDC. T. The Strength of Coal in Mine Pillars, Proceedings of the Sixth Symposium on Rock Mechanics (E. M. Spokes and C. R. Christiansen, Eds.) University of Missouri, Rolla, pp. 450-466 (1964). 5. GREENWALDH. P., HOW~TH H. C. and HARTMANN1. Experiments on Strength of Small Pillars of Coal in the Pittsburgh Bed, Report of Investigation of the U.S. Bureau of Mines, No. TP605 April 1939, 12p. and No. RI3575, June (1941). 6. PROTODYAKONOVM. M. and KOJFMANM. I. Uber den Massstabseffect bei Untersuchung von Gestein und Kohle--5. Landertreffen des lnternationalen Buroy fiir Gebirgsmechanik, Deutsche Akademie der Wissenschaften, Berlin, No. 3, pp. 97-108 (1964). 7. WEIBULLW. The phenomenon of rupture in solids. IngvetenskAkad. Handl. No. 153 (1939). 8. DENKHAUSH. G. A critical review of the present state of scientific knowledge related to the strength of coal pillars. Jl S. Afr. Inst. Min. Metall. 63, 59-75 (1962). 9. BENIAWSKIZ. T. An Analysis of Results from Underground Tests Aimed at Determining the in situ Strength of Coal Pillars. Report of the Council of Scientific and Industrial Research, South Africa, No. MEG 569, July (1967).
Pergamon Press 1968. Printed in Great Britain
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE S T R E N G T H O F COAL Z. T. National
BIENIAWSKI
Mechanical Engineering Research Institute, Pretoria, South Africa (Received 10 October 1967)
Abstract--This paper describes the results of extensive underground investigations aimed at establishing the in situ strength of coal. Details are given of the equipment and experimental techniques used for testing cubical coal specimensmeasuring from 0" 75 in.-6.6 ft (2 m) in size. Based on over sixty underground test results, an empirical relationship between the strength of coal and the size of the specimens tested is established and its significance for practical applications is outlined. 1. INTRODUCTION KNOWLEDGEof the in s i t u strength of coal is of vital importance to the coal mining industry in designing safe and economical colliery excavations. In South Africa, where extraction of coal is mainly by the bord-and-pillar method, data on compressive strength of coal is essential for the design of coal pillars. It may seem that the compressive strength of a material is a property easy enough to determine and that this can be done during a simple laboratory test. While this may be so, to a certain extent, in the case of such materials as steel, the compressive strength of coal is one of the most difficult properties to establish experimentally. It has been shown [1] that there are many factors that influence properties of coal, such as anisotropy, behaviour of cracks and fissures, pore pressure, environment, rate of loading, time effects and specimen size and shape. Although the compressive strength of coal has been measured since the beginning of this century* and some research has already been done related to factors influencing the compressive strength of coal [2--4], little reliable data exist for estimating the actual in s i t u strength of coal. This is particularly due to the influence of two important factors, namely: (i) The influence of environment. Coal samples, obtained either in the form of drilled cores or large lumps, weather and crack so rapidly after removal from the mine atmosphere, due to changes in temperature and humidity, that results of laboratory tests on coal can hardly be considered as representing in s i t u strength data. (ii) The effect of specimen size. Since coal is not a continuous solid material but contains various discontinuities such as cracks, cleat and bedding planes, the strength of coal is of necessity a statistical value depending on how many and what types of discontinuities are present. * See Louts H. Trans. Am. Inst. Min. Engrs 28 (1904-05).
325
326
z.T. BIEN1AWSKI These discontinuities give rise to a scatter of results. In addition, however, in a smaller specimen the probability of finding discontinuities is smaller and the strength is thus higher. Consequently, the strength of coal is known to decrease with increasing specimen size.
Therefore, to determine the strength data of coal not only a large number of specimens must be tested, to account for the scatter of results, but also the specimens should be as large as possible to account for the size effect. In addition, all the tests should be carried out underground and not in a laboratory in order to eliminate the influence of environment. It is clear that such underground tests must be very expensive and time-consuming and it is not surprising therefore that, as far as the author is aware, no such investigations have as yet been reported in the literature. With the exception of one series of large-scale tests on coal conducted in situ by GREENWALD et al. [5] in 1939, all the previous investigations were conducted in the laboratory. The tests by Greenwald although pioneering were, however, limited in their number with one specimen only tested for each specimen size and no experimental correlation with smaller specimen sizes was obtained. Thus, up to now no reliable experimental evidence exists for the effect of specimen size on strength of coal under conditions which prevail underground. Since this information is essential for the design of coal pillars efforts to obtain it are warranted. Extensive underground investigations were therefore conducted in South Africa by the Council for Scientific and Industrial Research on behalf of the Coal Mining Research Controlling Council and this paper reports on the findings.* 2. EXPERIMENTAL TECHNIQUES Since specimens of a wide range of sizes were tested, different methods of specimen preparation and loading had to be employed. These will now be briefly discussed for the three classes of cubical coal specimens, namely:
(a)
small-size specimens (b) medium-size specimens (c) large-size specimens
: up to 3 in. : up to 18 in. : up to 6.6 ft (2 m).
It should be emphasized that all specimen preparation and testing, including the small sizes, was done underground using specially developed portable equipment, to eliminate effects of environmental changes on the specimens.
2.1 Specimen preparation The small-size specimens were prepared either from loose coal pieces or from drill cores obtained by means of a heavy-duty hydraulic feed drilling machine. Drilling was done dry with compressed air as the cooling medium. The specimens were sawn to approximate sizes using a special specimen preparation unit fitted onto the drilling machine. This unit, driven by a vee-belt from the main motor of the drilling machine was fitted with a diamond saw for cutting the coal into suitable sizes. Final grinding of the specimens was achieved by means of a sanding machine illustrated in Fig. 1. The medium-size specimens were cut directly to the required size by means of the equipment developed for this purpose. The equipment consisted of a modified forestry chain * The locality for the tests was chosen at Witbank Colliery, Transvaal, South Africa.
FIG. 1. Underground grinding of small cubical specimens of coal.
FIG. 2. Chain saw in underground cutting operation of medium-size coal specimens facing page 326 Rock
FIG. 3. A large coal specimen prepared for in situ testing.
FIG. 4. Experimental set-up for testing small coal specimens underground.
FIG. 5. Loading arrangement for underground testing of medium-size coal specimens.
FIG. 6. A large-size coal specimen with loading system and deformation measuring equipment in place
FIG. 7. Violent failure of a 1-ft cube coal specimen.
FIG. 8. T h e double pyramid failure m o d e of a m e d i u m size coal specimen which had failed in a non-violent manner.
FIG. 9. A large coal specimen after failure.
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
327
saw in which the petrol engine was replaced by an air motor supplied from a compressor. The chain was driven through a flexible shaft and a 5 : 1 reduction gear box. The cutting links of the chain saw were provided with tungsten carbide tipping to prevent their wear and to enable efficient cutting. The chain saw thus modified and shown in Fig. 2 was mounted on a specially designed rig to facilitate underground cutting in both horizontal and vertical directions. The large-size specimens were prepared by means of a Simpson universal coal cutter specially modified to give as smooth a cut as possible. The first step in the preparation of the specimen was to take vertical cuts parallel to existing faces of the coal pillar in order to remove weathered and damaged coal and to expose fresh coal from which the specimen can be cut. It was normally required to remove a thickness of approximately 2 ft. Having thus exposed the corner of the coal specimens, the vertical cuts which separate the coal specimen from the remainder of the coal pillar comprised the next stage in the cutting of the specimen. The specimen faces were then instrumented with extensometer units and while the final horizontal cut was taken measurements of specimen expansion were made. Since a certain amount of unevenness in the cut surface is unavoidable, in order to minimize the creation of point loading at surface irregularities, the loaded surface of the specimen was capped with a cement and sand mix of approximately 3 in. thickness. The cap was then allowed to set for a minimum of two weeks. In order to simulate the effects of roof on the coal pillar a lateral restraint was provided in the form of either steel shuttering fastened around the upper part of the specimen or reinforced concrete capping. A specimen thus prepared is illustrated in Fig. 3. It should be noted that the specimens remain attached to the floor and will thus be tested in situ. 2.2 Loading of specimens All the specimens were loaded in uniaxial compression and in the direction perpendicular to the bedding planes. The small-size specimens were loaded underground in a 100-ton compression machine illustrated in Fig. 4. The load was measured by means of a strain gauged load cell accurately calibrated beforehand in the laboratory. For testing of the medium-size specimens a horizontal cut of 30 in. high and 6 ft deep was made in one of the pillars by means of the universal coal cutter. The specimens were then loaded against the top and bottom surfaces of the cut using hydraulic jacks designed for this purpose as shown in Fig. 5. All the large-size specimens were tested in situ* by the hydraulic jacks placed on top of each specimen which was then loaded against the roof and the floor of the seam as depicted in Fig. 6. The jacks had the following design features: (a) Maximum pressure exerted over the loading area--5000 lb/in 2. (b) Load area of one square foot. (c) Overall height of 11½ in. with a stroke of 5 in.--sufficient to meet practically any testing situation. (d) A spherical seat in order to accommodate slight imperfections on the loaded surfaces. * The cost of these tests was appreciable, amounting to some £2000 per o n e test as determined on the basis of expenses involved during two years of development, equipment manufacture, specimen preparation and testing including salaries of 5-man strong full-time research team. Rocx
514,----D
328
z. T. BIENIAWSKI (e) Total weight limited to approximately 150 lb so that one jack could be handled by two persons. (f) Simple design enabling the jack to be stripped and reassembled under conditions prevailing underground. (g) Overall cost kept to minimum (£225).
The pressure was generated by means of a variable-volume reversible flow pump incorporated into a hydraulic pump station specially designed for compactness and ease of transport. All the connexions between the pump station and the jacks were done using self-sealing quick coupling units which enabled efficient underground operations. Since a large number of jacks was to be used in loading the specimens (36 in the case of 2-m cube specimens), it was necessary to ensure that a uniform load distribution was achieved over the entire surface of the specimen and that possibilities of corner failures were eliminated. For this purpose the jacks were not placed directly on the concrete capped surface of the specimen but on a rigid steel channel arrangement. The channels, 4 in. high and of the length equal the width of the specimens, were welded in pairs to form rectangular 'pipes', all such 'pipes' being held in position by high tensile steel bars. Two sets of so-joined channels, placed on top of each other and mutually perpendicular were used (see Fig. 6). Steel plates were then placed on top of the jacks, the remaining space between the plates and the seam being filled with wooden packings. It should also be noted from Fig. 6 that measurement of deformation was achieved by means of vertical and lateral surface extensometer units together with an axial deformation gauge unit placed in the centre of the specimen, the output of this gauge being read off on a strain indicator. 3. EXPERIMENTAL RESULTS The results of the underground tests are summarized in Table 1. It should be seen from this table that, usually, a number of specimens was tested for the same specimen size in order to cross-check the results. TABLE L RESULTS OF UNDERGROUND TESTS ON CUBICAL COAL SPECIMENS LOADED IN UNIAXIAL COMPRESSION PERPENDICULAR TO THE BEDDING PLANES
Cubic size (in.) 0-75 1 2 2"7 3 6 12 18 24 28 36 48 60
Number off tested
Strength (lb/in2)
10 10 8 5 6 7 4 2 1 1 2 2 2
4260 4760 4880 4575 4070 1850 1158 910 800 774 709 650 643"5
Deviation (lb/in~) 814 700 1070 1250 400 435 115 12 --2 20 20
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
329
Generally, pressure on specimens was increased in steps of about 100 lb/in ~. Constant pressure was maintained after each increment until no further deformation was indicated by the deformation measuring instruments. It may be noted that, for large specimens, the time-interval between readings ranged from 15 minutes to a few hours. On the average each in situ test was completed within 8-10 hr. 3.1
Mode offailure
Failure of all specimens occurred in the direction of the applied compressive load. A characteristic of all small-size specimens and some medium-size specimens was a violent mode of failure, in some cases in a manner tantamount to an explosion as shown in an unusual photograph in Fig. 7. In the case of most medium and all large specimens, however, failure occurred in a distinctly non-violent fashion leading to a double pyramid type of failure as shown in Figs. 8 and 9. In these cases, failure was invariably associated with opening of vertical cleats and spalling from one or more faces, usually near a comer of the specimens, this phenomenon being concentrated in the mid-height of the specimen and gradually spreading throughout the specimen width. It should be observed from Figs. 8 and 9 that although each specimen had failed, a central block usually remained, sometimes fairly solid, even when all the loose pieces were removed as seen in Fig. 9. 3.2
Strength characteristics
The relationship between the strength and size of cubical coal specimens, as determined experimentally from the present tests is graphically depicted in Fig. 10. It may be seen from this figure that the strength decreases with increasing specimen size but that the curve flattens out, eventually approaching an asymptotic value at the specimen size of about 5 ft (1.5 m). It is also interesting to note that the scatter of the results is gradually reduced as the size of the tested specimens increases. Since the present underground investigations have provided what is believed to be the first reliable results obtained for coal for such a wide range of specimen sizes, these experimental results may now be compared with the PROTOOYAKONOVformula [6], given by equation (1) below, and describing the effect of size on strength of rock. The Protodyakonov formula reads as follows:
~d
d ÷ mb d+ b
) (1)
with m = ao/eM where o, is strength of cubical rock specimen with side length d a,~ is strength of the rock mass, i.e. d = a0 is strength of the specimen with d = 0 b is distance between discontinuities in the rock mass. For comparison with the experimental results equation (1) is also plotted in Fig. 10. The constants for this equation were determined as follows. It can be seen from Fig. 10
330
Z.T. BIENIAWSKI
that the value of 0-0 may be taken as 4575 lb/in 2 (321.65 kp/cm 2) and that the strength of coal mass 0-Mis 643.5 lb/in 2 (45.2 kp/cm2). Thus, m = o0/0-M= 7-1. The value of parameter b = 2.6 in. (6.6 cm) was obtained directly from in situ geological measurements.* It will be seen from the experimental results plotted in Fig. 10 that for specimen sizes of less than about 2.5 in. (6.35 cm), the strength becomes constant, i.e. independent of the specimen size. This interesting finding may be explained by the fact that if the size of the specimen is smaller than the least distance between the discontinuities, the strength of such specimens cannot be affected any more by these discontinuities. Consequently in determining parameter b the minimum rather than the average values were used. It is evident from Fig. 10 that a marked discrepancy exists between PROTODYAKONOV'S curve [6] and the experimental data. Calculations also show that Protodyakonov's curve will approach the strength of coal mass, to within 5 per cent only, for the specimen size of about 26 ft (8 m) ! This discrepancy between the experimental data and the values predicted by the Protodyakonov formula [6] stems from the fact that the constants for the formula were determined experimentally, independently of equation (1). However, determination of parameters b, 0-Mand m from equation (1) by solving three simultaneous equations yield the unrealistic values b = 60 in. (1.5 m), 0-M ---- 80 lb/in ~ (5-6 kp/cm 2) and m ---- 13.5. A further comparison between the theoretical predictions and the present experimental results can be made by considering the statistical theory of WEIBULL [7]. Weibull's relationship reads as follows: rn log ~ = log V~ 0" 3 v1
(2)
where al 0-3is the strength of specimens with volume II1 and V2 respectively, and m is a constant representing the slope of a straight line. In Fig. 11, the experimental results are plotted t as log 0- vs. log 1I. It will also be seen from this figure that a straight line, for m = 2.5, can be fitted according to equation (2) to a portion of the experimental data. It is clear, however, that equation (2) does not hold for all the results, the discrepancy being clearly observed at the beginning and at the end of the experimental curve. In both these instances the strengths are constant but the theory indicates a continuous decrease in strength with increasing specimen volume. An important finding of the present investigation is the fact that there appears to be no more change in strength once the specimen size of 5 ft (1.5 m) is reached. Consequently, from this size onwards the strength of specimens will be the same in spite of their increasing dimensions. Thus, a 20 ft cube coal pillar would have the same strength as a 5-ft cube coal specimen which means that results of such in situ tests are directly applicable to the design of full-size coal pillars. It also means that the strength of coal mass is reached by relatively small finite size coal specimens and not specimens of infinitely large dimensions as predicted theoretically. While this experimental finding throws an entirely new light on the strength-size relationship in rock, it is interesting to note that this has actually been anticipated by DENKHAUS * These measurements involved determinations of so-called 'cleat body' sizes by measurements of thicknesses of bed separation layers and distances between the cleat planes within the different cleat systems. The distance between discontinuities is the cubical root of the volume of a cleat body which is formed by two bed separation planes and two cleat planes. ~ For convenience,metric system was used to construct this graph.
331
THE EFFECT OF SFECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
¢
450
6500 -
&25 " 6000 ~00 55OO 375
b"
350 ~ 5000
I
325 " 4~-?s ~;5oo
.
THE EFFECT OF SPECIMEN SIZE ON
300
275
STRENGTH
400O
250 " 3500
OF
COAL
[
8 225 2OO
3OOO l i
1"/5 - 2500
i/
150
EXPERIMENTAL RELATIONSHIP
2000 125 100
/
PROTOOYAKONOVFORMULA 0"d
/
1500
t ~
b = 2.6ins
O. : 6.6cn
m = 0o/0", = 7.1 0'o= 4575 tb/sq.in=321.? kplcmt 0M= 643-51b/sq.in=~5,2 kp/cmt
"/5 " 1000 "~"~
~
--=---...~
~"'
500
2s
] 0
10
5
10
20
15 30
~0.
20 "50
25 60
30 70
35 80
90
~0 100
~5 110
50 120
CUBE SIZE d
FIG. 10. "1"heeffect of specimen size on strength o£ coal.
55 130
1/,0
60 150
65 inch~=s 150
1"/0 c m
332
z . T . BIENIAWSKI
W :::3
z
/ 0
0 u
E u
E .o 0
o
0
/
0
(~ld~)
~
5o3
THE EFFECT OF SPECIMEN SIZE ON COMPRESSIVE STRENGTH OF COAL
333
[8] even as early as 1962 on the basis of considerations of the influence of specimen size on strength of metallic materials. He pointed out that from the size range of the order of centimeters onwards, the strength of steel remains constant. This is the reason, he suggested, why the design of large structures (boilers, bridges, ships) may be based on strength tests on the well-known standard steel specimens of much smaller sizes. DENKHAUS [8] has then reasoned that a similar situation may exist in the case of rock but it is only now that this has been confirmed experimentally. Since the above reasoning has not been taken into account by either PROTODYAKONOV [6] or WEIBOLL [7] it is not surprising therefore, that the present experimental results do not confirm their respective theories. It is clear from Figs. 10 and 11 that the strength-size relationship for coal (and perhaps for other rock, for that matter) cannot be described by one mathematical expression such as given by equations (1) or (2) but that three different equations are needed for this purpose in order to describe: (a) the initial constant strength relationship; (b) the subsequent strength reduction relationship; and (c) the final constant strength relationship. From the point of view of practical applications only the last two cases are of interest, of which the third case is by far of the greatest importance. Thus, no attention will be given in this study to the first case. As far as the remaining two cases are concerned it must be emphasized that the present paper deals only with the effect of specimen size which must be distinguished from the effect of specimen shape, the knowledge of the influence of which is considered essential for a derivation of a pillar strength formula for practical applications. Space does not permit to consider the shape effect here, but it will suffice to say that this has been done elsewhere [9] and that the following relationships were derived on the basis of 25 large-scale in situ tests.* W0-16
For case (b):
o = 1100 -
For case (c):
o = 400 + 220 w/h
where
h0.55
.
(3)
(4)
~ is the specimen strength in lb/in ~ w is the width in feet h is the height in feet.
Equation (4) above is valid for w/h greater than unity and for specimen sizes equal or greater than five feet. For w/h ratios as well as specimen width dimensions of less than those specified above equation (3) will apply. This is, however, of academic interest only since such ratios and such pillar widths are not used in South African collieries. Equation (4) can now be used for design purposes of full-size square pillars. The opinion is often expressed [8], however, that pillars of rectangular cross section or barriers are not much stronger than square pillars with side length equal to the shortest side of the rectangular pillars. Since no scientific evidence is available to indicate the extent of any possible differences in the strength for such pillar types, the above statement was checked experimentally during the present study. It was found that, for 13 coal specimens tested underground, the average strength of rectangular coal specimens 3- 5 in. × 2 in. cross section and of 2 in. in * See footnote on page 327.
334
z . T . BIENIAWSKI
height was a b o u t 2 2 . 5 p e r cent higher t h a n that o f 2-in. cube specimens. This finding, while n o t fully conclusive, can nevertheless serve as an indication for strength d a t a o f some rect a n g u l a r pillars. I n o r d e r to determine if the results o b t a i n e d at W i t b a n k Colliery, where the present tests were conducted, are generally representative o f South A f r i c a n coal mines, a survey o f coal strength d a t a for other collieries was u n d e r t a k e n . I n Table 2 a s u m m a r y o f test results on 579 1-in. cube coal specimens from all the m a j o r coalfields in S o u t h A f r i c a is given. While these results d o n o t represent the in situ strength o f coal they can, however, be used for c o m p a r i s o n purposes. I n t r o d u c i n g a reference strength index o f 100 for the strength o f coal specimens f r o m the locality in W i t b a n k Colliery where the in situ tests were conducted, strength indices for o t h e r collieries can be determined as given in the last c o l u m n o f Table 2. It m a y be noted f r o m this c o l u m n that, with the exception o f one locality only, all the results are within 4- 12.5 ~ o f the reference strength index. This finding, therefore, justifies the use o f equation (4) for design purposes on South A f r i c a n collieries with accuracies sufficient for practical purposes. TABLE 2.
COMPARISON OF COAL STRENGTH DATA FOR COLLIERIES REPRESENTING ALL MAJOR COALFIELDS IN
SOUTH AFRICA
-o ~_ o O C
Colliery
Coalbrook North
Uniaxial compressive strength (lb/in2)
Coal density (lb/fts)
Mean
94'5
5920
14.1~
54
3
103-9
89"5 Bertha Section Section 5 89-6 Section 40 84-6 84"0 94"7 91-5
6350
16"9~
60
3
111.5
5000 5575 6400 5500 5875
10"0~ 16"7~ 22"6~ 14.3~ 16.9~
52 72 40 39 100
2 3 2 2 5
87-7 97.8 112"4 96-5 103"0
Locality
Second
Number Number of of Strength Standard specimens batches index deviation tested
seam
C
Cornelia
B B A C D
Durban Navigation Durban Navigation Kendal Sigma Springfield
A
Witbank, Wolvekrans
No. 4 Seam
90"7
5700
18-1~
35
6
100.0
A
Witbank, Wolvekrans
93" 1
6200
27.4 ~
78
9
108" 9
A
Witbank, Wolvekrans
No. 2 Seam No. 1 Seam
92- 9
8200
16" 3 ~
49
4
143" 9
Note: (i) All specimens were 1 in. cube in size and were loaded normal to the bedding planes at a constant rate of 100 lb/in2/sec in a standard laboratory testing machine. (ii) Coalfields: A --Witbank-Breyten Coalfield, Transvaal B --Klip River Coalfield, Natal C --Vereeniging Coalfield, Orange Free State D --Balfour Coalfield, Transvaal. 4. C O N C L U S I O N S . Experimental d a t a on the strength o f coal as a function o f the size o f cubical specimens tested u n d e r g r o u n d indicates that f r o m a certain specimen size onwards, a b o u t 5 ft, the strength remains constant.
THE EFFECTOF SPECIMEN SIZE ON COMPRESSIVESTRENGTH OF COAL
335
2. Theoretical relationships by PROTODYAKONOV [6] a n d by WEIBtrLL [7] are f o u n d to be inapplicable since their equations did n o t take into a c c o u n t the possibility of the above finding. 3. Proposed o n the basis of experimental evidence, a new pillar strength f o r m u l a is f o u n d to be generally applicable for design purposes in South African collieries. Acknowledgements The author wishes to thank Dr. H. G. DENKI-~US,Director of the Institute, for his valuable criticism and encouragement and to Messrs. D. MULLIGAN,U.W.O.L. Vogler and W. L. VAN HEEPa~ENof the Rock Mechanics Division for assistance in carrying out the tests. The Coal Mining Research Controlling Council of South Africa, who sponsored the major part of this study, is gratefully acknowledged for permission to publish this work. A part of this paper was presented in German at the Ninth Meeting of the International Bureau for Rock Mechanics, Leipzig, October 1967 under the title "Eine in situ studie des Bruchmechanismus yon Kohle". REFERENCES 1. BIENIAWSKIZ. T. Mechanism of Brittle Fracture of Rock, D.Sc.(Eng.) Thesis, University of Pretoria, August 1967. 2. GADDYF. L. A study of the ultimate strength of coal as related to the absolute size of the cubical specimens tested. VaPolytech. exp. Sta. Tech. Bull. No. 112, 63-76 (1956). 3. EVANSI., POMEROYC. D. and BERENBAUMR. The compressive strength of coal. Colliery Engng 38, 75-80, 123-217, 172-178 (1961). 4. HOLLANDC. T. The Strength of Coal in Mine Pillars, Proceedings of the Sixth Symposium on Rock Mechanics (E. M. Spokes and C. R. Christiansen, Eds.) University of Missouri, Rolla, pp. 450-466 (1964). 5. GREENWALDH. P., HOW~TH H. C. and HARTMANN1. Experiments on Strength of Small Pillars of Coal in the Pittsburgh Bed, Report of Investigation of the U.S. Bureau of Mines, No. TP605 April 1939, 12p. and No. RI3575, June (1941). 6. PROTODYAKONOVM. M. and KOJFMANM. I. Uber den Massstabseffect bei Untersuchung von Gestein und Kohle--5. Landertreffen des lnternationalen Buroy fiir Gebirgsmechanik, Deutsche Akademie der Wissenschaften, Berlin, No. 3, pp. 97-108 (1964). 7. WEIBULLW. The phenomenon of rupture in solids. IngvetenskAkad. Handl. No. 153 (1939). 8. DENKHAUSH. G. A critical review of the present state of scientific knowledge related to the strength of coal pillars. Jl S. Afr. Inst. Min. Metall. 63, 59-75 (1962). 9. BENIAWSKIZ. T. An Analysis of Results from Underground Tests Aimed at Determining the in situ Strength of Coal Pillars. Report of the Council of Scientific and Industrial Research, South Africa, No. MEG 569, July (1967).