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Stability Evaluation of Multi-pillar and Roof System Based on Instability Theory

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Abstract

Pillars are often used to support the roof in underground mining. The multi-pillar and roof system is regarded as a multi-pillar and rock beam model. For a system composed of n pillars, the interaction force between the roof and pillars is obtained by a semi-analytical and semi-numerical method. Pillar failure may be progressive or sudden, depending on the equilibrium stability of the system in the post-peak stage. The initial conditions of multi-pillar failure and influence of one pillar progressive failure or unstable failure on adjacent pillars are analyzed based on the instability theory. If one pillar fails gradually, the stress transfer between pillars is also progressive. Once the first pillar fails suddenly, part of the stress is transferred to adjacent pillars, which may lead to further unstable failure of adjacent pillars and cascading failure of multiple pillars. The factors of pillar unstable failure mainly include geometric and mechanical parameters of the system. The mechanical parameters cannot be changed; however, entry (or stope in metal mine) geometrical parameters can be adjusted to reduce the possibility of unstable failure. The variations of pillar stability with entry widths are analyzed according to the factor of safety (FoS) and roof-to-pillar stiffness ratio rk. If the pillars are arranged in a properly concentrated manner, FoS increases, rk decreases and the tendency of pillar unstable failure increases. Conversely, if the pillars are scattered close to the barrier pillars, the possibility of pillar unstable failure is reduced but the overall strength of all pillars is also reduced. Therefore, for a multi-pillar and roof system, the entry widths should be properly adjusted from an overall system perspective to ensure that both FoS and rk values are sufficiently large to minimize the possibility of pillar unstable failure.

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Abbreviations

q :

Uniformly distributed force acting on rock beam

M A,  R A :

Bending moment and shear force of rock beam at point A, respectively

w o_i :

Width of the ith entry

w p_i :

Width of the ith pillar

M B,  R B :

Bending moment and shear force of rock beam at point B, respectively

R i :

Supporting force of the ith pillar against the roof

F i :

Force of the roof acting on the ith pillar

a i,  b i :

The distances from the ith pillar center to the boundary of barrier pillars on both sides

E p,  A :

Young’s modulus and cross-sectional area of pillar, respectively

σ,  ε, u :

Stress, strain and deformation of pillar, respectively

ε 0, u 0 :

Average strain and deformation of pillar, respectively

u i, u 0_i :

Deformation and average deformation of the ith pillar, respectively

m,  k p :

Shape parameter and initial stiffness of pillar, respectively

w p,  h :

Width and height of pillar, respectively

λ :

Slope of pillar constitutive curve

u c :

Pillar deformation corresponding to the peak strength, uc = (1/m)1/m·u0

u c_i :

Deformation of the ith pillar corresponding to peak strength

u t,  λ t :

Deformation and stiffness at the inflection point of pillar F–u curve, ut = (1 + 1/m)1/m·u0

h b, l :

Thickness and total span of rock beam (or roof), l = a i+ bi

E b, I b :

Young’s modulus and inertia moment of rock beam, respectively

y,  θ :

Deflection and rotation angle of rock beam, respectively

u q_i, u Rj_i :

Roof deformation at the ith pillar position caused by distributed force q and force Rj, respectively

C ij :

Roof deformation at the ith pillar position caused by per unit force at the jth pillar position, i.e., flexibility

K ij :

Stiffness influence coefficient

U :

Potential energy function of a multi-pillar and roof system

ΔW e :

External work acting on the pillar during unstable pillar failure process

ΔW p :

Dissipated energy of pillar during unstable pillar failure process

ΔE :

Excess energy that is not completely dissipated during unstable pillar failure process

λ m :

The minimum slope value (negative value) in the post-peak range of all pillar constitutive curves

η c :

The minimum eigenvalue of matrix K

k i :

Local stiffness of the roof at the ith pillar position

λ i :

Slope of the ith pillar constitutive curve

λ t_i :

Post-peak stiffness of the ith pillar at the inflection point

r k_i :

Stiffness ratio of the system at the ith pillar position, defined as rk_i = ki / λi. rk_i equals to the panel factor of stability (PFS).

FoS:

Factor of safety

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Acknowledgements

This study was supported by Postdoctoral Science Foundation of China (No. 2020M670782), Fundamental Research Funds for Central Universities of China (No. 170104026) and National Natural Science Foundation of China (Nos. U1710253, 51904057, 52004053). The authors are very grateful for the financial contributions and convey their appreciation to the organizations for supporting this basic research.

Funding

Postdoctoral Science Foundation of China, 2020M670782, Kai Guan, Fundamental Research Funds for Central Universities of China, 170104026, **nrong Wang, National Natural Science Foundation of China, U1710253, Tianhong Yang, 51904057, **ge Liu, 52004053, Kai Guan.

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Authors and Affiliations

  1. School of Mining Engineering, University of Science and Technology Liaoning, Anshan, 114051, China

    **nrong Wang

  2. Center for Rock Instability and Seismicity Research, School of Resource and Civil Engineering, Northeastern University, Shenyang, 110819, China

    Tianhong Yang, Kai Guan, **ge Liu & Yong Zhao

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  1. **nrong Wang
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Correspondence to **nrong Wang.

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Wang, X., Yang, T., Guan, K. et al. Stability Evaluation of Multi-pillar and Roof System Based on Instability Theory. Rock Mech Rock Eng 55, 1461–1480 (2022). https://doi.org/10.1007/s00603-021-02712-8

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  • DOI: https://doi.org/10.1007/s00603-021-02712-8

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