# Generalised Analytical Models for the Strength of the Inclined as well as the Flat Coal Pillars using Rock Mass Failure Criterion

## Abstract

The assessment of the strength of the coal pillars is essential for the safe extraction of the coal seam. All the pillar strength formulae used worldwide are developed for the flat coal pillars. Therefore, their adoption in evaluating the strength of the inclined coal pillars may endanger the workings of the inclined coal seams. The strength of an inclined coal pillar should be estimated by considering the inclination of the coal seam and its associated behaviour, because the shearing effect along the true dip aggravates the instability of the inclined coal pillars. In this paper, generalised analytical solutions have been developed to estimate the strength of the coal pillars which can be applied for both the inclined and flat coal pillars. The mathematical models are derived to obtain the confining stress in the coal pillar and the corresponding peak stress at the time of its failure using a rock mass failure criterion. The mathematical expressions are also developed for the stress distribution over the pillar considering the increase of the shearing effect with the dip of the coal seam. The Mohr–Coulomb criterion is considered for the shear characteristics at the interfaces of pillar–floor and pillar–roof. The asymmetrical stress distribution and failure along the dip-rise and the strike directions of the inclined coal pillars are addressed in this study. The concept of the confined core and three-dimensional stress distribution over the coal pillar are used to derive the strength formulae for the square, rectangular, and very long pillars. The performance of the derived strength formulae is assessed by the stable and failed cases of the flat and the inclined coal pillars. It is observed that all the inclined and flat coal pillars cases are correctly predicted by the derived generalised strength formulae. According to the derived strength formulae, the strength of the inclined coal pillar decreases with the increase of the inclination of the coal pillar. This paper also describes the variation of the strength of the inclined coal pillars with respect to the coal seam inclinations and the frictional properties of the contact planes for the different width-to-height ratios.

## Keywords

Yield zone Rock mass failure criterion Inclined coal pillar strength Flat coal pillar strength## List of Symbols

- \({\sigma _{1{\text{sm}}}}\;{\text{and}}\;{\sigma _{1{\text{bm}}}}\)
Strength of the solid coal mass and broken coal under confinement

- \({\sigma _{3{\text{sm}}}}\;{\text{and}}\;{\sigma _{3{\text{bm}}}}\)
Confining stress of the solid coal mass and broken coal

- \({\sigma _{{\text{cbm}}}}\)
Uniaxial compressive strength (UCS) of the broken coal

- \(a\;{\text{and}}\;b\)
Constants

- \({f_{{\text{sm}}}}({\sigma _3})\;{\text{and}}\;{f_{{\text{bm}}}}({\sigma _3})\)
Failure envelops of the solid coal mass and the broken coal

- \({\sigma _{1{\text{T}}}}\)
Maximum peak stress in the pillar

- \({\sigma _{3{\text{T}}}}\)
Confining stress in the pillar corresponding to \({\sigma _{{\text{1T}}}}\)

- \({\varphi _{{\text{bm}}}}\)
Internal friction angle of the broken coal

- \(\sigma\)
Vertical stress during loading

- \(X,{\text{ }}Y{\text{ and }}Z{\text{ axes}}\)
Directions along the dip-rise, the strike, and the normal to the coal seam

- \({\sigma _z}\;{\text{and}}\;{\sigma _s}\)
Normal and shear components of the vertical stress

- \(\alpha\)
Inclination of the coal seam

- \(h\)
Height of the pillar

- \(\gamma\)
Bulk weight of coal

- \(k\)
In-situ major horizontal and vertical stress ratio

- \({\sigma _{\text{r}}}\)
Resultant stress on the inclined pillar

- \({\sigma _x}\;{\text{and}}\;{\sigma _y}\)
Confinement of the pillar along the dip-rise and the strike direction

- \(\tau\)
Shear stress acting on the contact plane between the pillar/floor and pillar/roof

- \(\mu\)
Coefficient of friction of the contact surface between the pillar/floor and pillar/roof

- \({L_{\text{D}}}\)
Distance from the edge of the inclined coal pillar along the dip-rise direction when \({\sigma _x}={\sigma _{{\text{3T}}}}\) and \({\sigma _z}={\sigma _{{\text{1T}}}}\)

- \({L_{\text{S}}}\)
Distance from the edge of the coal pillar along the strike direction when \({\sigma _y}={\sigma _{{\text{3T}}}}\) and \({\sigma _z}={\sigma _{{\text{1T}}}}\).

- \(s\)
Strength of the pillar

- \({A_{\text{p}}}\)
Plan area of the pillar

- \({A_z}\)
Plan area of the small strip of height d

*σ*_{z}.- \(w\)
Width of square pillar

- \({w_1}\;{\text{and}}\;{w_2}\)
Shorter side and longer side of rectangle

- \({\text{L}}{{\text{i}}_2}\)
Dilogarithm

- \({\sigma _{{\text{cm}}}}\)
UCS of the cubical pillar

- \({\sigma _{\text{c}}}\)
UCS of the coal obtained from laboratory

- \(H\)
Depth of cover

- \(B\)
Width of the gallery

## Notes

### Acknowledgements

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 views expressed in this paper are that of the authors and not necessarily of the organisations to which they belong.

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