UNSW PILLAR STRENGTH DETERMINATIONS FOR AUSTRALIAN AND SOUTH AFRICAN CONDITIONS

 

JM Galvin1, BK Hebblewhite2 and MDG Salamon3

 

1,2School of Mining Engineering, The University of New South Wales

3 Department of Mining Engineering, Colorado School of Mines

 

 

Abstract: A series of mine design mishaps and accidents in the late 1980s resulted in a major research program at the University of New South Wales (UNSW) aimed at developing pillar and mine design guidelines. A database of both failed and unfailed Australian underground coal mine pillar case studies was compiled. A procedure was developed to enable the effective width of rectangular pillars to be taken into account. The database was analysed statistically using the maximum likelihood method, both independently and as a combined data set with the more extensive South African database. Probabilities of failure were correlated to Factors of Safety. It was found that there was less than a 4% variance in pillar design extraction ratios resulting from each of these approaches. There is a remarkable consistency between the design formulae developed from back-analysis of the two separate national pillar databases containing many different coal seams and geological environments.

 

 

 

1.         INTRODUCTION

 

In the three year period to 1992, 60 continuous miners were trapped by falls of strata for more than seven hours in collieries in New South Wales. In the preceding two years, eight coal miners were killed in pillar extraction operations in the state. In the New South Wales (NSW) and Queensland (Qld) coalfields at least 15 extensive collapses of bord and pillar workings occurred unexpectedly in the 15 years to 1992. Six of these collapses occurred in working panels, fortuitously five of which were during shut-down periods and the sixth while the continuous miner was being flitted to the surface for repairs.

 

One contributor to these mishaps was the lack of a comprehensive pillar design procedure.  NSW legislation at the time simply required coal pillars to have a minimum width of one-tenth depth or 10m, whichever was the greater.  The influence of pillar height on strength received no recognition.

 

This set of circumstances lead to the NSW Joint Coal Board funding a major research project into pillar design and behaviour.  The research was undertaken by the School of Mining Engineering at the University of New South Wales (UNSW). The primary objectives of the research were to improve the understanding of coal pillars and associated floor and roof strata behaviour under various loading conditions and to incorporate these outcomes into the mine design knowledge base.

 

 

2.         RESEARCH METHODOLOGY

 

The approach adopted to pillar design was based on that developed for square pillars by Salamon and Munro (1966, 1967). However, the extensive use of rectangular and diamond shaped pillars in Australia required more detailed consideration as to the effective width of parallelepiped pillars and the effect of this width on pillar strength.

 

Firstly, an adequate Australian database of failed and unfailed pillar case histories was established. A relationship was then developed to factor in the influence of rectangular pillar and diamond shaped pillars that comprised just over 50% of the database. This database was then subjected to rigorous statistical analysis using a range of techniques in order to quantify parameters associated with each of two generally accepted empirical formulae for describing pillar strength.  This facilitated the establishment of correlation, for all strength expressions, between the probability that a formula would yield a successful design versus the respective design factor of safety.

 

The Australian database was also combined with the much larger and long-established South African database and the analysis repeated in order to determine if the two population bases could be considered as one.  A close correlation was obtained, leading to an increased level of confidence in this methodology and to a number of more universal conclusions concerning pillar design.

 

 

 

3.         EMPIRICAL COAL PILLAR STRENGTH ESTIMATIONS.

 

The development of computer and numerical technologies in recent decades has facilitated, at least in principle, the analysis of stresses in pillars and their foundations, that is, the roof and floor strata. Unfortunately, physical experimentation has not advanced equally rapidly.  Hence, the understanding of the intrinsic constitutive laws controlling the behaviour of yielding rocks is still unsatisfactory. More immediate problems include the significant discrepancies between the physical properties exhibited by rocks in situ and those measured in the laboratory by testing small specimens. These problems relate to the effects of size and shape on rock strength.

 

Many investigators have proposed simple empirical formulae to describe the strength of coal pillars. The most common feature of most of these empirical relationships is that they define strength ostensively only in terms of the linear dimensions of the pillars and a multiplying constant, representing the strength of the unit volume of coal. Investigators over the years have proposed formulae that belong to one of two types. One type defines pillar strength simply as a linear function of the width to height ratio:

 

                                                                                                                                                  (1)

 

where K1 is the compressive strength of a cube and r is a dimensionless constant. The quantities of w and h are width and height of the pillar respectively.


If the notation:

                                   

                                                                                                                                                (2)

 

is introduced then (1) becomes:

 

                                                                                                                                                    (3)

 

According to this formula, geometrically similar pillars have the same strength regardless of their actual dimensions.

 

A second commonly used pillar strength formula takes the form of:

 

                                                                                                                                                    (4)

 

which is expressed in a dimensionally correct form.  and ß are dimensionless parameters, w and h are the linear dimensions of the pillar. Multiplier K2 is the strength of a reference body of coal of height h0 and a square cross-section with side length w0..

 

In most instances, the reference body is taken to be cube of unit volume for convenience sake, in which case h0 and w0 are both unity and can be omitted from the formula. Expressions belonging to this family are referred to as power law strength formulae. In contrast to formulae of the form of (1), these formulae are also volume sensitive.

 

 

 

4.         EFFECTIVE WIDTH OF PARALLELEPIPED PILLARS.

 

The development of statistically based pillar design formulae rests upon the minimum premise that a fairly large and tolerably reliable database of unfailed and failed pillar panels can be compiled.  Salamon et al (1996) have identified a number of strict criterion that must be satisfied before a case can be included in the database.  One of these that must be appreciated when applying the outcomes of this pillar design research is that these outcomes only apply to competent roof and floor environments.  That is, the database relates only to failures of the coal pillar element of the pillar system and not to the roof or floor elements.

 

Against this background, an Australian database of 19 failed and 16 unfailed cases was assembled. Rectangular pillars comprised 8 of the failed and 9 of the unfailed cases.  Diamond shape pillars comprised one failed case. In order to preserve in these circumstances the availability of the strength formulae derived for square pillars, many researchers have proposed the introduction of an effective width.

 

One of the most basic approaches is to define the effective width, we as:

 


                                                                                                                                                (5)


                       

where w1  =  minimum pillar width ( measured along roadway).

where w2  =  maximum pillar width ( measured along roadway).

 

 

In situations where w2 is not extremely different to w1, thus approach has merit.  However, when w2 >> w1, the equation produces an unrealistic effective pillar width, Table 1.

 

Table 1.           Application of Various Effective Pillar

Width Formulae.

 

w1

w2

h


Öw1 w2

4Ap/Cp

w
Q

100

100

3

100.0

100

100

80

100

3

89.4

88.9

88.9

50

100

3

70.7

66.7

66.7

30

100

3

54.7

46.2

46.2

20

100

3

44.7

33.3

33.3

15

100

3

38.7

26.1

21.7

10

100

3

31.6

18.2

10.7

1

100

3

10.0

2.0

1

 

 

The most promising recommendation has came from Wagner (1974, 1980) who, making use of the concept of hydraulic radius, suggested that the effective width be defined as: 

                                                                                                                               (6)

where  and  are the cross sectional area and the circumference of the pillar, respectively.

 

Application of (6) produces similar effective pillar widths to that of (5) when w1 is greater than about 0.5w2, Table 1.  At moderate to low values of w1  (0.4w2 £ w1  £ 0.2w2), (6) predicts a smaller effective width, which is more sensible from a mechanistic viewpoint.  However, at very low values w1 (w1 < 0.2w2), the equation is still considered to overestimate the effective pillar width.  This is because when a pillar is narrow, failure is likely to occur across the narrow dimension before sufficient confinement is generated in the longitudinal direction to be of benefit.

 

This leads to the concept that rectangular and irregular pillars need to be of a critical minimum width before benefit is gained from confinement generated in the longitudinal direction.  This benefit can be expected to ramp up to a plateau level as the minimum width increases.  Furthermore, it is reasonable to expect that this minimum critical width will be a function of mining height, increasing with increasing mining height.

 

The need to nominate a minimum critical pillar width has been incorporated into the analysis by modifying (6) on the basis that almost all pillars can be regarded as parallelepiped, the base of which is a parallelogram, Figure 1. Pillars therefore have side lengths w1 and w2 (w1 £ w2) and an internal angle . (6) then becomes:

 

                                                                                                                              (7)

 

where w is the minimum width of the pillar, that is;

 

                                                                                                                           (8)

 

and the dimensionless factor  is defined by

 

                                                                                                                         (9)

 

The range of this factor is , which is encountered as the aspect ratio moves from unity towards infinity.  Experience indicates that much before the complete failure of a pillar, its edges are already yielding.  Thus, if the width to height ratio in one direction of a rectangular pillar is low, one of the principal stresses confining its core will remain small and this stress, together with the maximum stress, will control failure.

 

Hence, the extra confinement that may arise from the aspect ratio will have little or no effect.  It is suggested that such apprehension may be catered for by postulating that the effective width is the minimum width, that is,  as long as  and it becomes  when .

 

In the intermediate range, that is when , the effective width changes smoothly in accordance with:

 

                                                                                                             (10)

 

Here the choice of the limiting width to height ratios is open to judgement.  It appears reasonable however, to use the following values:

 

                                                                                             (11)

 

Table 1 and Figure 2 show the effects of the various approaches when applied to calculating the effective pillar width of a 100m long, 3m high rectangular pillar.

 

Using the concept of effective width, the power law in (4) can be rewritten for pillars that have a general parallelogram shape.

 

                                                                                                                 (12)

 

An alternative form of this formula expresses the strength as the function of the pillar volume V and the width-height ratio R:

 

                                                                                                                 (13)

 

where the volume refers to a dummy square pillar of width w and height h, and the width-height ratio is calculated from the minimum pillar width.

 

                                                                               (14)

 

The new constants a and b can be defined in terms of constants a and b:

 

                                                                           (15)

 

 

Experience has shown that the original power law formula (4) tends to underestimate the strength of squat pillars, that is, of pillars which have a width to height ratio in excess of about five.  To cater for this problem, Salamon (1982) suggested an extension of (4) into the range of higher width to height ratios.  This extension, after adaptation to pillars of parallelogram shape, is as follows:

 

                                                                                 (16)

 

which is valid if  and where  is defined in (10).  This particular form was chosen to ensure that there is a smooth transition between this and the formula in (13) at  (Salamon and Wagner, 1985).  Here  are appropriately chosen constants.  The expression is often referred to as the squat pillar strength formula.  Since its inception, it has been applied widely in South Africa, using the following pair of constants:

 

                                                                                               (17)

 

In critical situations, the judgement exercised in deriving the effective pillar width relationship may be regarded as too speculative.  This concern can be addressed by either choosing an elevated design factor of safety to account for this level of uncertainty or by reverting to the use of the minimum pillar width in pillar strength calculations.

 

Another aspect to the use of rectangular pillars is the calculation of pillar load. In calculating the tributary load, the true dimensions need to be employed.  Thus, the pillar load assumes the following form:

                                                                                         (18)

In this relationship  is a modifier.  It is unity in all cases where the pillar burden is the conventional tributary load.  If, however, due to secondary extraction the pillar load is believed to differ from this value, then the load can be adjusted by applying this factor.  Moreover, to remain consistent with earlier calculations  is taken to be:

                                     

 

 

 

5.         UNSW INITIAL DESIGN FORMULAE

 

In 1992, following a number of serious incidents related to there being no restriction on pillar height, the Chief Inspector of Coal Mines in New South Wales required operators to obtain approval to mine at heights exceeding 4m. To address the need for a pillar design methodology, the UNSW research team undertook in 1995 a preliminary analysis of its database (Hocking et al, 1995).

 

At the time, the database comprised 14 collapsed cases and 16 stable cases that satisfied the selection criteria. The database was analysed statistically using the full Maximum Likelihood Method. Galvin and Hebblewhite (1995) subsequently published the following pillar design formulae which find current application in Australia:

 

                                                                                                (MPa)                                                                            (19a)

 

and its squat pillar version (R>5):

 

                                                                         (MPa)                                (19b)

 

A conservative approach was adopted and the minimum pillar width was proposed as the effective width. It follows therefore, that =1 in these expressions. There was little difference in the pillar strength obtained by allowing all parameters to float in the statistical analysis as opposed to allowing only the K values to float and fixing the other parameters to be the same as those used for many years in South Africa.  To avoid confusion and to facilitate the introduction of the formulae therefore, only those formulae derived by allowing the K values to float were presented to operators. The formula for strength based on the linear relationship took the form:

 

                                                                                       (MPa)                                         (20)

 

6.         UNSW REFINED (RECTANGULAR) FORMULAE.

 

In 1996, a more comprehensive statistical analysis was completed of the expanded Australian database incorporating the effective width of rectangular pillars as defined earlier. (Salamon et al, 1996). Statistical methods utilised included Least Squares, Limited Maximum Likelihood and Full Maximum Likelihood. Both power law models and linear law models were evaluated and all parameters were allowed to float. In all instances, the power law model gave better correlations.

 

The following strength formula were found to best describe the observed behaviour of pillars in NSW and Queensland:

 

                                                     (MPa)                                         (21a)

 

The corresponding expression for squat pillars is given by:

 

                                                                         (21b)

 

In these expressions, w = w1 sinq and the effective width factor  is as defined in (10).

 

The relationship between pillar strength and pillar load produced by these equations for each point in the database is shown in Figure 3.  Design factors of safety associated with the probability of achieving a stable design are shown in Table 2.

 

Table 2            Probability of Failure vs Factor of Safety

 

Probability of Failure

 

Factor of Safety

8 in 10

0.87

5 in 10

1.00

1 in 10

1.22

5 in 100

1.30

2 in 100

1.38

1 in 100

1.44

1 in 1,000

1.63

1 in 10,000

1.79

1 in 100,000

1.95

1 in 1,000,000

2.11

 

7.         RE-ANALYIS OF SOUTH AFRICAN DATABASE.

 

The original extensive South African coal pillar database used by Salamon and Munro in 1966 has since been updated and supplemented by Madden and Hardman (1992). This combined South African database comprises 44 failed cases and 98 unfailed cases. It has also been re-analysed using the same statistical techniques as for the Australian database. Two failed cases were later omitted from the data set (see Salamon et al, 1996).

 

This analysis has produced the following strength formulae.

                                                                                              (MPa)                                                                      (22a)

 

The corresponding expression for squat pillars () is given by the expression:

 

                                                                   (MPa)                             (22b)

 

The linear version of the strength estimator is simply:

 

                                                                                  (MPa)                                        (23)

Figure 4 shows the comparison between the pillar strength produced by (22a) and (22b) and that predicted by the original Salamon and Munro formula and its modified squat pillar form. In the case of a mining height of 2m, the figure shows that for a given pillar strength, pillars designed with the updated formulae may need to be some 2m more in width.  For a bord width of 6m at a width to height ratio of 10, this results in about 3% less resource recovery.  For similar circumstances in a 4m mining height environment, the increase in pillar size is of the order of 3.2m.

 

 

 

8.         COMBINED AUSTRALIAN AND SOUTH AFRICAN DATABASES

 

 

A further step in the research program was to combine the South African and Australian databases and to analyse them as a combined population and then compare and contrast them with the two independent data populations for each country.

 

This combined database comprised 177 cases of pillar systems including 61 collapsed cases. This produced the following formulae:

 

                                                                                            (MPa)                                           (24a)

 

For R > 5, the squat version of this expression takes the form:

                                                                          (MPa)                                 (24b)

 

The corresponding linear formula can be expressed simply as:

 

                                                                                     (MPa)                                        (25)

 

Figure 5 shows failed and unfailed cases in the load plane.  The figure illustrates a fairly good discrimination between the two sets of points.  Only one unfailed point occurs on the wrong side of the s=1 line and the median failed cases is 1.039.

 

Figures 6(a) and (b) shows a comparison between pillar strengths using power law estimators derived from the Australian, South African and the combined Australian and South African databases.  The closeness of the predictions is remarkable considering the geographical separation of the Australian and South African coalfields.

 

 

 

9.         CONCLUSIONS

 

 

The statistical analysis of the Australian database indicates that the method proposed for calculating the effective width of parallelepiped pillars produced sensible outcomes.  However, it must be remembered that, although of sufficient size as to statistically significant, the parallelepiped database is small. The method should therefore be used with caution.

 

In order to enhance confidence in the pillar design procedure, including the use of the effective pillar width method, additional research was undertaken.  It was noted that the formula derived from the initial Australian database resembled closely the original Salamon and Munro expression.  This somewhat surprising resemblance prompted further research and the enlargement of the database.  The larger database yielded pillar strengths that again were similar to those obtained from the initial UNSW research and by Salamon and Munro.  The combination of the Australian and South African databases reinforced the original impression, namely that the underlying pillar strengths in these countries resembled each other closely.

 

The outcome of the investigation lends support to the view expressed by Mark and Barton (1996).  They suggested that strength values obtained in the laboratory cannot be utilised in a meaningful way in pillar design and the variation in the strength of pillars of the same size can be disregarded in many instances.  Other investigators have come close to making a similar statement (Mark, 1990, Salamon, 1991, Galvin et al, 1995b).

 

Mark and Barton emphasise that they do not claim that the in situ strength of all US coal is the same.  Their study merely showed that a uniform strength is a better approximation than one based on laboratory testing.  Whilst the UNSW research conclusions are encouraging, complacency is not justified.  The formulae are based on competent roof and floor conditions.  Significantly different pillar strengths may be associated with abnormal strata behaviour mechanisms. Since pillars with w/h ratios greater than 10 have not been tested to destruction, it must also be recognised that neither linear nor power law formulae have been validated at w/h ratios greater than about 8.

 

It cannot be over-emphasised that because the design formulae have been developed on a probabilistic basis, they need to be reviewed periodically as the database expands and the understanding of pillar mechanics advances. A fundamental rule of empirical research is that the results should be used within the range of data used in their derivation. Extrapolation with empirical formulae is always fraught with danger.

 

 

 

REFERENCES

 

Galvin, J.M., Hebblewhite, B.K. and Wagner, H., 1995a  Roadway and pillar mechanics workshop: Stage 2 - Design principles and practice. Dept of Mining Eng, Univ. of New South Wales.

 

Galvin, J.M. and Hebblewhite, B.K., 1995b. UNSW pillar design methodology, Department of Mining Engineering, The University of New South Wales, Research Release No. 1.

 

Hocking, G., Anderson, I. and Salamon, M.D.G., 1995.  Coal pillar strength formulae and stability criteria.  Department of Mining Engineering, University of New South Wales.  Research Report 1/95.

 

Madden, BJ and Hardman, DR., 1992.  Long term stability of bord and pillar workings.  Symp on Construction over Mined Areas. Pretoria, South Africa

 

Mark, C., 1990.  Pillar design methods for longwall mining. BuMines IC 9247, 53p.

 

Mark, C. and Barton, T., 1996. The uniaxial compressive strength of coal: Should it be used to design pillars? Proc. 15th International Conf. On Ground Control in Mining. Golden, Colorado, USA.

 

Salamon, M.D.G., 1991.  Behaviour and design of coal pillars. Australian Coal J. No 32, pp. 11-22.

 

Salamon, M.D.G., Galvin, J.M., Hocking, G. and Anderson, I., 1996. Coal pillar strength from back calculation, Department of Mining Engineering, The University of New South Wales, RP 1/96. ISBN 0 7334 1489 3.

 

Salamon, M.D.G. and Munro, A.H., 1966. A study of the strength of coal pillars.  Transvaal and Orange Free State Chamber of Mines, Research Report No. 71/66.

 

Salamon, M.D.G. and Munro, A.H., 1967. A study of the strength of coal pillars. J.S.Afr. Inst. Min. Metall., V.67.

 

Salamon, MDG and Wagner, H.., 1985. Practical experiences in the design of coal pillars. Safety in Mines Research. Proc 21st Int Conf. Sydney, Australia.

 

Wagner, H., 1974. Determination of the complete load deformation characteristics of coal pillars. Proc. 3rd Cong. Int. Soc. Rock Mech., Denver, CO, USA. pp1076-81.

 

Wagner, H., 1980. Pillar design in coal mines. J. S. Afr Min. Metall., V80, pp 37-45         


 

w1
 

b2
 
 


b1
 

Figure 1.  Definition of mining variables associated with a parallelogram shaped pillar.

 

 


Figure 2.  Comparison between the various proposals for calculating

the effective width of rectangular pillars.

 


Figure 3.  Pillar strength and pillar load relationship for both the failed (o) and unfailed (+) Australian cases

 

 

 

 

(a)   h = 2 m

 

 

 

(b)  h = 4 m

 

Figure 4.  Comparison between South African Power Formulae - 1966/82 and 1996.

 


Figure 5.  The failed (o) and unfailed (+) cases in a pillar strength versus pillar load plot, sing the combined Australian and South African databases.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)   h = 2 m

 

 

(b)  h = 4 m

 

 

Figure 6.  Comparison between power law strength formulae derived for Australian, South African and combined data bases - pillar heights 2m and 4m.