Rock Mechanics and Rock Engineering

, Volume 52, Issue 10, pp 3691–3717 | Cite as

Assessment of the Strength of Inclined Coal Pillars through Numerical Modelling based on the Ubiquitous Joint Model

  • Arka Jyoti Das
  • Prabhat Kumar MandalEmail author
  • Partha Sarathi Paul
  • Rabindra Kumar Sinha
  • Subhashish Tewari
Original Paper


The inclined coal pillars, formed during the underground extraction of the inclined coal seam, are different from the flat coal pillars due to the high strength anisotropy of the inclined bedded rock, and the asymmetrical stress distribution. To ease the manoeuvring of the men and machinery in the inclined coal mine, the pillars are developed along the apparent dip. Thus, the pillars become rhombus-shaped with acute-angled corners which crush rapidly due to the high stress concentration. No suitable formula is found to estimate the strength of the inclined coal pillar that incorporates all these factors. Thus, the use of the available coal pillar strength formulae may endanger the extraction of the inclined coal seams. This study elucidates the procedures to estimate the strength of the inclined coal pillars by the numerical modelling technique. The ubiquitous joint model is used to simulate the shearing characteristics of the inclined strata. The parametric study shows that the strength of the inclined coal pillars decreases with the increase of the dip of the coal seams. It is also obtained that the strength of the pillars decreases with the decrease of the value of the acute angle of the corners. The peak stress distribution and the strain accumulation over the inclined pillars at the time of failure are plotted in three-dimensional graphs which show the asymmetrical characteristics of the inclined coal pillars. The analysis of variance shows that the dip of the coal seams and the acute angles of the corners have a statistically significant effect on the strength of the inclined coal pillars. Based on the results of the simulation, the best fit relation is established by the multivariate non-linear regression technique to estimate the strength of the inclined coal pillars. The coefficient of determination (R2) and the root mean square error of the model are obtained as 0.92 and 0.065, respectively. The validation of the developed model has been carried out by the stable and failed inclined pillar cases of different underground inclined coal mines in India. Comparisons of the safety factors, obtained from the developed model and the flat pillar strength formula, indicate that the flat pillar strength formula overestimates the strength of the inclined coal pillars. The developed model can be used to design the inclined coal pillars for safe extraction of inclined coal seams.


Ubiquitous joint model Apparent dip Rhombus-shaped pillar Inclined coal pillar strength Multivariate non-linear regression 

List of Symbols


Total increment of the strain


Elastic part of strain increment


Plastic part of strain increment


Stress increment


Elastic constitutive matrix

\(D_{ijkl}^{\text{ep}}\)\(D_{{i^{\prime } j^{\prime } k^{\prime } l^{\prime } }}^{\text{ep}}\)

Elasto-plastic constitutive matrix of rock matrix and weak planes


Function of the failure envelope


Plastic potential function


Plastic multiplier


Major principal stress


Minor principal stress

c, cw

Cohesion of the rock matrix and weak plane

ϕ, ϕw

Internal friction angle of the rock matrix and friction angle of the weak plane


Tension cut-off of Mohr–Coulomb criterion for the rock matrix


Dilation angle of the rock matrix

\(\partial \bar{\varepsilon }^{\text{p}}\)

Accumulated equivalent plastic strain

t and (t − Δt)

Current and previous calculation step


Transformation matrix


Tri-axial strength of the rock mass


Confining stress of the rock mass

σci, σcm

Compressive strength of the intact rock and rock mass

σti, σtm

Tensile strength of the intact rock and rock mass


Shear strength of rock mass


Cohesion of the rock mass


Hardening or softening parameter

μ0m, ϕ0m

Coefficient of friction and internal friction angle of the rock mass


Depth of cover


Vertical in situ stress


Major horizontal in situ stress


Minor horizontal in situ stress


Coefficient of thermal expansion


Geothermal gradient

Sflat, Sinclined

Strength of the flat and inclined coal pillar


Intact strength of the cubic coal of side 25 cm


Height of the coal pillar


Width of the pillar


Null hypothesis


Alternative hypothesis


Mean pillar strength derived from the Sheorey’s formula


Mean pillar strength obtained from the numerical modelling


Strength ratio between inclined and flat coal pillar


Dip of the coal seam


Acute angle of the corners of the coal pillar


Error function

σn, τ, σr

Normal, shear, and resultant stress on the inclined pillar


Unit weight of the overlying rock strata


Length of the pillar


Width of the gallery


Ratio of the major horizontal and the vertical in situ stresses



The authors are obliged to the Directors, CSIR-Central Institute of Mining and Fuel Research, Dhanbad and Indian Institute of Technology (ISM), Dhanbad for their kind encouragement and support to publish this paper. The authors are thankful to the different colliery management for the co-operation and help provided at the different stages of this study. The views expressed in this paper are that of the authors and not necessarily of the organisations to which they belong.


  1. Agapito JFT, Hardy MP (1982) Induced horizontal stress method for pillar design in oil shale. In: Proceedings of the 15th oil shale symposium, Denver, Colorado, USAGoogle Scholar
  2. Bieniawski ZT (1968) The effect of specimen size on the strength of coal. Int J Rock Mech Min Sci 5:325–335CrossRefGoogle Scholar
  3. Bieniawski ZT (1976) Rock mass classifications in rock engineering. In: Bieniawski ZT (ed) Exploration for rock engineering. Balkema, Rotterdam, pp 97–106Google Scholar
  4. Bieniawski ZT (1982) Improved design of room and pillar coal mines for us conditions. In: Brawner CO (ed) Proceedings of the first international conference on stability in underground mining. Vancouver, AIME, New York, pp 19–51Google Scholar
  5. Chen CS, Pan E, Amadei B (1998) Determination of deformability and tensile strength of anisotropic rock using Brazilian tests. Int J Rock Mech Min Sci Geomech Abstr 35:43–61CrossRefGoogle Scholar
  6. Ching J, Yang ZY, Shiau JQ, Chen CJ (2013) Estimation of rock pressure during an excavation/cut in sedimentary rocks with inclined bedding planes. Struct Saf 41:11–19CrossRefGoogle Scholar
  7. Clausen J, Damkilde L, Andersen L (2006) Efficient return algorithms for associated plasticity with multiple yield planes. Int J Numer Methods Eng 66:1036–1059CrossRefGoogle Scholar
  8. Das AJ, Mandal PK, Bhattacharjee R, Tiwari S, Kushwaha A, Roy LB (2017a) Evaluation of stability of underground workings for exploitation of an inclined coal seam by the ubiquitous joint model. Int J Rock Mech Min Sci 93:101–114CrossRefGoogle Scholar
  9. Das AJ, Mandal PK, Paul PS, Sinha RK, Kushwaha A, Tewari S (2017b) Effect of the strata inclination during underground extraction of the inclined coal seams. In: Proceedings of the 7th Asian mining congress, 8–10 November, Kolkata, India, pp 223–238Google Scholar
  10. Das AJ, Mandal PK, Paul PS, Sinha RK (2018a) Strategies for underground extraction of the inclined coal seams by continuous miner. MGMI Trans 114:1–12Google Scholar
  11. Das AJ, Mandal PK, Paul PS, Sinha RK, Tewari S (2018a) Rock mechanics considerations for mechanised extraction of an inclined coal seam. In: Proceedings of the conference on recent challenges in mining industry (RCMI-2018), 28th April, 2018, CSIR-CIMFR, Dhanbd, pp 121–130Google Scholar
  12. Davis RO, Selvadurai AP (2005) Plasticity and geomechanics. Cambridge University Press, CambridgeGoogle Scholar
  13. Deb D (2010) Finite element methods: concepts and applications in geomechanics. PHI Learning Private Limited, New DelhiGoogle Scholar
  14. Elmo D, Stead D (2010) An integrated numerical modelling-discrete fracture network approach applied to the characterisation of rock mass strength of naturally fractured pillars. Rock Mech Rock Eng 43:3–19CrossRefGoogle Scholar
  15. Esterhuizen E, Mark C, Murphy MM (2010) Numerical model calibration for simulating coal pillars, gob and overburden response. In: Proceeding of the 29th international conference on ground control in mining, Morgantown, WV, pp 46–57Google Scholar
  16. Garza-Cruz T, Pierce M, Board M (2018) Effect of shear stresses on pillar stability—a back-analysis of the troy mine experience to forward predict pillar performance at Montanore. In: Proceeding of the 52nd US rock mechanics/geomechanics symposium. American Rock Mechanics AssociationGoogle Scholar
  17. Ghasemi E, Shahriar K (2012) A new coal pillars design method in order to enhance safety of the retreat mining in room and pillar mines. Saf Sci 50:579–585CrossRefGoogle Scholar
  18. Greenwald HP, Howarth HC, Hartmann I (1941) Experiments on strength of small pillars of coal in the pittsburgh bed. US Bureau of Mines, R.I P. 3575Google Scholar
  19. Hartman HL (1987) Introductory mining engineering. Wiley, New YorkGoogle Scholar
  20. He M (2011) Physical modeling of an underground roadway excavation in geologically 45° inclined rock using infrared thermography. Eng Geol 121:165–176CrossRefGoogle Scholar
  21. Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34:1165–1186CrossRefGoogle Scholar
  22. Holland CT (1962) Design of pillars for overburden support—Part 1. Min Congr J 48:24–28 (and Part 2, 48:66–71) Google Scholar
  23. Hustrulid WA (1982) Underground mining methods handbook. SME-AIME, New YorkGoogle Scholar
  24. Itasca (2017) Itasca Consulting Group, Inc., FLAC3D (fast Lagrangian analysis of continua in 3 dimensions). Version 5.0. Minneapolis, MNGoogle Scholar
  25. Jaiswal A, Shrivastva BK (2009) Numerical simulation of coal pillar strength. Int J Rock Mech Min Sci 46:779–788CrossRefGoogle Scholar
  26. Kazakidis VN, Diederichs M (1993) Understanding jointed rock mass behaviour using a ubiquitous joint approach. Int J Rock Mech Min Sci Geomech Abstr 30:163–172CrossRefGoogle Scholar
  27. Kushwaha A, Banerjee G (2005) Exploitation of developed coal mine pillars by shortwall mining—a case example. Int J Rock Mech Min Sci 42:127–136CrossRefGoogle Scholar
  28. Kushwaha A, Singh SK, Tewari S, Sinha A (2010) Empirical approach for designing of support system in mechanized coal pillar mining. Int J Rock Mech Min Sci 47:1063–1078CrossRefGoogle Scholar
  29. Lorig LJ, Cabrera A (2013) Pillar strength estimates for foliated and inclined pillars in schistose material. In: Proceedings of the 3rd international FLAC/DEM symposium. Hangzhou, China, vol 7, pp 1–7Google Scholar
  30. Mandal PK, Singh R, Maiti J, Singh AK, Kumar R, Sinha A (2008) Underpinning-based simultaneous extraction of contiguous sections of a thick coal seam under weak and laminated parting. Int J Rock Mech Min Sci 45:11–28CrossRefGoogle Scholar
  31. Mandal PK, Das AJ, Kumar N, Bhattacharjee R, Tewari S, Kushwaha A (2018) Assessment of roof convergence during driving roadways in underground coal mines by continuous miner. Int J Rock Mech Min Sci 108:169–178CrossRefGoogle Scholar
  32. McLamore R, Gray KE (1967) The mechanical behavior of anisotropic sedimentary rocks. J Eng Ind 89:62–73CrossRefGoogle Scholar
  33. Mohan GM, Sheorey PR, Kushwaha A (2001) Numerical estimation of pillar strength in coal mines. Int J Rock Mech Min Sci 38:1185–1192CrossRefGoogle Scholar
  34. Obert L, Duvall WI (1967) Rock mechanics and the design of structures in rock. Wiley, New YorkGoogle Scholar
  35. Pariseau WG (1982) Shear stability of mine pillars in dipping seams. In: Proceedings of the 23rd US symposium on rock mechanics (USRMS), American Rock Mechanics AssociationGoogle Scholar
  36. Plesha ME (1987) Constitutive models for rock discontinuities with dilatancy and surface degradation. Int J Numer Anal Methods 11:345–362CrossRefGoogle Scholar
  37. Ran JQ, Passaris EKS, Mottahed P (1994) Shear sliding failure of the jointed roof in laminated rock mass. Rock Mech Rock Eng 27:235–251CrossRefGoogle Scholar
  38. Salamon MDG, Munro AH (1967) A study of the strength of coal pillars. J S Afr Inst Min Metall 68:55–67Google Scholar
  39. Sheorey PR (1992) Pillar strength considering in situ stresses. In: Iannacchione AT et al (eds) In: Proceedings of the workshop on coal pillar mechanics and design, USBM IC 9315, Santa Fe, pp 122–127Google Scholar
  40. Sheorey PR (1994) A theory for in situ stresses in isotropic and transversely isotropic rock. Int J Rock Mech Min Sci 31:23–34CrossRefGoogle Scholar
  41. Sheorey PR (1997) Empirical rock failure criteria. Balkema, RotterdamGoogle Scholar
  42. Sheorey PR, Das MN, Bordia SK, Singh B (1986) Pillar strength approaches based on a new failure criterion for coal seams. Int J Min Geol Eng 4:273–290CrossRefGoogle Scholar
  43. Sheorey PR, Das MN, Barat D, Parasad RK, Singh B (1987) Coal pillar strength estimation from failed and stable cases. Int J Rock Mech Min Sci 24:347–355CrossRefGoogle Scholar
  44. Sheorey PR, Mohan MG, Sinha A (2001) Influence of elastic constants on the horizontal in situ stresses. Int J Rock Mech Min Sci 38:1211–1216CrossRefGoogle Scholar
  45. Singh M, Rao KS, Ramamurthy T (2002) Strength and deformational behaviour of a jointed rock mass. Rock Mech Rock Eng 35:46–64CrossRefGoogle Scholar
  46. Singh AK, Singh R, Maiti J, Kumar R, Mandal PK (2011) Assessment of mining induced stress development over coal pillars during depillaring. Int J Rock Mech Min Sci 48:805–818CrossRefGoogle Scholar
  47. Steart FA (1954) Strength and stability of pillars in coal mines. J Chem Metall Min Soc S Afr 54:309–325Google Scholar
  48. Wang TT, Huang TH (2009) A constitutive model for the deformation of a rock mass containing sets of ubiquitous joints. Int J Rock Mech Min Sci 46:521–530CrossRefGoogle Scholar
  49. Wang TT, Huang TH (2013) Anisotropic deformation of a circular tunnel excavated in a rock mass containing sets of ubiquitous joints: theory analysis and numerical modeling. Rock Mech Rock Eng 47:643–657CrossRefGoogle Scholar
  50. Wilson AH (1972) A hypothesis concerning pillar stability. Min Eng 131:409–417Google Scholar
  51. Zhou YY, Feng XT, Xu DP, Fan QX (2017) An enhanced equivalent continuum model for layered rock mass incorporating bedding structure and stress dependence. Int J Rock Mech Min Sci 97:75–98CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CSIR-Central Institute of Mining and Fuel ResearchDhanbadIndia
  2. 2.Indian Institute of Technology (Indian School of Mines)DhanbadIndia

Personalised recommendations