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

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## Abstract

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 (*R*^{2}) 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.

## Keywords

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

- \({\text{d}}\varepsilon_{ij}\)
Total increment of the strain

- \({\text{d}}\varepsilon_{ij}^{\text{e}}\)
Elastic part of strain increment

- \({\text{d}}\varepsilon_{ij}^{\text{p}}\)
Plastic part of strain increment

- \({\text{d}}\sigma_{ij}\)
Stress increment

- \(D_{ijkl}\)
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

*F*Function of the failure envelope

*g*Plastic potential function

*λ*Plastic multiplier

*σ*_{11}Major principal stress

*σ*_{33}Minor principal stress

*c*,*c*_{w}Cohesion of the rock matrix and weak plane

*ϕ*,*ϕ*_{w}Internal friction angle of the rock matrix and friction angle of the weak plane

*σ*_{t}Tension cut-off of Mohr–Coulomb criterion for the rock matrix

*ψ*Dilation angle of the rock matrix

- \(\partial \bar{\varepsilon }^{\text{p}}\)
Accumulated equivalent plastic strain

*t*and (*t*− Δ*t*)Current and previous calculation step

*R*Transformation matrix

*σ*_{1m}Tri-axial strength of the rock mass

*σ*_{3m}Confining stress of the rock mass

*σ*_{ci},*σ*_{cm}Compressive strength of the intact rock and rock mass

*σ*_{ti},*σ*_{tm}Tensile strength of the intact rock and rock mass

*τ*Shear strength of rock mass

*τ*_{sm}Cohesion of the rock mass

*H*_{p}Hardening or softening parameter

*μ*_{0m},*ϕ*_{0m}Coefficient of friction and internal friction angle of the rock mass

*H*Depth of cover

*σ*_{v}Vertical in situ stress

*σ*_{H}Major horizontal in situ stress

*σ*_{h}Minor horizontal in situ stress

*β*Coefficient of thermal expansion

*G*Geothermal gradient

*S*_{flat},*S*_{inclined}Strength of the flat and inclined coal pillar

*σ*_{c}Intact strength of the cubic coal of side 25 cm

*h*Height of the coal pillar

*w*Width of the pillar

*H*_{0}Null hypothesis

*H*_{1}Alternative hypothesis

*μ*_{1}Mean pillar strength derived from the Sheorey’s formula

*μ*_{2}Mean pillar strength obtained from the numerical modelling

*A*Strength ratio between inclined and flat coal pillar

- \(\theta\)
Dip of the coal seam

- \(\varphi\)
Acute angle of the corners of the coal pillar

*E*Error function

*σ*_{n},*τ*,*σ*_{r}Normal, shear, and resultant stress on the inclined pillar

*γ*Unit weight of the overlying rock strata

*L*Length of the pillar

*B*Width of the gallery

*k*Ratio of the major horizontal and the vertical in situ stresses

## 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 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.

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