A Practical Nonprobabilistic Reliability Assessment Method on Key Strata Shear Failure in Longwall Mining

Roof cutting has long been a potential hazard factor in longwall panels in some diggings in China. Meanwhile, the key strata structural reliability, which provides an assessment on the stability of overlying roof strata,may be a significant reference for support design in underground coal mines.This paper aims to investigate a practical nonprobabilistic reliability assessment method on key strata.Themechanical tests and the hollow inclusion triaxial strain tests were conducted tomeasure relevantmechanical parameters and in situ stress. Furthermore, against the typical failure features in Datong Diggings, China, a shear failure mechanical model of key strata is proposed. Then, an allowable-safety-factor based nonprobabilistic stability probability assessment method is given. The sensitivity of geometrical dimensions and uncertainty levels of friction angle and cohesion are further studied. It is found that thickness and span of key strata havemore dominative effect on key strata’s stability compared with the other factor and the increase of uncertainty levels results in decrease of stability probability.


Introduction
Longwall (LW) mechanized mining methodology, contributing most of Chinese raw coal production, has been extensively applied all over the world.At the same time, roof collapse has long been one of the greatest hazards faced by underground coal miners.The hazardous nature of roof collapse can be illustrated from the national statistics of coal mine accidents.From the Chinese accident statistics, 274 accidents, involving 325 fatalities in 2013, occurred related to roof collapse.Among a diverse range of roof failure, roof cutting, resulting in massive roof strata instability and violent air blast, has been a serious threat in some diggings in China.In recent 30 years, more than 30 accidents triggered by roof cutting, which caused huge economic losses and heavy casualties, have been recorded in Datong Diggings, China.It can be evident from the statistics that roof cutting has a significant impact on the safety conditions.To minimize the risks, a clear understanding of roof strata structural stability and failure model is extremely necessary.
To date, extensive studies in these areas have been carried out.Roof collapse evolution process and stress distribution in overlying strata have been studied and characterized by many researchers [1,2].In recent decade, the key strata (KS), contributing dramatically to the roof strata structure stability, have been the focus of investigative emphasis and hot spot.The most important outcome is the "key strata" theory [3,4], which provides the theoretical basis for studying overlying strata structural stability.Further researches by Xu et al. [5,6], Miao et al. [7], and Ju et al. [8,9] refined the KS theory theoretically and practically.Towards the typical hazardous roof cutting in Datong Diggings, China, relevant researches also have been launched.Xiong [10] applied energy theory to approach the mechanism of roof cutting.Hou et al. [11] summarized roof cutting characteristics and analyzed possible influencing factors based on sufficient fieldwork experience in Datong Diggings.Liu et al. [12] set up a complex rock beam model to analyze the overlying strata shear slipping and applied it in engineering design.At the same time, as uncertainty extensively exists in engineering problems, which are commonly related to material properties, loads, and so forth, the probability model has been widely adopted in engineering design with the aim of quantifying the uncertainty with respect to the characteristics of uncertain variables [13][14][15][16].Since entering 1990s, Ben-Haim [17,18] and Elishakoff et al. [19] suggested using convex models to represent uncertainty.Based upon the convex model theory, nonprobabilistic reliability principle was first introduced by Ben-Haim [20] and then Ben-Haim and Elishakoff [21] proposed the possibility measure methodology.Based on interval arithmetic, a new nonprobabilistic reliability model for structural stability was presented by Guo et al. [22][23][24].In their studies, the nonprobabilistic reliability index for structural systems was defined and some algorithms were proposed to calculate the nonprobabilistic reliability index.Compared to the traditional probabilistic model, the nonprobabilistic method requires less data known and needs no objective distribution function.This may alleviate the computational burden and may be a valid and reasonable alternative when available information on the uncertain parameters is limited.
This paper aims to give a theoretical solution for KS shear failure assessment in longwall mining panels and hereby propose a practical nonprobabilistic reliability assessment method.Through the present work, we expect to promote an allowable-safety-factor based nonprobabilistic reliability assessment method.Four main parts are included in the following text.Firstly, based on some coal mine in Datong Diggings, China, density and mechanical tests and hollow inclusion (HI) triaxial strain test in KS roof strata were carried out.Secondly, a KS shear failure model directing against the failure features in Datong Diggings is proposed.Thirdly, an allowable-safety-factor based reliability assessment method is proposed and applied in the coal mine.Effects of KS geometrical dimensions and parameter uncertainty levels are further studied.Finally, a conclusion is given.

Engineering Background
2.1.Site Description.The project case in this paper is located in Datong Diggings, Shanxi Province, China, which features hard roof strata and KS massive collapse.It currently produces approximately 10 million ton raw coal annually.The LW mechanized retreat mining method is applied in the working panel.
The 5 # coal seam is being exploited in number 8301 working panel whose length is 2000 m and width is 200 m.The roof strata of 5 # coal seam are mainly composed of 13 m thick sandstone with a fraction of mudstone.Core samples are taken from different positions in the 5 # coal seam roof strata to conduct density and shear box testing.

Density Test.
Specimens were firstly prepared into standard cylinders with a diameter of 50 mm and length of 100 mm.Then density was measured by electronic balance and vacuum pump unit.According to test result, the density interval is given by  ∈ [2203.89kg/m 3 , 2415.56 kg/m 3 ].1(a) and 1(b).Four sets of beveled dies with different angles, 40 ∘ , 50 ∘ , 60 ∘ , and 70 ∘ , respectively, were employed in this study.Two groups of specimens were tested on a YAD-2000 testing machine.Some of the broken specimens are shown in Figure 1(c).The results are illustrated in Figure 2.

Shear Box Test. The loading setup of shear box tests is shown in Figures
The friction angle and cohesion were estimated by applying least square method.As shown in Figure 2, the lines are well fit with the test results.Then, intervals of friction angle and cohesion are, respectively, given by  ∈ [37.304 ∘ , 38.938 ∘ ];  ∈ [3.371 MPa, 3.722 MPa].

Hollow Inclusion Triaxial Strain Test.
As shown in Figure 3, a system composed of CSIRO hollow inclusion triaxial strain cells, a KJ327-F in situ stress monitoring station, and a calibration machine was applied to test the in situ stress in 5 # coal seam roof strata.Three HI triaxial strain cells were distributed in the roof strata of KS.Through postprocessing of measured data, the in situ stresses are shown in Table 1.
The lateral stress coefficient is defined as the ration between the maximum principal stress and the vertical stress: We substituted relevant parameters into (1) and obtained the calculated later stress coefficient as 1.45.

KS Shear Failure Model
The key strata dominate the stability of the whole or most of the overlying strata and carry most of the overlying load.With the mining excavation advancing, the span of the KS keeps extending until internal stress reaches tensile strength or shear strength and the failure occurs [4].Accordingly, the instability of KS may be triggered by shear failure or tensile failure.The geospatial location is shown in Figure 4.
It is observed that, in Datong Diggings, China, the massive roof collapses are usually caused by vertical cutting at both ends of short edge of LW panels.Shear failure is considered as the leading cause of roof cutting [11].A simplified mechanical model is proposed, as shown in Figure 5.As shown in Figure 5, we analyze a sheet element in the KS.Considering the limit state of KS, the sheet element attains equilibrium under the combined effect of self-gravity, pressure from upper surface, support from lower surface, and friction at both ends.The vertical stress equilibrium is given by where  is the half length of LW panel short edge,  is the density,  V is the vertical stress, and   is the shear strength.
The relationship between the vertical stress and the horizontal stress is given by where  is the lateral pressure coefficient and it differs from site to site due to diverse tectonic geology.The shear strength is given by the Coulomb criterion as where  is the friction angle and  is the cohesion.Substituting (3) and ( 4) into (2), the vertical stress ( V ) is obtained by Considering the boundary condition, when  = 0,  V = , where  is the load from overlying strata.Thus, we get the particular solution of vertical stress: Then the resisting force is given by where  is the perimeter of KS and  is the thickness of KS.The driving force is given by where  and  are the horizontal area and volume of KS, respectively.

Reliability Assessment on KS Shear Failure
Taking a new nonprobabilistic reliability analysis method by Wang et al. [25] and Liu et al. [26] for reference, which takes the ratio of the volume of safe domain to the total volume of the domain constructed by the basic interval variables as the measure of structural nonprobabilistic stability probability, the nonprobabilistic stability probability is defined as where  stability and  total are the stability domain area and total domain area, respectively, as shown in Figure 6.

An Allowable-Safety-Factor Based Method.
The conception of allowable safety factors is widely employed in structure design and slope stability analysis.Here we define the allowable safety factor of KS as 1 + .It indicates that the slope of the threshold line in Figure 6 is equal to 1 + .According to different positional relation between the domain constructed by intervals and the threshold line, as shown in Figure 7, the allowable-safety-factor based nonprobabilistic stability probability is given as follows.

Stability Probability
Assessment for the Existing KS.Since there has not been a generally accepted stability probability standard for KS, we take correlative rock engineering standard [27] for reference.We take  = 0.3 in this study and substitute relevant parameters into (11a), (11b), (11c), (11d), (11e), and (11f).The calculated nonprobabilistic stability probability of the existing KS is 97.14% under the condition of no earthquake and no water.Overall, the KS are considered to be reliable for the threshold value according to the Grade I engineering requirement [27].be specific, the increase of KS span causes a steady decrease of nonprobabilistic stability probability with the stability probability dropping from nearly 100% to approximately 0 when KS span increases from 28 to 33 m.Similarly, Figure 9 indicates a gradually decreasing trend.However, the growth of KS length shows a minor weakening effect on nonprobabilistic stability which declines merely 13% with KS length increasing from 200 m to 300 m.On the contrary, the rise of KS thickness enhances the KS stability probability as shown in Figure 10.Comparatively, thickness, boosting the nonprobabilistic stability probability from 0 to 100% by merely 2-meter growth, plays a more dominative role in the stability of KS.  respectively. and  are defined as the uncertain levels of friction angle and cohesion, respectively.We can figure it out that the nonprobabilistic stability probabilities in Figures 11 and 12 both experience a decline with the increase of uncertain levels.Compared with the effect of the uncertain level of friction angle, the uncertain level of cohesion shows a more sustainable weakening effect on KS nonprobabilistic stability probability.

Conclusion
Based on the typical KS failure features in Datong Diggings, China, the following conclusions can be reached through applying the nonprobabilistic reliability method to analyze the stability of KS.  (2) Relating to the mechanical tests and in situ stress test, an allowable-safety-factor based nonprobabilistic stability assessment method is proposed and applied to evaluate the KS reliability.
(3) The effects of geometrical dimensions and uncertainty levels are further studied.While the increase of thickness significantly enhances the stability, the trends for the length and the span of KS both witness a steady decline.Besides, the increase of uncertainty levels of friction angle and cohesion both lead to decrease of KS stability probability.
(4) The sensitivity analysis indicates that thickness and span of KS have a great effect on the reliability of KS.The rock mass, however, is irregularly distributed in strata.Therefore, an intensive geological survey on the KS is essential for minimizing the hazardous roof cutting.One should also curtail the exposure time of the KS and guarantee timely support.

Figure 5 :
Figure 5: Mechanical model for roof shear failure.

Figure 6 :
Figure 6: Schematic diagram of stability and failure domains.

5. 1 .
Geometrical Dimensions Analysis.The effects of KS span and length were studied and are shown in Figures8 and 9. To

Figure 7 :
Figure 7: Positional relations between interval domain and threshold line.

Figure 8 :
Figure 8: Variation relationship between nonprobabilistic stability probability and KS span.

Figure 9 :
Figure 9: Variation relationship between nonprobabilistic stability probability and KS length.

Figure 10 :
Figure 10: Variation relationship between nonprobabilistic stability probability and KS thickness.

Figure 11 :
Figure 11: Variation relationship between nonprobabilistic stability probability and uncertain level of friction angle.

( 1 )
A shear failure model is proposed in this paper based on the typical failure model of KS in Datong Diggings.

Figure 12 :
Figure 12: Variation relationship between nonprobabilistic stability probability and uncertain level of cohesion.

Table 1 :
HI triaxial strain testing result.