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Numerical estimation of pillar strength in coal mines
International Journal of Rock Mechanics & Mining Sciences 38 (2001) 1185–1192
Numerical estimation of pillar strength in coal mines G. Murali Mohan, P.R. Sheorey*, A. Kushwaha Central Mining Research Institute, Barwa Road, Dhanbad 826 001, Jharkhand, India Accepted 19 October 2001
Abstract Results of numerical modelling of failed and stable cases of pillars from Indian coal mines are reported in this paper. The complete procedure for modelling using FLAC3D in the strain-softening mode is given. It is shown that estimation of pillar strength is possible using numerical modelling and it may provide a viable, and perhaps better, alternative to earlier conventional pillar strength approaches. Methods are given to estimate the large number of input parameters required in the models. Because of the paucity of rock characterisation and in situ stress data, some approximations and assumptions were made in all the cases based on earlier experience. The modelling work leads to the conclusion that such data are necessary for the success of this method. Areas for further research related to this subject are given in the end. r 2002 Published by Elsevier Science Ltd.
1. Introduction Pillars in coal mines serve various purposes e.g., protection of gate roadways or entries, panel isolation to guard against spontaneous heating, protection of mine shafts and surface subsidence control. Considering the fact that the structural integrity of a coal mine largely depends on pillars, considerable research in the area of pillar strength and design has been done over the last few decades. Even so, some identifiable grey areas in defining pillar strength still remain. The large number of pillar strength approaches proposed so far has been comprehensively reviewed earlier [1]. These approaches can be classified into four types: empirical, semi-empirical, statistical and analytical. As far as the authors are aware, a generalised formulation employing numerical modelling for estimating pillar strength has not been tried as yet. This paper attempts to show that numerical modelling may perhaps be considered as a viable alternative method to define pillar strength. This approach will have the advantage of duly accounting for the in situ stresses as well as roof and floor strata properties. The earlier strength formulae clearly do not possess this advantage. In addition, they disregard the fact that roof and floor rocks may themselves be in a state of failure at the time of, or even before, pillar failure. *Corresponding author. Tel.:/fax: +91-326-204-339. E-mail address: [email protected] (P.R. Sheorey). 1365-1609/01/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 1 3 6 5 - 1 6 0 9 ( 0 1 ) 0 0 0 7 1 - 5
Because of the large number of parameters involved in such modelling, as seen later, it was thought best to model actual cases of failed and stable pillars to make the procedure for estimating these parameters reasonably realistic.
2. Failed and stable pillar cases Any method for estimating pillar strength should satisfy the following two statistical conditions [2]: (a) All failed pillar cases should have a safety factor of 1.0. In statistical terms, this merely means that the line of safety factor=1.0 should be the best fit for a plot of pillar strength vs. pillar load. (b) All stable cases must have a safety factor>1.0. Both conditions implicitly refer to the point of strength failure or the peak stress sustainable by a pillar. It may, however, be necessary in the case of very squat pillars to consider their post-failure resistance, if the stress–strain characteristic rises after failure. Such a rise in the post-failure curve is seen to exist in laboratory squat specimens of coal [3]. Details of the failed and stable cases of pillars [4] from Indian coalfields are presented in Tables 1 and 2. All the cases except those from Jitpur are of pillars approximately square in plan.
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Nomenclature sh sv E n b G H s1 ; s3 sc st b
scm ; stm ; bm mean horizontal in situ stress vertical in situ stress elastic modulus Poisson’s ratio coefficient of linear thermal expansion geothermal gradient depth of cover major and minor principal stresses intact rock compressive strength intact rock tensile strength exponent in the failure criterion
3. In situ stresses The Fast Lagrangian Analysis of Continua (FLAC) softwares developed by ITASCA Consultants of USA [5], provide for elasto-plastic analysis of rock excavations with strain softening using the linear Mohr– Coloumb failure criterion. The three-dimensional version of these softwares was employed for modelling the cases of Tables 1 and 2. The estimation of the different parameters to be input besides in situ stresses and the modelling procedure are described in subsequent sections. A few preliminary model runs of some of the cases indicated that the pillar strength was sensitive to the in situ horizontal stress. It was therefore necessary to try
RMR tsm m0m f0m S P w h B
rock mass strength constants corresponding to the intact rock constants above Bieniawski’s rock mass rating (1976) rock mass shear strength rock mass coefficient of internal friction rock mass angle of internal friction pillar strength pillar load pillar width pillar height roadway width
and estimate the horizontal stresses within the coal seam as realistically as possible. A theory for in situ stresses [6] shows that the mean in situ horizontal stress (mean of the major and minor horizontal stresses) depends on the elastic constants (E; n), the coefficient of thermal expansion (b) and the geothermal gradient (G). This theory gives the mean horizontal stress as sh ¼
n bEG sv þ ðH þ 1000Þ: 1n 1n
ð1Þ
In a recent work [7], this equation is shown to fit stress measurement data from different parts of the world quite well, if they are not unduly affected by surface topography or major geological features. It may thus be
Table 1 Failed pillar case data Case no.
Mine (seam)
H (m)
h (m)
W (m)
B (m)
w=h
sc (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Amritnagar (Nega Jamehari) Amritnagar (Nega Jamehari) Begonia (Begonia) Amlai (Burhar) Sendra Bansjora (X) W. Chirimiri (Main) Birsingpur (Johilla top) Pure Kajora (Lower Kajora) Pure Kajora (Lower Kajora) Shankarpur (Jambad bottom) Ramnagar (Begunia) Ramnagar (Begunia) Kankanee (XIII) Kankanee (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV)
30 30 36 30 23 90 129 54 56 42 70 51 160 140 450
4.5 6.0 3.0 5.4 8.1 3.75 3.6 3.6 3.6 4.8 1.8 1.8 6.6 8.4 3.6
3.6 3.6 3.9 4.5 4.65 5.4 7.5 5.4 4.95 4.5 2.85 3.0 19.8 18.6 18 34.5 10.5 12 12 21 18 21 18 25.5 15 63 18 30 18 30 15 33
5.7 5.4 6.0 4.5 5.55 6.0 6.0 6.0 6.45 4.5 3.15 3.6 4.2 5.4 6.0
0.8 0.6 1.3 0.83 0.57 1.44 2.08 1.5 1.38 0.94 1.58 1.67 3.0 2.2
45 45 26 25 24 45 38 33 33 47 26 26 27 25 19
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Table 2 Data for stable pillar cases Case no.
Mine (seam)
H (m)
h (m)
w (m)
B (m)
w=h
sc (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Bellampalli (Ross) Nimcha (Nega) Morganpit (Salarjung) Ramnagar (Ramnagar) Lachhipur (Lower Kajora) N. Salanpur (X) Bankola (Jambad top) Bankola (Jambad top) Surakacchar (G-I) Lachhipur (Lower Kajora) Sripur (Koithee) E. Angarapatra (XII) Kargali Incline (Kathara) Jamadoba 6 and 7 Pits (XVI) Topsi (Singharan) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV)
36 48 270 75 38 30 102 75 106 38 266 30 36 80 85 450
3.0 6.0 3.0 2.7 5.1 5.1 4.8 3.0 3.5 5.1 4.8 2.1 3.6 2.0 1.8 3.6
5.4 9.9 8.1 9.9 7.2 9.0 10.1 6.3 16.0 18.3 40.0 6.0 9.3 5.8 7.0 21 39 18 48 19.5 30 18 31.5 18 42
6.0 6.0 3.6 6.6 3.9 6.0 2.4 4.2 4.0 4.2 5.0 6.0 5.7 5.5 3.9 6.0
1.8 1.7 2.7 3.7 1.4 1.8 2.1 2.1 4.6 3.6 8.3 2.9 2.6 2.9 3.9
48 50 46 28 33 21 35 35 29 33 43 19 40 29 41 19
seen that in a sedimentary formation, consisting of beds of different properties, it will not be proper to assign a common equation for horizontal stress variation with depth. This also implies that the stress values obtained in coal measure rocks will not represent the in-seam values. The vertical stress can be taken for coal measure rocks as sv ¼ 0:025H
ðMPaÞ:
ð2Þ
Then putting Eq. (2) and the following values in Eq. (1): n ¼ 0:25;
b ¼ 3 105 =1C;
E ¼ 2000 MPa;
G ¼ 0:031C=m;
stresses used in the roof, seam and floor were different, the pillar strength remained virtually the same. Therefore, Eq. (3) was adopted for the roof and floor rocks as well along with Eq. (2). It may be mentioned that many of the cases of Tables 1 and 2 are quite old and it was not possible to determine the elastic modulus of the coals concerned, though this was desirable. Similarly, rock classification studies, which are required for estimating some of the input parameters described in the next section, could not be undertaken for the same reason. However, the uniaxial compressive strength of coal was fortunately determined at the time of collecting the details of these cases in the 1960s and 1970s. These values are included in the tables.
we obtain the mean horizontal stress as sh ¼ 2:4 þ 0:01H
ðMPaÞ:
ð3Þ
In Eqs. (2) and (3), the depth of cover H is to be taken in metres. The value of the coefficient of thermal expansion b for coal could be obtained from only one reference [8]. The thermal gradient G is given for Indian coal measures. Important deviations from Eq. (3) can exist since the elastic modulus of coal can vary from about 800 to 5000 MPa. In fact, the value E ¼ 2000 MPa was chosen as a rough average only after trial model runs of a few cases from Table 1. Thus, Eqs. (2) and (3) were adopted for all the cases. Application of Eq. (3) indicates that the two horizontal in situ stresses are equal. However, significant differences in pillar strength can exist in the presence of high stress anisotropy. A brief investigation by numerical modelling showed that if the in situ horizontal
4. Estimation of input parameters The various strength and elastic constants necessary for numerical modelling using FLAC3D in the strainsoftening mode are given below: * *
*
*
Elastic constants; peak and residual shear strength and the variation in between with the shear strain; peak and residual angle of internal friction and the variation with the shear strain and angle of dilation.
All the above properties are to be specified for both the coal seam and the roof and floor rocks. In the absence of test data, the elastic constants were simply taken as given in Table 3. Since published data on the dilation angle of coal are non-existent it was simply
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Table 3 Elastic and dilation constants used in the models
Coal Rock
τ
E (GPa)
n
Dilation angle
2 7
0.25 0.25
0 0
Sheorey criterion Φ0m - 5
o
Mohr-Coloumb criterion
assumed to be zero, though undoubtedly it will have a bearing on the post-failure behaviour of the pillar, but perhaps not on the peak strength. The compressive strength of the roof and floor is assumed to be the same as that of the coal seam for cases where the strength of coal is >30 MPa. When it is less, it is taken as 30 MPa for the roof and floor rocks. This was necessary as the roof and floor failure zone was unrealistically high and yielded unacceptable pillar strength values when the compressive strength was o30 MPa. The shear strength and friction angle were estimated as rock mass parameters using a failure criterion for rock masses [9]. This criterion uses the 1976 rock mass rating (RMR) of Bieniawski [10] for reducing the laboratory strength parameters to give the corresponding rock mass values. For this purpose, an Indian average value of RMR ¼ 55 was taken, since the individual RMR values were not known. This criterion is defined as s3 bm s1 ¼ scm 1 þ ; stm where s1 and s3 are major and minor principal stresses at failure and the rock mass strength parameters are defined by RMR 100 scm ¼ sc exp ; 20 RMR 100 ; stm ¼ st exp 27 bm ¼ bRMR=100 : In the above equations, the subscript m stands for the rock mass. For estimating these parameters, only the value of the compressive strength sc was known (Tables 1 and 2). Then the tensile strength st ¼ sc =15 and b ¼ 0:5 were taken as the most representative values, as seen from a large number of test data published earlier [9]. From these, the rock mass shear strength tsm ; the coefficient and the angle of internal friction m0m and f0m are obtained as 1=2 bbmm ; tsm ¼ scm stm ð1 þ bm Þ1þbm m0m ¼
t2sm ð1 þ bm Þ2 s2tm ; 2tsm stm ð1 þ bm Þ
f0m ¼ tan1 ðm0m Þ:
Φ0m 1.1 τsm
τsm
σtm 0
σ
Fig. 1. Schematic showing the non-linear Sheorey criterion as against the linear Mohr–Coloumb criterion adopted in FLAC3D.
Table 4 Change in tsm and f0m with shear strain Shear strain
Cohesion (tsm ) (MPa)
Friction angle (f0m ) (1)
0.000 0.005 0.010 0.050
1.1tsm 1.1tsm =5 0 0
f0m 5 f0m 7.5 f0m 10 f0m 10
It was however found that the peak values of shear strength tsm and friction angle f0m so determined had to be changed slightly to account for the fact that the strain-softening mode in FLAC3D uses the linear Mohr– Coloumb criterion while the Sheorey criterion is nonlinear. The value of tsm obtained from the Sheorey criterion was increased by 10% and that of f0m was reduced by 51 to use them as Mohr–Coloumb parameters (Fig. 1). The residual values were then taken as tsm ðresidualÞ ¼ 0; f0m ðresidualÞ ¼ f0m 2101: The values in between with the corresponding shear strains were taken as explained in Table 4.
5. Numerical modelling procedure Estimation of pillar strength has been made in a way analogous to that of laboratory estimation of uniaxial compressive strength under servo-controlled testing conditions. However, apart from differences of size between real pillars and laboratory specimens so tested, laboratory tests disregard in situ stresses as well as roof
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and floor behaviour. The method of modelling consists of the following steps: * * *
*
*
*
Grid generation; selection of appropriate material behaviour model; incorporation of material properties, in situ stresses and boundary conditions; solution to the equilibrium of initial elastic model to generate the in situ stresses in the model; development of roadway excavations in the model and application of a constant vertical velocity at the top of the model and continuous monitoring of the average vertical stress and strain in the pillar at each solution ‘step’.
Except the cases of pillars from Jitpur colliery, those given in Tables 1 and 2 represented arrays over large areas consisting of more or less similar pillars. The Jitpur pillars, on the other hand, were located in a shaft pillar area. During the initial trial runs mentioned earlier, the Jitpur cases, probably due to the great depth and poor coal strength, gave erratic failure curves. Considering that perhaps more research is required for such pillars, these cases have not been further included in this study. In the other cases, because of their arrayed nature, only a quarter of the pillar was modelled taking advantage of symmetry. One example grid is shown in Fig. 2. To optimise the memory and runtime requirements, only 30 m of cover has been modelled for the cases whose depth of cover is more than 30 m. This distance was considered to be free from the influence of roadways. The FLAC3D software models the continua by brick shaped ‘zones’. It is well known that the solution converges to the true solution as the number of zones increases or the zone size reduces in a model. For this reason, convergence studies were made for a select few cases and the zone size was optimised at a relative solution error of 5% or less. To define the ultimate load bearing capacity or the peak stress and to obtain the post-failure behaviour of the pillar, strain-softening material model has been chosen. The relevant material properties and in situ stresses employed have already been discussed. After making the roadway excavations, the top of the model was fixed in the vertical direction to maintain a constant vertical velocity. The other boundary conditions included zero vertical displacements at the model bottom and zero normal displacements at the four vertical symmetry planes. Even for a static problem, FLAC3D seeks the solution by explicitly solving the dynamic equations of motion with appropriate damping. For this reason, if a large excavation is made in a single go, transient stresses are developed in the model during the first few steps of the solution. In order to minimise the effect of these stresses
30 m
h = 4.8 m
30 m
Fig. 2. An example of a FLAC3D grid showing a quarter of a pillar used for modelling the Bankola case (case 7 of Table 2).
on the model results, the initial model and the first few steps after the roadway development were run under elastic conditions. After the effect of transient stresses was minimised, the coal seam and 10 m of roof and floor were declared as strain-softening materials. The 10 m limit was fixed after a few trial runs in which the yield zones were found to be within this limit. At this stage, the top of the model was fixed in the vertical direction and a constant velocity was applied which, after a few experimental runs, was fixed as 105 m/s. This value has been used in all the models.
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The average vertical stresses in all pillar zones and the average vertical strain in the pillar were continuously monitored and plotted. The vertical strain in the pillar was calculated as the average roof-to-floor movement (convergence) over the pillar area divided by the nominal pillar height.
6. Modelling results Figs. 3 and 4 show typical plots of the stress–strain behaviour of two pillar cases from Tables 1 and 2, respectively. It may be noted from these plots, that the starting ordinate value is not zero. Instead, it starts at a 5.0
P ¼ 0:025H
ðw þ BÞ2 w2
ðMPaÞ:
ð5Þ
Eq. (5) simply defines the tributary area pressure. Here, w and h are pillar width and height and B is the roadway width in metres. In order to examine the validity of the estimations, the model strength is plotted against the load as shown in Figs. 5 and 6 for the failed and stable cases, respectively. It may be seen that the Salamon and Munro conditions given earlier are satisfied for all cases except for one stable pillar case of Morgan Pit, which is also underestimated by the pillar strength equation (4).
Pillar strength = 4.68 MPa
4.5 4.0 3.5
Stress, MPa
value near about the cover pressure corresponding to that depth. It may further be noted that though the postfailure characteristics look reasonable, these will be somewhat different if a realistic dilation angle and its variation with strain are input. The strength values obtained from the models of the cases of Tables 1 and 2 are given in Tables 5 and 6. Also given are the pillar strength S [4] and load P estimated from the equations given below. H w 0:36 S ¼ 0:27sc h þ1 1 ðMPaÞ; ð4Þ þ 250 h
3.0 2.5 2.0
1.5 1.0
7. Discussion and conclusion
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
-3
Normal Strain (x 10 )
Fig. 3. Complete stress–strain behaviour of failed Amritnagar pillar (case 2 of Table 1). 8.0 7.5
Pillar Strength = 7.52 MPa
7.0
Stress, MPa
6.5
6.0 5.5 5.0 4.5 4.0
3.5 3.0 2.0
4.0
6.0
8.0
10.0
12.0
-3
Normal Strain (x 10 )
Fig. 4. Complete stress–strain behaviour of stable Surakacchar pillar (case 9 of Table 2).
From the studies reported, it can be concluded that numerical modelling with strain softening can be a possible, and perhaps more powerful, alternative to other pillar strength approaches. This can be seen from the results of Figs. 5 and 6 for the failed and stable cases. In order to use this tool successfully, however, it is necessary to obtain the various input parameters realistically. In the pillar cases, which are quite old, many such data are based on experience. Even so, this modelling technique has a good potential for more studies in this subject. In fact, while dealing with an important pillar design or stability problem, proper testing and characterisation should be undertaken not only for the coal seam but also for the roof and floor rocks. In situ stress measurement should also be done in such cases. It is known to practising mining engineers, that coal pillars with a soft roof and/or floor often become unstable. This numerical technique can be used to estimate the stability of such pillars. Most pillar strength formulae assume that the effects of size and width-to-height ratio (w=h) predominate in defining pillar strength. As far as the modelled pillar cases in this work are concerned, it was not found necessary to account for the influence of size on coal strength separately. This finding agrees with that of Bieniawski [11] that the effect of size becomes negligible after the size of 1.5 m.
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Table 5 Comparison of pillar strength and pillar load for failed cases Case no.
Mine (seam)
Pillar load (MPa)
Pillar strength by Eq. (4)
Model strength (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Amritnagar (Nega Jamehari) Amritnagar (Nega Jamehari) Begonia (Begonia) Amlai (Burhar) Sendra Bansjora (X) W. Chirimiri (Main) Birsingpur (Johilla top) Pure Kajora (Lower Kajora) Pure Kajora (Lower Kajora) Shankarpur (Jambad bottom) Ramnagar (Begunia) Ramnagar (Begunia) Kankanee (XIII) Kankanee (XIV)
5.01 4.68 5.80 3.61 2.77 8.12 10.45 6.02 7.43 4.20 7.76 6.17 5.88 5.83
6.85 5.93 5.07 3.49 2.59 8.15 8.11 6.23 6.08 5.46 6.43 6.48 6.98 5.03
5.57 4.68 5.67 3.57 2.45 7.59 8.52 6.52 6.64 6.05 6.88 7.27 6.46 5.20
Table 6 Comparison of pillar strength and pillar load for stable cases Case no.
Mine (seam)
Pillar load (MPa)
Pillar strength by Eq. (4)
Model strength (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Bellampalli (Ross) Nimcha (Nega) Morganpit (Salarjung) Ramnagar (Ramnagar) Lachhipur (Lower Kajora) N. Salanpur (X) Bankola (Jambad top) Bankola (Jambad top) Surakacchar (G-I) Lachhipur (Lower Kajora) Sripur (Koithee) E. Angarapatra (XII) Kargali Incline (Kathara) Jamadoba 6 and 7 Pits (XVI) Topsi (Singharan)
4.01 3.09 14.08 5.20 2.25 2.08 3.90 5.20 4.14 1.43 8.41 3.00 2.34 7.59 5.15
9.64 7.85 11.89 8.75 5.43 4.01 6.92 7.79 10.07 7.93 21.73 6.00 8.62 8.60 12.83
8.79 7.89 10.40 8.79 5.62 5.02 7.08 7.41 7.52 7.42 8.62 6.05 8.06 10.84 14.03
12
16 14
10
Model Strength, MPa
Model Strength, MPa
12 8
6
4
10 8 6 4
2 2 0
0 0
2
4
6
8
10
12
Pillar Load, MPa Fig. 5. Pillar load vs. model strength for the failed pillar cases of Table 1.
0
2
4
6
8
10
12
14
16
Pillar Load, MPa Fig. 6. Pillar load vs. model strength for the stable pillar cases of Table 2.
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The effect of width-to-height ratio on pillar strength, as we know it, has been described in the past on the basis of test results as a linear or exponential rise in strength with w=h: This concept may be in question, particularly if failure in the roof and floor is initiated before the pillar goes into the post-failure stage. Further research is being conducted by the authors in this area. Further research is also required for estimating pillar strength in deep coal seams. Acknowledgements The authors are thankful to the director, CMRI for granting permission to publish this work. The opinions expressed in the paper are of the authors and not necessarily of the institution to which they belong. References [1] Sheorey PR. Design of coal pillar arrays, chain pillars. In: Hudson JA, et al., editors. Comprehensive rock engineering, vol. 2. Oxford: Pergamon, 1993. p. 631–70.
[2] Salamon MDG, Munro AH. A study of the strength of coal pillars. J S Afr Inst Min Metall 1967;68:55–67. [3] Das MN. Influence of width/height ratio on post-failure behaviour of coal (Short Communication). Int J Min Geol Eng 1986;4:79–87. [4] Sheorey PR. Pillar strength considering in situ stresses. In: Iannacchione AT, et al., editors. Proceedings of the Workshop on Coal Pillar Mechanics and Design, USBM IC 9315, Santa Fe, 1992. p. 122–7. [5] Itasca Consulting Group Inc. FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions). Version 2.0. Minneapolis, MN, 1997. [6] Sheorey PR. A theory for in situ stresses in isotropic and transversely isotropic rock. Int J Rock Mech Min Sci Geomech Abstr 1994;31:23–34. [7] Sheorey PR, Murali Mohan G, Sinha A. Influence of elastic constants on the horizontal in situ stresses. Int J Rock Mech Min Sci Geomech Abstr, Pergamon, Oxford, UK. 2001; in press. [8] Van Krevelen DW. Coal. Amsterdam: Elsevier, 1961. p. 514. [9] Sheorey PR. Empirical rock failure criteria. Rotterdam: Balkema, 1997. p. 176. [10] Bieniawski ZT. Rock mass classifications in rock engineering. In: Bieniawski ZT, editor. Exploration for rock engineering, vol. 1. Rotterdam, Balkema, 1976. p. 97–106. [11] Bieniawski ZT. The effect of specimen size on compressive strength of coal. Int J Rock Mech Min Sci 1968;5:325–35.
Numerical estimation of pillar strength in coal mines G. Murali Mohan, P.R. Sheorey*, A. Kushwaha Central Mining Research Institute, Barwa Road, Dhanbad 826 001, Jharkhand, India Accepted 19 October 2001
Abstract Results of numerical modelling of failed and stable cases of pillars from Indian coal mines are reported in this paper. The complete procedure for modelling using FLAC3D in the strain-softening mode is given. It is shown that estimation of pillar strength is possible using numerical modelling and it may provide a viable, and perhaps better, alternative to earlier conventional pillar strength approaches. Methods are given to estimate the large number of input parameters required in the models. Because of the paucity of rock characterisation and in situ stress data, some approximations and assumptions were made in all the cases based on earlier experience. The modelling work leads to the conclusion that such data are necessary for the success of this method. Areas for further research related to this subject are given in the end. r 2002 Published by Elsevier Science Ltd.
1. Introduction Pillars in coal mines serve various purposes e.g., protection of gate roadways or entries, panel isolation to guard against spontaneous heating, protection of mine shafts and surface subsidence control. Considering the fact that the structural integrity of a coal mine largely depends on pillars, considerable research in the area of pillar strength and design has been done over the last few decades. Even so, some identifiable grey areas in defining pillar strength still remain. The large number of pillar strength approaches proposed so far has been comprehensively reviewed earlier [1]. These approaches can be classified into four types: empirical, semi-empirical, statistical and analytical. As far as the authors are aware, a generalised formulation employing numerical modelling for estimating pillar strength has not been tried as yet. This paper attempts to show that numerical modelling may perhaps be considered as a viable alternative method to define pillar strength. This approach will have the advantage of duly accounting for the in situ stresses as well as roof and floor strata properties. The earlier strength formulae clearly do not possess this advantage. In addition, they disregard the fact that roof and floor rocks may themselves be in a state of failure at the time of, or even before, pillar failure. *Corresponding author. Tel.:/fax: +91-326-204-339. E-mail address: [email protected] (P.R. Sheorey). 1365-1609/01/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 1 3 6 5 - 1 6 0 9 ( 0 1 ) 0 0 0 7 1 - 5
Because of the large number of parameters involved in such modelling, as seen later, it was thought best to model actual cases of failed and stable pillars to make the procedure for estimating these parameters reasonably realistic.
2. Failed and stable pillar cases Any method for estimating pillar strength should satisfy the following two statistical conditions [2]: (a) All failed pillar cases should have a safety factor of 1.0. In statistical terms, this merely means that the line of safety factor=1.0 should be the best fit for a plot of pillar strength vs. pillar load. (b) All stable cases must have a safety factor>1.0. Both conditions implicitly refer to the point of strength failure or the peak stress sustainable by a pillar. It may, however, be necessary in the case of very squat pillars to consider their post-failure resistance, if the stress–strain characteristic rises after failure. Such a rise in the post-failure curve is seen to exist in laboratory squat specimens of coal [3]. Details of the failed and stable cases of pillars [4] from Indian coalfields are presented in Tables 1 and 2. All the cases except those from Jitpur are of pillars approximately square in plan.
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Nomenclature sh sv E n b G H s1 ; s3 sc st b
scm ; stm ; bm mean horizontal in situ stress vertical in situ stress elastic modulus Poisson’s ratio coefficient of linear thermal expansion geothermal gradient depth of cover major and minor principal stresses intact rock compressive strength intact rock tensile strength exponent in the failure criterion
3. In situ stresses The Fast Lagrangian Analysis of Continua (FLAC) softwares developed by ITASCA Consultants of USA [5], provide for elasto-plastic analysis of rock excavations with strain softening using the linear Mohr– Coloumb failure criterion. The three-dimensional version of these softwares was employed for modelling the cases of Tables 1 and 2. The estimation of the different parameters to be input besides in situ stresses and the modelling procedure are described in subsequent sections. A few preliminary model runs of some of the cases indicated that the pillar strength was sensitive to the in situ horizontal stress. It was therefore necessary to try
RMR tsm m0m f0m S P w h B
rock mass strength constants corresponding to the intact rock constants above Bieniawski’s rock mass rating (1976) rock mass shear strength rock mass coefficient of internal friction rock mass angle of internal friction pillar strength pillar load pillar width pillar height roadway width
and estimate the horizontal stresses within the coal seam as realistically as possible. A theory for in situ stresses [6] shows that the mean in situ horizontal stress (mean of the major and minor horizontal stresses) depends on the elastic constants (E; n), the coefficient of thermal expansion (b) and the geothermal gradient (G). This theory gives the mean horizontal stress as sh ¼
n bEG sv þ ðH þ 1000Þ: 1n 1n
ð1Þ
In a recent work [7], this equation is shown to fit stress measurement data from different parts of the world quite well, if they are not unduly affected by surface topography or major geological features. It may thus be
Table 1 Failed pillar case data Case no.
Mine (seam)
H (m)
h (m)
W (m)
B (m)
w=h
sc (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Amritnagar (Nega Jamehari) Amritnagar (Nega Jamehari) Begonia (Begonia) Amlai (Burhar) Sendra Bansjora (X) W. Chirimiri (Main) Birsingpur (Johilla top) Pure Kajora (Lower Kajora) Pure Kajora (Lower Kajora) Shankarpur (Jambad bottom) Ramnagar (Begunia) Ramnagar (Begunia) Kankanee (XIII) Kankanee (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV)
30 30 36 30 23 90 129 54 56 42 70 51 160 140 450
4.5 6.0 3.0 5.4 8.1 3.75 3.6 3.6 3.6 4.8 1.8 1.8 6.6 8.4 3.6
3.6 3.6 3.9 4.5 4.65 5.4 7.5 5.4 4.95 4.5 2.85 3.0 19.8 18.6 18 34.5 10.5 12 12 21 18 21 18 25.5 15 63 18 30 18 30 15 33
5.7 5.4 6.0 4.5 5.55 6.0 6.0 6.0 6.45 4.5 3.15 3.6 4.2 5.4 6.0
0.8 0.6 1.3 0.83 0.57 1.44 2.08 1.5 1.38 0.94 1.58 1.67 3.0 2.2
45 45 26 25 24 45 38 33 33 47 26 26 27 25 19
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Table 2 Data for stable pillar cases Case no.
Mine (seam)
H (m)
h (m)
w (m)
B (m)
w=h
sc (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Bellampalli (Ross) Nimcha (Nega) Morganpit (Salarjung) Ramnagar (Ramnagar) Lachhipur (Lower Kajora) N. Salanpur (X) Bankola (Jambad top) Bankola (Jambad top) Surakacchar (G-I) Lachhipur (Lower Kajora) Sripur (Koithee) E. Angarapatra (XII) Kargali Incline (Kathara) Jamadoba 6 and 7 Pits (XVI) Topsi (Singharan) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV) Jitpur (XIV)
36 48 270 75 38 30 102 75 106 38 266 30 36 80 85 450
3.0 6.0 3.0 2.7 5.1 5.1 4.8 3.0 3.5 5.1 4.8 2.1 3.6 2.0 1.8 3.6
5.4 9.9 8.1 9.9 7.2 9.0 10.1 6.3 16.0 18.3 40.0 6.0 9.3 5.8 7.0 21 39 18 48 19.5 30 18 31.5 18 42
6.0 6.0 3.6 6.6 3.9 6.0 2.4 4.2 4.0 4.2 5.0 6.0 5.7 5.5 3.9 6.0
1.8 1.7 2.7 3.7 1.4 1.8 2.1 2.1 4.6 3.6 8.3 2.9 2.6 2.9 3.9
48 50 46 28 33 21 35 35 29 33 43 19 40 29 41 19
seen that in a sedimentary formation, consisting of beds of different properties, it will not be proper to assign a common equation for horizontal stress variation with depth. This also implies that the stress values obtained in coal measure rocks will not represent the in-seam values. The vertical stress can be taken for coal measure rocks as sv ¼ 0:025H
ðMPaÞ:
ð2Þ
Then putting Eq. (2) and the following values in Eq. (1): n ¼ 0:25;
b ¼ 3 105 =1C;
E ¼ 2000 MPa;
G ¼ 0:031C=m;
stresses used in the roof, seam and floor were different, the pillar strength remained virtually the same. Therefore, Eq. (3) was adopted for the roof and floor rocks as well along with Eq. (2). It may be mentioned that many of the cases of Tables 1 and 2 are quite old and it was not possible to determine the elastic modulus of the coals concerned, though this was desirable. Similarly, rock classification studies, which are required for estimating some of the input parameters described in the next section, could not be undertaken for the same reason. However, the uniaxial compressive strength of coal was fortunately determined at the time of collecting the details of these cases in the 1960s and 1970s. These values are included in the tables.
we obtain the mean horizontal stress as sh ¼ 2:4 þ 0:01H
ðMPaÞ:
ð3Þ
In Eqs. (2) and (3), the depth of cover H is to be taken in metres. The value of the coefficient of thermal expansion b for coal could be obtained from only one reference [8]. The thermal gradient G is given for Indian coal measures. Important deviations from Eq. (3) can exist since the elastic modulus of coal can vary from about 800 to 5000 MPa. In fact, the value E ¼ 2000 MPa was chosen as a rough average only after trial model runs of a few cases from Table 1. Thus, Eqs. (2) and (3) were adopted for all the cases. Application of Eq. (3) indicates that the two horizontal in situ stresses are equal. However, significant differences in pillar strength can exist in the presence of high stress anisotropy. A brief investigation by numerical modelling showed that if the in situ horizontal
4. Estimation of input parameters The various strength and elastic constants necessary for numerical modelling using FLAC3D in the strainsoftening mode are given below: * *
*
*
Elastic constants; peak and residual shear strength and the variation in between with the shear strain; peak and residual angle of internal friction and the variation with the shear strain and angle of dilation.
All the above properties are to be specified for both the coal seam and the roof and floor rocks. In the absence of test data, the elastic constants were simply taken as given in Table 3. Since published data on the dilation angle of coal are non-existent it was simply
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Table 3 Elastic and dilation constants used in the models
Coal Rock
τ
E (GPa)
n
Dilation angle
2 7
0.25 0.25
0 0
Sheorey criterion Φ0m - 5
o
Mohr-Coloumb criterion
assumed to be zero, though undoubtedly it will have a bearing on the post-failure behaviour of the pillar, but perhaps not on the peak strength. The compressive strength of the roof and floor is assumed to be the same as that of the coal seam for cases where the strength of coal is >30 MPa. When it is less, it is taken as 30 MPa for the roof and floor rocks. This was necessary as the roof and floor failure zone was unrealistically high and yielded unacceptable pillar strength values when the compressive strength was o30 MPa. The shear strength and friction angle were estimated as rock mass parameters using a failure criterion for rock masses [9]. This criterion uses the 1976 rock mass rating (RMR) of Bieniawski [10] for reducing the laboratory strength parameters to give the corresponding rock mass values. For this purpose, an Indian average value of RMR ¼ 55 was taken, since the individual RMR values were not known. This criterion is defined as s3 bm s1 ¼ scm 1 þ ; stm where s1 and s3 are major and minor principal stresses at failure and the rock mass strength parameters are defined by RMR 100 scm ¼ sc exp ; 20 RMR 100 ; stm ¼ st exp 27 bm ¼ bRMR=100 : In the above equations, the subscript m stands for the rock mass. For estimating these parameters, only the value of the compressive strength sc was known (Tables 1 and 2). Then the tensile strength st ¼ sc =15 and b ¼ 0:5 were taken as the most representative values, as seen from a large number of test data published earlier [9]. From these, the rock mass shear strength tsm ; the coefficient and the angle of internal friction m0m and f0m are obtained as 1=2 bbmm ; tsm ¼ scm stm ð1 þ bm Þ1þbm m0m ¼
t2sm ð1 þ bm Þ2 s2tm ; 2tsm stm ð1 þ bm Þ
f0m ¼ tan1 ðm0m Þ:
Φ0m 1.1 τsm
τsm
σtm 0
σ
Fig. 1. Schematic showing the non-linear Sheorey criterion as against the linear Mohr–Coloumb criterion adopted in FLAC3D.
Table 4 Change in tsm and f0m with shear strain Shear strain
Cohesion (tsm ) (MPa)
Friction angle (f0m ) (1)
0.000 0.005 0.010 0.050
1.1tsm 1.1tsm =5 0 0
f0m 5 f0m 7.5 f0m 10 f0m 10
It was however found that the peak values of shear strength tsm and friction angle f0m so determined had to be changed slightly to account for the fact that the strain-softening mode in FLAC3D uses the linear Mohr– Coloumb criterion while the Sheorey criterion is nonlinear. The value of tsm obtained from the Sheorey criterion was increased by 10% and that of f0m was reduced by 51 to use them as Mohr–Coloumb parameters (Fig. 1). The residual values were then taken as tsm ðresidualÞ ¼ 0; f0m ðresidualÞ ¼ f0m 2101: The values in between with the corresponding shear strains were taken as explained in Table 4.
5. Numerical modelling procedure Estimation of pillar strength has been made in a way analogous to that of laboratory estimation of uniaxial compressive strength under servo-controlled testing conditions. However, apart from differences of size between real pillars and laboratory specimens so tested, laboratory tests disregard in situ stresses as well as roof
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and floor behaviour. The method of modelling consists of the following steps: * * *
*
*
*
Grid generation; selection of appropriate material behaviour model; incorporation of material properties, in situ stresses and boundary conditions; solution to the equilibrium of initial elastic model to generate the in situ stresses in the model; development of roadway excavations in the model and application of a constant vertical velocity at the top of the model and continuous monitoring of the average vertical stress and strain in the pillar at each solution ‘step’.
Except the cases of pillars from Jitpur colliery, those given in Tables 1 and 2 represented arrays over large areas consisting of more or less similar pillars. The Jitpur pillars, on the other hand, were located in a shaft pillar area. During the initial trial runs mentioned earlier, the Jitpur cases, probably due to the great depth and poor coal strength, gave erratic failure curves. Considering that perhaps more research is required for such pillars, these cases have not been further included in this study. In the other cases, because of their arrayed nature, only a quarter of the pillar was modelled taking advantage of symmetry. One example grid is shown in Fig. 2. To optimise the memory and runtime requirements, only 30 m of cover has been modelled for the cases whose depth of cover is more than 30 m. This distance was considered to be free from the influence of roadways. The FLAC3D software models the continua by brick shaped ‘zones’. It is well known that the solution converges to the true solution as the number of zones increases or the zone size reduces in a model. For this reason, convergence studies were made for a select few cases and the zone size was optimised at a relative solution error of 5% or less. To define the ultimate load bearing capacity or the peak stress and to obtain the post-failure behaviour of the pillar, strain-softening material model has been chosen. The relevant material properties and in situ stresses employed have already been discussed. After making the roadway excavations, the top of the model was fixed in the vertical direction to maintain a constant vertical velocity. The other boundary conditions included zero vertical displacements at the model bottom and zero normal displacements at the four vertical symmetry planes. Even for a static problem, FLAC3D seeks the solution by explicitly solving the dynamic equations of motion with appropriate damping. For this reason, if a large excavation is made in a single go, transient stresses are developed in the model during the first few steps of the solution. In order to minimise the effect of these stresses
30 m
h = 4.8 m
30 m
Fig. 2. An example of a FLAC3D grid showing a quarter of a pillar used for modelling the Bankola case (case 7 of Table 2).
on the model results, the initial model and the first few steps after the roadway development were run under elastic conditions. After the effect of transient stresses was minimised, the coal seam and 10 m of roof and floor were declared as strain-softening materials. The 10 m limit was fixed after a few trial runs in which the yield zones were found to be within this limit. At this stage, the top of the model was fixed in the vertical direction and a constant velocity was applied which, after a few experimental runs, was fixed as 105 m/s. This value has been used in all the models.
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The average vertical stresses in all pillar zones and the average vertical strain in the pillar were continuously monitored and plotted. The vertical strain in the pillar was calculated as the average roof-to-floor movement (convergence) over the pillar area divided by the nominal pillar height.
6. Modelling results Figs. 3 and 4 show typical plots of the stress–strain behaviour of two pillar cases from Tables 1 and 2, respectively. It may be noted from these plots, that the starting ordinate value is not zero. Instead, it starts at a 5.0
P ¼ 0:025H
ðw þ BÞ2 w2
ðMPaÞ:
ð5Þ
Eq. (5) simply defines the tributary area pressure. Here, w and h are pillar width and height and B is the roadway width in metres. In order to examine the validity of the estimations, the model strength is plotted against the load as shown in Figs. 5 and 6 for the failed and stable cases, respectively. It may be seen that the Salamon and Munro conditions given earlier are satisfied for all cases except for one stable pillar case of Morgan Pit, which is also underestimated by the pillar strength equation (4).
Pillar strength = 4.68 MPa
4.5 4.0 3.5
Stress, MPa
value near about the cover pressure corresponding to that depth. It may further be noted that though the postfailure characteristics look reasonable, these will be somewhat different if a realistic dilation angle and its variation with strain are input. The strength values obtained from the models of the cases of Tables 1 and 2 are given in Tables 5 and 6. Also given are the pillar strength S [4] and load P estimated from the equations given below. H w 0:36 S ¼ 0:27sc h þ1 1 ðMPaÞ; ð4Þ þ 250 h
3.0 2.5 2.0
1.5 1.0
7. Discussion and conclusion
0.5
1.0
2.0
3.0
4.0
5.0
6.0
7.0
-3
Normal Strain (x 10 )
Fig. 3. Complete stress–strain behaviour of failed Amritnagar pillar (case 2 of Table 1). 8.0 7.5
Pillar Strength = 7.52 MPa
7.0
Stress, MPa
6.5
6.0 5.5 5.0 4.5 4.0
3.5 3.0 2.0
4.0
6.0
8.0
10.0
12.0
-3
Normal Strain (x 10 )
Fig. 4. Complete stress–strain behaviour of stable Surakacchar pillar (case 9 of Table 2).
From the studies reported, it can be concluded that numerical modelling with strain softening can be a possible, and perhaps more powerful, alternative to other pillar strength approaches. This can be seen from the results of Figs. 5 and 6 for the failed and stable cases. In order to use this tool successfully, however, it is necessary to obtain the various input parameters realistically. In the pillar cases, which are quite old, many such data are based on experience. Even so, this modelling technique has a good potential for more studies in this subject. In fact, while dealing with an important pillar design or stability problem, proper testing and characterisation should be undertaken not only for the coal seam but also for the roof and floor rocks. In situ stress measurement should also be done in such cases. It is known to practising mining engineers, that coal pillars with a soft roof and/or floor often become unstable. This numerical technique can be used to estimate the stability of such pillars. Most pillar strength formulae assume that the effects of size and width-to-height ratio (w=h) predominate in defining pillar strength. As far as the modelled pillar cases in this work are concerned, it was not found necessary to account for the influence of size on coal strength separately. This finding agrees with that of Bieniawski [11] that the effect of size becomes negligible after the size of 1.5 m.
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Table 5 Comparison of pillar strength and pillar load for failed cases Case no.
Mine (seam)
Pillar load (MPa)
Pillar strength by Eq. (4)
Model strength (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Amritnagar (Nega Jamehari) Amritnagar (Nega Jamehari) Begonia (Begonia) Amlai (Burhar) Sendra Bansjora (X) W. Chirimiri (Main) Birsingpur (Johilla top) Pure Kajora (Lower Kajora) Pure Kajora (Lower Kajora) Shankarpur (Jambad bottom) Ramnagar (Begunia) Ramnagar (Begunia) Kankanee (XIII) Kankanee (XIV)
5.01 4.68 5.80 3.61 2.77 8.12 10.45 6.02 7.43 4.20 7.76 6.17 5.88 5.83
6.85 5.93 5.07 3.49 2.59 8.15 8.11 6.23 6.08 5.46 6.43 6.48 6.98 5.03
5.57 4.68 5.67 3.57 2.45 7.59 8.52 6.52 6.64 6.05 6.88 7.27 6.46 5.20
Table 6 Comparison of pillar strength and pillar load for stable cases Case no.
Mine (seam)
Pillar load (MPa)
Pillar strength by Eq. (4)
Model strength (MPa)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Bellampalli (Ross) Nimcha (Nega) Morganpit (Salarjung) Ramnagar (Ramnagar) Lachhipur (Lower Kajora) N. Salanpur (X) Bankola (Jambad top) Bankola (Jambad top) Surakacchar (G-I) Lachhipur (Lower Kajora) Sripur (Koithee) E. Angarapatra (XII) Kargali Incline (Kathara) Jamadoba 6 and 7 Pits (XVI) Topsi (Singharan)
4.01 3.09 14.08 5.20 2.25 2.08 3.90 5.20 4.14 1.43 8.41 3.00 2.34 7.59 5.15
9.64 7.85 11.89 8.75 5.43 4.01 6.92 7.79 10.07 7.93 21.73 6.00 8.62 8.60 12.83
8.79 7.89 10.40 8.79 5.62 5.02 7.08 7.41 7.52 7.42 8.62 6.05 8.06 10.84 14.03
12
16 14
10
Model Strength, MPa
Model Strength, MPa
12 8
6
4
10 8 6 4
2 2 0
0 0
2
4
6
8
10
12
Pillar Load, MPa Fig. 5. Pillar load vs. model strength for the failed pillar cases of Table 1.
0
2
4
6
8
10
12
14
16
Pillar Load, MPa Fig. 6. Pillar load vs. model strength for the stable pillar cases of Table 2.
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The effect of width-to-height ratio on pillar strength, as we know it, has been described in the past on the basis of test results as a linear or exponential rise in strength with w=h: This concept may be in question, particularly if failure in the roof and floor is initiated before the pillar goes into the post-failure stage. Further research is being conducted by the authors in this area. Further research is also required for estimating pillar strength in deep coal seams. Acknowledgements The authors are thankful to the director, CMRI for granting permission to publish this work. The opinions expressed in the paper are of the authors and not necessarily of the institution to which they belong. References [1] Sheorey PR. Design of coal pillar arrays, chain pillars. In: Hudson JA, et al., editors. Comprehensive rock engineering, vol. 2. Oxford: Pergamon, 1993. p. 631–70.
[2] Salamon MDG, Munro AH. A study of the strength of coal pillars. J S Afr Inst Min Metall 1967;68:55–67. [3] Das MN. Influence of width/height ratio on post-failure behaviour of coal (Short Communication). Int J Min Geol Eng 1986;4:79–87. [4] Sheorey PR. Pillar strength considering in situ stresses. In: Iannacchione AT, et al., editors. Proceedings of the Workshop on Coal Pillar Mechanics and Design, USBM IC 9315, Santa Fe, 1992. p. 122–7. [5] Itasca Consulting Group Inc. FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions). Version 2.0. Minneapolis, MN, 1997. [6] Sheorey PR. A theory for in situ stresses in isotropic and transversely isotropic rock. Int J Rock Mech Min Sci Geomech Abstr 1994;31:23–34. [7] Sheorey PR, Murali Mohan G, Sinha A. Influence of elastic constants on the horizontal in situ stresses. Int J Rock Mech Min Sci Geomech Abstr, Pergamon, Oxford, UK. 2001; in press. [8] Van Krevelen DW. Coal. Amsterdam: Elsevier, 1961. p. 514. [9] Sheorey PR. Empirical rock failure criteria. Rotterdam: Balkema, 1997. p. 176. [10] Bieniawski ZT. Rock mass classifications in rock engineering. In: Bieniawski ZT, editor. Exploration for rock engineering, vol. 1. Rotterdam, Balkema, 1976. p. 97–106. [11] Bieniawski ZT. The effect of specimen size on compressive strength of coal. Int J Rock Mech Min Sci 1968;5:325–35.