Introduction

Room-and-pillar method is widely practiced in many countries such as USA, Russia, China, South Africa, Korea, and India (Choi et al. 2017; Mathey and Van der Merwe 2016; Peng 2007; Shao et al. 2009). The method provides high productivity and low mining costs (Choochang 2013; Kun et al. 2014; Luo 2015; Yu et al. 2017). In this method, pillars are left between the rooms to maintain their stability and act as primary or temporary support systems (Idris et al. 2015; Jessu et al. 2018; Yang et al. 2012). Therefore, pillar stability is the major requisites for safe working conditions. Pillar instability can result in rock falls from pillar ribs and consequently lead to roof collapse if one pillar or more fails (Clarke et al. 2006; Esterhuizen et al. 2007). Alternatively, pillar failure incurs if the neighboring pillars are unable to carry the transferred load from the collapsed ones or if the roof span is too large (Dehghan et al. 2013; Li et al. 2013; Ma et al. 2012; Poulsen and Shen 2013; Sahu and Lokhande 2015).

Ground falls causes about 36% of fatalities and 12% of all injuries related lost working days between 1996 and 2005 (Esterhuizen et al. 2006). The performance of pillars is governed by several factors such as characteristics of rock mass (i.e., strength, friction angle, cohesion, etc.), present of discontinuities, pillar geometry (i.e., shape and size), pillar contact conditions with roof and floor, mining activity, and in situ stress field (Gaede et al. 2014). Pillars are often partially or completely mineralized at a later stage after their service is no longer needed. For economic reasons, an optimum-sized pillar is the smallest one satisfying safety requirements. Pillar size is usually determined based on experience or using strength equations ((Esterhuizen et al. 2008) & 2006). As mining extends at greater depths (i.e., > 300 m), larger pillars will be needed. Thus, room-and-pillar method is not preferable due to less ore recovery.

Methods of analysis

The most challenging task in underground mines is to develop safe and economic pillar design (Ünlü 2001). Pillars will remain stable if their strength (i.e., bearing capacity) exceeds the load acting upon it (i.e., overburden stress). However, these two principal parameters (i.e., ultimate pillar strength and acting stress) are never precisely known, because they are functions of stress state, ground conditions, orebody properties, and mining activities. Thus, there is no unique (i.e., standard) pillar’s geometry (i.e., shape and size) which exists in underground mines (Kendorski 2007). Consequently, the design process is usually carried out by trial-and-error based on experience, existed mine case history, and field observations. The pillar’s shape, strength, and load (i.e., overburden stress) acting upon it are the main parameters to be considered in pillar design process (Tzalamarias et al. 2018). Analytical and numerical modelling are the main methods of analysis which are used to estimate pillar stability (Kortnik 2015). These methods are presented and discussed in the next sections.

Analytical methods of analysis

Usually, empirical equations are applied to determine the factor of safety (FOS), where strength of pillar is divided by full tributary area (i.e., full cover load) (Chikande and Zvarivadza 2016). Pillar’s width-to-height ratio is the principal design factor which is included into these formulae. However, other important factor such as thickness of seam is seldom directly considered (Guy et al. 2017). Despite these methods not giving an exact result, however, they are generally simple and useful. The assumption of stresses imposed on pillars being uniformly distributed is the main weakness of the analytical methods. Tributary area method that is used to estimate the overburden stress acting upon pillar is presented and discussed in the following section. As well, different formulae are used to estimate strength of pillar.

Estimating strength of pillar

In this section, pillar strength is calculated using different empirical equations. Such equations incorporate pillar dimensions and uniaxial compressive strength of intact rock. The majority of pillar strength analysis has been conducted for coal pillars. Consequently, “back calculation” approach has been employed to establish hard rock pillar strength formulae (Martin and Maybee 2000). A review of the empirically developed pillar strength equations for hard rock mines reveals that the equations can be placed into three groups (Esterhuizen 2007):

  • Power equations

Power equations such as Hedley and Grant (Hedley and Grant 1972) equation has been applied extensively to the design of pillar in hard rock as given in Eq. (1). It follows the form of the coal pillar strength equation that has been developed by Salamon and Munro (Salamon and Munro 1967; Bieniawski, Z. T.1967).

$$ {S}_{\mathrm{p}}=k\frac{W^{\alpha }}{h^{\beta }} $$
(1)

where k is the strength of unit cube of rock material forming the pillar, w and h are the pillar width and height, respectively, and α and β are parameters related to the geomechanical conditions of the rock mass. For massive quartzite and conglomerates, Eq. (1) is rewritten as follows (Martin and Maybee 2000):

$$ {S}_{\mathrm{p}}=133\frac{W^{0.50}}{h^{0.75}} $$
(2)

For metasediments rock mass, Eq. (2) can be rewritten as shown in Eq. (3) (Kimmelmann et al. 1984; Martin and Maybee 2000):

$$ {S}_{\mathrm{p}}=65\frac{W^{0.46}}{h^{0.66}} $$
(3)

Based on statistical database which represents our real case study and assuming that a pillar with no defects has a decent size, whereas a demolished pillar is too small to derive a pillar strength equation, Eq. (3) can be re-written as given in Eq. (4):

$$ {S}_{\mathrm{p}}=43\frac{W^{0.46}}{h^{0.66}} $$
(4)

  • Linear equations

The first linear equation is originally proposed by Obert and Duval (Obert and Duval 1967) (Bunting 1991) based on laboratory tests on rock samples. Equation (5) shows that strength of pillar ( Sp) depends on uniaxial compressive strength of a specimen (σ1) and the geometry of pillar (i.e., width, w and height, h).

$$ {S}_{\mathrm{p}}={\sigma}_1\left(0.778+0.222\ \frac{w}{h}\right) $$
(5)

Equation (5) is valid when pillar’s width-to-height ratio, w/h, is between 0.25 and 4.0 under the weight condition of a dead load. Obert-Duvall recommended that the factor of safety, FOS, should be 2 for short-term stability and 4 for long-term pillar stability. The strength of pillar also was determined by conducting field tests on pillars in West Virginia (Hosseini et al. 2012). The same equation was proposed as above, but σ1 was defined only as the maximum strength of a specimen in the forward direction when the specimen is larger than a critical size. Moreover, Krauland and Soder (Krauland and Soder 1987) determine pillar strength based on pillar failure observations in limestone mines, as per Eq. (6) (Martin and Maybee 2000):

$$ {S}_{\mathrm{p}}=35.4\left(0.778+0.222\ \frac{w}{h}\right) $$
(6)

Sjöberg (Sjöberg 1992) modified Eq. (6) based on failure modes and behavior of pillars in limestones/skarn mines, as presented per Eq. (7):

$$ {S}_{\mathrm{p}}=74\left(0.778+0.222\ \frac{w}{h}\right) $$
(7)

Potvin et al. (Potvin et al. 1989) suggest the pillar strength in Canadian Shield is calculated based on rock mass compressive strength as presented in Eq. (8):

$$ {S}_{\mathrm{p}}=0.42\ {\sigma}_c\left(\ \frac{w}{h}\right) $$
(8)

Lunder and Pakalnis (Lunder and Pakalnis 1997) analyzed 178 case histories from hard rock mines, 98 of which were located in the Canadian Shield, compiled an extensive database of hard rock pillar failure and proposed Eq. (9) to obtain pillar strength:

$$ {S}_{\mathrm{p}}=0.44\ {\sigma}_c\ \left(0.68+0.52\ k\right) $$
(9)

where k is pillar friction term and is calculated as given in Eq. (11) below.

  • Pillar confinement equations (Esterhuizen 2007; Jessu et al. 2018)

These equations were developed by Lunder and Pakalnis (Lunder and Pakalnis 1997) based on pillar confinement as given in Eq. (10):

$$ {S}_{\mathrm{p}}=\left(K. UCS\right)\ \left({C}_1+k\ {C}_2\right) $$
(10)

where K is the rock mass strength size factor (i.e., K = 0.44). C1 and C2 are empirically derived constants determined to be 0.68 and 0.52, respectively, and k is the pillar fiction term and is determined as given per Eq. (11):

$$ k=\tan \left( co{s}^{-1}\left(\frac{1-{C}_{pav}}{1+{C}_{pav}}\right)\right) $$
(11)

where Cpav is the average pillar confinement and is given from Eq. (12):

$$ {C}_{pav}=0.46{\left(\log \left(\frac{w_p}{h}+0.75\right)\right)}^{\frac{1.4}{\raisebox{1ex}{${w}_p$}\!\left/ \!\raisebox{-1ex}{$h$}\right.}} $$
(12)

where \( \frac{w_p}{h} \) is pillar width-to-height ratio.

Estimating average pillar stress

The average pillar stress (σP) can be estimated using tributary area method. Such method deals mainly with the divergence of stress and the concentration of the vertical stress acting upon pillars between close cavities. Tributary area method is not applicable where mineral deposits are irregular, dipping, mining pattern is inconsistent, and complex triaxial stress field (Lunder 1994). Stress at an arbitrary point inside a pillar is determined based on the following two factors:

  • Average pillar stress determined by the ratio between the shear area of a cavity and the remaining shear area

  • Stress concentration determined by the pillar arrangement between adjoining cavities

Suppose a pillar is part of a large group of pillars, and the weight of the bedrock is evenly distributed across these pillars. The average stress (σp) of a pillar applied to the rock is derived as per Eq. (13), where Pz is the load imposed on pillar, γ is the specific weight of the overlying rocks, z is the depth of cover, and Wo, Wp are the width of cavity and pillar respectively.

$$ {\sigma}_{\mathrm{P}}={P}_{\mathrm{z}}{\left(1+\frac{W_{\mathrm{o}}}{W_{\mathrm{P}}}\right)}^2=\upgamma \mathrm{z}{\left(1+\frac{W_{\mathrm{o}}}{W_{\mathrm{P}}}\right)}^2 $$
(13)

The average pillar stress, based on pillar arrangement, is given as per Eq. (14) and depicted as shown in Fig. 1. It is calculated by dividing area of the activity supported by pillars, Ao and the cross-section area of each pillar, Ap.

Fig. 1
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Average vertical pillar stress in common pillar arrangements

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$$ {\sigma}_{\mathrm{P}}=\frac{A_{\mathrm{o}}}{A_{\mathrm{P}}}\times \upgamma \mathrm{z} $$
(14)

Factor of safety

The safety factor, FOS, is defined as the ratio of pillar strength (SP) to the load (i.e., average stress) carried by the pillar (σP). It can be obtained as per Eq. (15):

$$ \mathrm{FOS}=\frac{S_{\mathrm{P}}}{\sigma_{\mathrm{P}}} $$
(15)

The strength of pillar (SP) is function of its geometry (i.e., shape and size). Such strength is inversely proportional to its size. Alternatively, pillar strength decreases as its size increases, while it increases with the increase of width-to-height ratio (i.e., \( \raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$h$}\right. \)). Pillar stress (σP) relies on the density of overlying rock mass, mining depth, and width of cavity-to-pillar width ratio (i.e., \( \frac{W_0}{W_{\mathrm{p}}} \)). The pillar can therefore be designed (i.e., dimensioned) by imposing that the minimum safety factor (FOSmin) in the pillar is above a certain threshold value (FOSthreshold) (Fahimifar et al. 2013). Table 1 gives the calculation of safety factor (FOS) at various empirical equations.

Table 1 Calculated average pillar strength, pillar stress, and safety factor using analytical methods
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$$ {\mathrm{FOS}}_{\mathrm{min}}\ge {\mathrm{FOS}}_{\mathrm{threshold}} $$
(16)

Based on the analytical results, there is a significant discrepancy in the values of safety factor ranging from 7.35 to 36.36. The reason is that each empirical equation has been developed for certain conditions (i.e., equations depend basically on several constants (k, α, β) and therefore are site specific). Although these equations are useful, however, they do not explain pillar loading and failure mechanisms. In addition, they disregard roof and floor conditions, geomechanical properties of rock mass, dipping of orebody, and mining sequences that have a crucial effect in rock pillar bearing capacity (Peng 2007). Thus, they are replaced by numerical modelling methods.

Numerical methods of analysis

Numerical techniques provide adequate, robust, and reliable analysis. Thus, they replace analytical methods particularly when mine geometry is very complex. Various codes are available and well accepted by rock mechanics practitioners in mining engineering. Fast Lagrangian Analysis of Continua (FLAC2D) is a two-dimensional explicit finite-difference method (FDM), employed to conduct this analysis and to estimate factor of safety, based on stress state and deformation, using FISH code (i.e., “solve fos” command). Such code is programming language embedded within FLAC that enables the user to define new variables and functions. The built-in function, FISH code, uses Mohr-Coulomb input rock mass properties (i.e., cohesion, friction angle, tensile strength, compressive strength, Young’s modulus, etc.) and in situ stresses (i.e., σ1, σ3) to compute mining induced stresses and therefore safety factor.

Case study

Nohyun limestone mine is located about 13 km south of Cheongju city, North Chungcheong Province, Korea. The mine is operated by Union Company and is currently mined out by underground mining (i.e., room-and-pillar method).

Mine geology

The soil is composed of a Midongsan layer covered by an Unkyori layer. The Unkyori layer is presumed to be widely distributed owing to folding, and is spread mainly in the N-NE and S-SW directions. Foliation is relatively steep at 60° to 80° NW, SE. The Midongsan layer is found around the eastern part of the mine in a zonal distribution to the N-NE and S-SW. This layer mainly consists of quartzite but has partly sandy schist. Both are medium-texture rocks with pinkish gray and brown colors and made up of quartz, apatite, biotite, muscovite, and others. Mining is being underway or is planned for three levels. Figures 2 and 3 present plan view of mining level #1 and level #2 respectively.

Fig. 2
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Plane map of level #1

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Fig. 3
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Plane map of level #2

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Research on mineshaft

The mining of mineshaft level #1 is already completed. As shown in Fig. 2, point “A” is the entrance of the shaft and point “B” is the ramp descending to level #2. The ground levels of two points are 66ML and 36ML with a 30 m difference between levels. The mining operation in mineshaft level #2 is currently underway, as depicted in Fig. 3. Mining in zone “A” has been completed for the upper and lower halves. However, the operation in zone “B” is done only in the upper half, and approximately 5 m in height remains in the lower half. The height of the pillars in zone “B” is 6.5–7.0 m and planned to be 12–14 m after mining of the lower half is completed. “Po” in zone “C” has a ground level of 22.37 ML, girth of 44 m, and area of 121 m2. The distance between pillars is 11–12 m and the height of the pillar measured from four surrounding spots (a, b, c, d) was 6.5–7.0 m. The first pillar of level #3 is located approximately 19 m under the “Po” pillar in level #2, as shown in Fig. 4.

Fig. 4
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Detailed map of zone C in level #2

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In mineshaft level #3, mining of the upper half is underway. The ground level of the area is 8.57 ML, and the space between level #2 and level #3 is 19.2 m. A pillar with a girth of 41.5 m and area of 106.9 m2 is installed in the direction towards the entrance of level #3, as shown per Fig. 5. The mining of shaft level #1 is completed and the mining operation on level #2 is almost done except for some parts in the lower half. On level #3, one pillar is installed and the mining of the upper half is being carried out simultaneously with operations on the lower half of level #2. During site investigation, the vertical locations of the pillars in level #2 and level #3 were found to be misaligned, as shown in Fig. 6. Cross-sections along limestone mineral vein are shown in Figs. 7 and 8.

Fig. 5
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Plane map of level #3

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Fig. 6
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Plane map of mining plan

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Fig. 7
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Cross section of A-A′

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Fig. 8
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Cross section of B-B′

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Figures 7 and 8 show that the limestone exists at an angle of 70°–80° to the ground. Since productivity decreases when an aligned pillar arrangement between levels is mined, the pillar arrangement on each level is designed according to the shape and condition of the remained limestone mineral deposit. This may lead to a misalignment between pillars on different levels, and this can ultimately threat the stability of the shaft. Structural safety is best assured when the positions of pillars on each level are well aligned.

Evaluation of rock pillar behavior

The pillars, located on level #2 and level #3, are found to be misaligned. Current research is conducted to examine the structural stability when pillars situated on different levels are misaligned, and when level #3 is subsequently mined out.

Numerical model

Numerical analysis is carried out using finite difference code, FLAC2D for the cross sections A-A′ and B-B′, which are orthogonal to the entrance ramp of level #3. A base point is set at the area where level #3 is being developed as a shaft mining zone, straight below the strip mining area. For level #3, where mining has not yet begun, an analysis was conducted considering a mining plan based on the remaining pillars on level #3. Factoring in the relaxation of rocks caused by shaft mining, the width and height were set at 340 m (W) × 330 m (H) for cross-section A-A′, and 290 m (W) × 190 m (H) for cross-section B-B′. An elastoplastic model of the Mohr-Coulomb yield criterion is applied as listed in Table 2.

Table 2 Geomechanical properties of limestone and in situ stresses used in numerical modelling analysis employing Mohr-Coulomb failure criterion
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Parametric factors

Parametric stability analysis has been conducted for the three conditions as listed in Table 3. Stability is evaluated under the current situation. Possible changes in stability owing to the mining sequence of level #3, change in mining depth, and mineshaft width are also examined.

Table 3 Mining conditions applied within numerical parametric analysis
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Results and discussion

Results of this study are presented and discussed in the following section in terms of stress state (i.e., max. and min. principal stresses), deformation (i.e., horizontal and vertical displacements, yielding), and factor of safety (i.e., FOS) with respect to mining conditions listed in Table 3 (i.e., mining sequences, mining depth, and mineshaft width).

Effect of mining sequence

Mining upper half of level #2 followed by mining upper half of level #3—Case I

Figure 9 displays the results when the upper half of level #2 is mined out as the first step, and subsequently followed by mining upper half of level #3, before mining starts on the lower half of level #2.

Fig. 9
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Numerical modelling analysis in terms of stress state, deformation, plastic zones, and safety factor—Case I

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When cavity mining is conducted, level #2 is stabilized after a certain period of time. It is found that the stress is concentrated around the corners of pillars located on level #3, as shown in Fig. 9: (i.e., (a) red and pink notches with max. stress varies between 3 and 4 MPa and (b) a pink notch with min. stress varies from 1.5 to 1.75 MPa). For pillars located on level #2, stress also is concentrated around the corners of the pillars but with less magnitudes (i.e., green, yellow, and pink notches with max. stress ranges between 2 and 3 MPa, Fig. 10a and light blue notches with min. stress varies from 0.75 to 1.0 MPa, Fig. 10b). In terms of deformation, horizontal and vertical displacements are given as shown in Fig. 9 (c, d). Figure 9 c shows that as mining starts to proceed (i.e., after excavating rooms), both sidewalls of pillars, on level #2 and level #3, move towards left (i.e., towards mined rooms and horizontal displacements are negative as indicated by arrows direction).

Fig. 10
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Numerical modelling analysis in terms of stress state, deformation, plastic zones, and factor of safety—Case II

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The maximum horizontal displacements occur on the right wall/side of the pillars located on level #2 (i.e., − 1.25 × 10−3 m), while maximum horizontal displacement occurs on the pillars located on level #3 is − 1 × 10−3 m. Figure 9 d shows that deflection or sag (i.e., negative y-sign means move downwards) occurs in the roof/back of pillars located on level #2 (i.e., maximum deflection ranges from − 5 × 10−4 to − 1 × 10−3 m). For pillars located on level #3, deflection occurs in the roof (i.e., downward vertical displacement is − 5 × 10−4 m represented by pink-colored contours), where heave occurs in the floor of the pillars (i.e., upward vertical displacement is + 5 × 10−4 m represented by yellow colored contours).

Mohr-Coulomb yielding is the most common criterion used in elasto-plastic numerical modelling. The condition of yielding is reached when the stress state reaches the surface of the yield function, which is when the rock is loaded beyond its elastic limit. Figure 9 e depicts yielding (i.e., failure) plastic zone (i.e., surrounded by blue circles) occurring around pillars on both level #2 and level #3 due to mining activity. The analysis reveals that the majority of shear failure occurs in the left sidewalls of pillars located on level #3 (i.e., red colored asterisks) compared with limited tensile failure occurs in the roof (i.e., pink colored circle). Also, Fig. 9 e shows that significant extent of yield zones around right and left walls of pillars for both levels (i.e., extent of yielding exceeds 0.5 m on both pillar sides).

Figure 9 f gives the calculated safety factors for pillars located on level #2 and level #3. As introduced previously (i.e., “Factor of safety”) that the safety factor is the ratio between pillar strength and induced stress that pillar has to carry due to mining activity. It can be shown that the safety factor ranges between 1 and 2 (i.e., red- and pink-colored contours) for pillars located on level #3, and ranges between 2 and 3 for pillars located on level #2 (i.e., pink-colored contours). Alternatively, if the average pillar strength is 2.33 MPa (i.e., UCS of rock mass), the average vertical stress applied on pillars located on level #3, as depicted in Fig. 9d, lies between 1.5 and 1.75. Therefore, the safety factor ranges from 1.33 to 1.55. For pillars located on level #2, safety factor ranges between 2.33 and 3.10 (i.e., pillar strength is 2.33 MPa and pillar stress ranges between 0.75 and 1 MPa).

Complete mining of level #2 followed by mining of upper half of level #3—Case II

Figure 10 depicts the results when the upper half of level #3 is mined out after level #2 is completely extracted. As in the first case, level #2 is stabilized, and the pillars on level #3 experienced a similar stress concentration on the corners. For pillars located on level #3, the maximum principal stress ranges from 2.5 to 3 MPa (i.e., Fig. 10a), while the minimum principal stress varies between 1.25 and 1.5 MPa (i.e., Fig. 10b). For pillars located on level #2, the maximum principal stress lies between 1.5 and 2.0 MPa, while the minimum principal stress varies between 0.25–0.50 MPa. Figure 11 c shows that the sidewalls of pillars, located on level #3, move towards left (i.e., maximum horizontal displacements vary between 2.5 × 10−4 and 1.25 × 10−3 m). For the pillars located on level #2, generally sidewalls of pillars move towards left (i.e., maximum horizontal displacements range from 7.5 × 10−4 to 1.5 × 10−3 m), except the left wall of one pillar moves towards the right side (i.e., displacement is + 2.5 × 10−4 m entire the mined room due to the effect of big cavity).

Fig. 11
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Numerical modelling analysis in terms of stress state, deformation, plastic zones, and factor of safety—Case III

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Figure 10 d depicts the vertical displacement occurs in the roof of the pillars located on level #2 (i.e., roof sag ranges from 5 × 10−4 to 1 × 10−3 m). For pillars located on level #3, there is no deflection occurs in the roof of the pillars (i.e., vertical displacement is almost zero as represented by yellow-colored contours), whereas, floor heave occurs (i.e., upward vertical displacement ranges between 5 × 10−4 to 1 × 10−3 m represented by light green–colored contours). The development of failure zones around pillars is displayed in Fig. 10e below. It can be seen that the pillars located on level #2 (i.e., pillar on right side) experienced significant yielding particularly on its left side (i.e., right side of the mined room).

The yield zones extend diagonally from right side of the mined room (i.e., pillar’s left side) towards the pillar’s right side. For pillars located on level #3, yielding occurs in shear in both sides of pillars (i.e., no yielding occurs in tensile). The safety factor (i.e., FOS = 2) is shown in Fig. 10e (i.e., pink-colored contours). Alternatively, the overburden stress on pillars located on level #3 varies between 1.25 to 1.5 MPa. Therefore, FOS ranges between 1.55 and 1.86.

Complete mining of level #2 followed by complete mining of level #3—Case III

Figure 11 shows the analysis when level #3 is mined out after level #2 is completely extracted. As in the first and second cases, level #2 is stabilized, but almost all pillars on level #3 experienced severe stress concentrations in the left and right diagonal directions. The maximum principal stress around pillars located on level #3 ranges from 3 to 3.5 MPa (i.e., Fig. 11a), whereas, minimum principal stress lies between 1.5 to 1.75 MPa (i.e., as depicted in Fig. 11b). The horizontal displacement contours are presented in Fig. 11c. It is obvious that the pillar walls are shifted towards the left side (i.e., maximum horizontal displacement is 1.5 × 10−3 m). The vertical displacement is displayed in Fig. 11d. It can be seen that roof sag occurs for the pillars located on level #2 (i.e., maximum roof deflection is 1 × 10−3 m). While floor heave occurs for pillars located on level #3 (i.e., maximum floor heave is 1 × 10−3 m).

The extent of yielding zones around pillars is given as shown in Fig. 11e. It can be seen that the extent of yield zones increases around pillars walls. The majority of yielding occurs in shear around pillars located on level #3, except for limited regions yield in tension. The safety factor ranges between 2 and 3 (i.e., brown-colored contours) with small red notches around pillars (i.e., FOS ranges from 1 to 2). The exact value of safety factor lies between 1.33 and 1.55 (i.e., pillar strength is 2.33 MPa and overburden stress varies between 1.5 and 1.75 MPa). Table 4 gives the summary of the effect of mining sequence on the performance stability of pillars located on level #3 in terms of stress state, deformation, and safety factor.

Table 4 Effect of mining pattern on the stability of pillars located on level #3 in terms of stress state, deformation, and safety factor
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Effect of mining depth

Numerical analysis has been conducted after assuming a mining depth on level #3 is 6 m for the upper half and 4 m for the lower half. These are similar to the depths of mining conducted on level #2. Although the possibility of local fractures is detected in some areas, the overall structural stability is guaranteed. An additional stability assessment is performed assuming a situation where the mining depth increases to 12 m and 15 m to boost production.

Mining depth of level #3 is 12 m—Case IV

Figure 12 presents the results of numerical analysis when mining depth is 12 m. As mining depth extends to 12 m, stress concentration intensifies on corners of pillars and spreads diagonally. The maximum principal stress around pillars located on level #3 lies between 3.5 and 4 MPa (i.e., Fig. 12a), while the overburden stress ranges from 1.5 to 1.75 MPa (i.e., Fig. 12b). The horizontal displacement around pillars located on level #2 ranges between 2.5 × 10−3 to 3 × 10−3 m, while it ranges between 2 × 10−3 to 2.5 × 10−3 m around pillars located on level #3 (i.e., Fig. 12c). The vertical displacement is depicted as shown in Fig. 12d. It is clear that roof sag occurs to the pillars located on level #2 (i.e., maximum deflection ranges between1 × 10−3 and 1.5 × 10−3 m), but no roof sag occurs to the pillars located on level #3. Also, it is noticed that floor heave occurs to pillars located on level #2 and level #3 (i.e., floor heave are 5 × 10−4 m and 1 × 10−3 to 1.5 × 10−3 m) respectively.

Fig. 12
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Numerical modelling analysis in terms of stress state, deformation, plastic zones, and factor of safety—Case IV

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The yielding zone is depicted as per Fig. 12e. It is clear that pillars located on level #3, yield in shear and tensile (i.e., the plastic zones extend diagonally along the entire pillars). Also, factor of safety (i.e., Fig. 12e) on the top and bottom of the pillars ranges between 2 and 3; however, it deteriorates in the middle of pillars (i.e., FOS lies between 1 and 2). Alternatively, safety factor varies from 1.33 to 1.55 (i.e., overburden pillar stress lies between 1.5 to 1.75 MPa and pillar strength is 2.33 MPa).

Mining depth of level #3 is 15 m—Case V

Figure 13 displays the results of numerical modelling when mining depth is 15 m. It can be seen that stress concentrates around corners and extends diagonally along pillars (i.e., Fig. 13a). The maximum and minimum stresses around pillars located on level #3 range from 4 to 4.5 MPa and 1.75 to 2 MPa respectively. The horizontal displacement contours are shown in Fig. 13c. It is obvious that pillars move towards left side due to mining activity (i.e., effect of cavity). Except for top left and middle pillar located on level #2 and level #3 respectively, their left walls move towards the entire cavity (i.e., to right). The maximum horizontal displacements are 3 × 10−3 m and 2 × 10−3 m for pillars located on level #2 and level #3 respectively. Vertical displacements are depicted as shown in Fig. 12d. Deflection obviously occurs for pillars located on both levels. The maximum roof sags are 2 × 10−3 m and 1 × 10−3 m for pillars located on level #2 and level #3 respectively. Floor heave only occurs for pillars located on level #3 (i.e., floor heave is 1 × 10−3 to 2 × 10−3 m).

Fig. 13
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Numerical modelling analysis in terms of stress state, deformation, plastic zones, and factor of safety—Case V

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The development of plastic yield zones is shown in Fig. 13e. It clears that pillars located on level #3 entirely yield in shear and some areas on the boundaries of pillars are found to be vulnerable to tensile fracture. Safety factor ranges between 1.16 and 1.33 for all pillars on the two levels as shown in Fig. 13f. This indicates extremely unstable condition. Based on the findings above, it can be concluded that stability is ensured at a mining depth of approximately 10 m, which is a current practice of the mine, and a safety risk can be posed if the mining depth increases. Table 5 gives the summary for the effect of mining depth on stability of pillars located on level #3.

Table 5 Summary of the effect of mining depth on the stability of pillars located on level #3 in terms of stress state, deformation, and safety factor
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Effect of mineshaft width

Pillar stability assessment has been carried out by assuming that the mineshaft width of level #3 is widened. The width is increased by 1 m in the first step, and 2 m in the second step. Since the total size of the mining area is remained unchangeable. Thus, the width of the pillars is assumed to decrease by 1 m and 2 m respectively (i.e., width of the mineshaft accordingly increased).

Width of mineshaft of level #3 increases by 1 m—Case VI

Figure 14 displays the results when the mineshaft width is increased by 1 m. It can be seen that stress concentrates around the corners of the pillars and spreads diagonally. The maximum and minimum principal stresses are 3.5 to 4 MPa (i.e., Fig. 14a) and 1.5 to 1.75 MPa (i.e., Fig. 14b) for the pillars located on level #3.

Fig. 14
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Numerical modelling analysis in terms of stress state, deformation, plastic zones, and factor of safety—Case VI

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As shown in Fig. 14c, the horizontal displacement contours show that the pillars shift towards left side. Except for the left sides of top left and middle pillars located on level #2 and level #3, respectively, where the left walls of pillars move towards the entire of cavity (i.e., mined room). Maximum horizontal displacements for pillars on both two levels are 3 × 10−3 m. Roof sag occurs for pillars located on level #2 with maximum roof deflection of 2 × 10−3 m. While floor heave (i.e., 2 × 10−3 m) took place for pillars located on level #3 (i.e., Fig. 14c). The development of failure zones is depicted in Fig. 14d. Shear and tensile failure is detected entire the pillars located on level #3. Safety factor ranges from 2 to 3 (i.e., top and bottom pillars) and 1 to 2 in the middle of pillars (i.e., Fig. 14f). The exact value of safety factor lies between 1.33 to 1.55 (i.e., pillar strength is 2.33 MPa, pillar stress 1.5 to 1.75 MPa).

Width of mineshaft of level #3 is increases by 2 m—Case VII

Figure 15 shows the results when the mineshaft width is increased by 2 m. The area of stress concentration expands to the entire area of the pillars. Maximum principal stress varies between 4 and 4.5 MPa (i.e., Fig. 15a), while the minimum principal stress (i.e., Fig. 15b) ranges between 1.75 and 2 MPa for pillars located on level #3. The horizontal displacement is depicted as shown in Fig. 15c. The maximum horizontal displacements are 3 × 10−3 and 2.5 × 10−3 m for pillars located on level #2 and level #3 respectively. The pillars on both levels move towards left side (i.e., towards cavity) except for the top left and the middle pillars located on level #2 and level #3 respectively (i.e., left wall moves towards right). Figure 15 d shows the contours of vertical displacement. It can be seen that roof sag occurs for pillars located on level #2 (i.e., roof sag is 2 × 10−3 m). Floor heave occurs for the pillars located on level #3 (i.e., floor heave ranges from 1 × 10−3 to 2 × 10−3 m).

Fig. 15
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Numerical modelling analysis in terms of stress state, deformation, plastic zones and factor of safety—Case VII

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Shear plastic zones significantly spread entire the pillars located on level #3 with limited tensile fracture zones found on the boundaries of the pillars (i.e., as shown in Fig. 15e). Factor of safety contours range between 2 and 3 on the top and bottom with big red notches in the middle (i.e., safety factor varies between 1 and 2) of the pillars located on level #3. Alternatively, safety factor ranges between 1.16 and 1.33 (i.e., pillar strength is 2.33 MPa and pillar stress ranges between 1.75 and 2 MPa). Table 6 summarizes the findings of the effect of mineshaft width on the stability of pillars located on level #3.

Table 6 Summary of the effect of mineshaft width on the stability of pillars located on level #3 in terms of stress state, deformation, and safety factor
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Conclusion

The stability of pillars is the major concern in room-and-pillar method. This paper presents the findings of parametric stability analysis. Three factors have been examined, namely, mining sequence, mining depth, and mineshaft width, using finite difference code, FLAC2D. The results reveal that high stress concentration occurs around the corners and diagonally along the pillars located on level #3 in all three simulated mining scenarios. The highest stress concentration is found when mining upper half of level #2 followed by mining upper half of level #3 (i.e., maximum principal stress ranges between 3 and 4 MPa and minimum principal stress ranges between 1.5 and 1.75 MPa). Similarly, the minimum pillar safety reaches when entirely mining level #3 after complete mining of level #2. As mining continues, the sidewalls of the pillars shift towards left (i.e., towards mined cavity) with maximum horizontal displacement of 1 × 10−3 m. Also, roof sag (i.e., 5 × 10−4 m) and floor heave occurs to the pillars. Moreover, the majority of yielding zones occurs in shear failure in the left sidewalls of pillars compared with limited yielding zones occur in tensile failure. The safety factor ranges from1.33 to 1.55.

When mining has been conducted at depth of 15 m, the intensity of stress concentration increases around corners of pillars located on level #3 (i.e., maximum principal stress ranges between 4 and 4.5 MPa and minimum principal stress ranges between 1.75 and 2 MPa). As a result of mining activity, pillars tend to move towards left side, except for the middle pillar (i.e., located on level #3) and the top left pillar (i.e., located on level #2), their left walls move towards the entire cavity (i.e., shifted to the right). The maximum horizontal displacement is found to be 2 × 10−3 m. Roof sag and floor heave also occur with small amounts of 1 × 10−3 m and 2 × 10−3 m respectively. Plastic yielding zones are significantly developed particularly along the pillars. Results show that the pillars located on level #3 are entirely yielded in shear failure with some areas found to be vulnerable to tensile fracture on the pillar boundaries. The safety factor also deteriorates (i.e., FOS lies between 1.16 and 1.33) for all pillars on the two levels. This indicates extremely unstable condition. Based on these findings, it can be concluded that the pillar stability is ensured at mining depths of approximately 10 m, which is the current practice of the mine, and a safety risk can be posed if the mining depth increases.

When pillar width is reduced by 2 m (i.e., mineshaft width increases by 2 m), the stress concentration clearly spreads from pillar corners to the entire area of pillars. Maximum and minimum principal stresses range from 4 to 4.5 MPa and from 1.75 to 2 MPa respectively. As mining advances, pillars walls shifted towards left (i.e., towards mined rooms), except for left walls of the top left and middle pillars located on level #2 and level #3 respectively (i.e., left walls move towards entire cavity). The maximum horizontal displacements are 3 × 10−3 m and 2.5 × 10−3 m for pillars located on level #2 and level #3 respectively. The results also show that little roof sag (i.e., 1 × 10−3 m) is detected. Floor heave occurs with limited amount ranges from 1 × 10−3 to 2 × 10−3 m. The analysis shows that most of yielding zones occurs in shear and significantly spreads entire the pillars with limited tensile fracture zones found on the boundaries of the pillars. Factor of safety deteriorates and ranges between 1.16 and 1.33.

Recommendations

Probabilistic methods of analysis should be employed to tackle the intrinsic uncertainty associated with rock mass properties, in situ stress field, and geological features.