Evaluation of the strength of

slender pillars 

G.S. Esterhuizen 

Mining Engineer, National Institute for Occupational Safety and Health, Pittsburgh, Pennsylvania 

Abstract 

Pillars with width-to-height ratios of less than 1.0 are frequently created in underground hard rock mines. The 

strength of slender pillars can be estimated using empirically developed equations. However, the equations can 

provide variable results when the width-to-height ratios approach 0.5. This paper investigates some of the 

issues affecting pillar strength at low width-to-height ratios in hard brittle rock. The investigation includes an 

evaluation of empirical pillar strength data presented in the literature and observations of pillar performance 

in underground limestone mines in the eastern United States, supplemented by numerical modeling in which 

failure processes and sensitivity of slender pillars to variations in rock mass properties are evaluated. The 

results showed that the strength of slender pillars is more variable than that of wider pillars. The numerical 

model results demonstrated the increasing role of brittle rock failure in slender pillar strength. The absence of 

confinement in slender pillars can result in a fully brittle failure process, while wider pillars fail in a combined 

brittle and shearing mode. The onset of spalling in slender pillars occurs at or near the ultimate strength, 

while this is not the case for wider pillars. Slender pillars are shown to be more sensitive to the presence of 

discontinuities than wider pillars, which can partly explain the increased variability of slender pillar strength. 

Two examples are presented that illustrate failure initiation by brittle spalling and the sensitivity of slender 

pillars to the presence of discontinuities. 

Introduction 

The National Institute for Occupational Safety and Health's 

(NIOSH's) Pittsburgh Research Laboratory has embarked on a 

project to develop pillar design guidelines for underground 

limestone mines. A survey of mining methods and pillar and room 

dimensions in 70 underground limestone mines (Iannacchione, 

1999) showed that the room-and-pillar method was used in 69 of 

the 70 mines surveyed. The average depth of cover was 80 m (260 

ft), varying between 7 m (23 ft) and 610 m (2,000 ft). Pillars were 

typically square in plan view, but rectangular or rib pillars are also 

used. During initial development, the average pillar width-toheight (w:h) ratio was 1.73 but was reduced to 0.92 after bench 

mining of the floor. The minimum and maximum w:h ratio 

observed in the study was 0.4 and 3.13, respectively. Nine cases 

of pillar failure were identified, with all of them at width-toheight ratios of less than or equal to 1.5. Because floor benching 

is conducted in more than half of the limestone mines, pillars that 

were previously stable could become unstable when their widthto-height ratio is reduced during benching. NIOSH has, therefore, 

initially focused the project on the strength of slender pillars. 

The design of stable pillars requires that both the strength and 

loading of the pillars be known. In addition, an appropriate safety 

factor should be selected to ensure that the variability 

and uncertainty of the pillar strength and loading is accounted for. 

In the case of regular arrays of flat lying pillars, the load can be 

estimated by the tributary area method (Salamon and Munro, 

1967), or if the layout is more complex., estimates of average 

pillar loading can be obtained from numerical models (Brady and 

Brown, 1985). 

Pillar strength can be estimated from empirical equations that 

have been developed by observing both failed and stable pillar 

configurations. The pioneering work in this field was carried out 

for coal mine pillar design (Salamon and Munro, 1967; 

Bieniawski and van Heerden, 1975). Several empirically based 

pillar strength equations have since been developed for hard rock 

mines (Hedley and Grant, 1972; von Kimmelman et al. (1984); 

Lunder and Pakalnis, 1997). 

Analytical methods to estimate pillar strength have been 

developed, such as Wilson's confined core model (Wilson, 1972) 

and a similar model by Barron (1986). Although these methods 

have assisted in understanding pillar failure mechanics, they have 

not found wide acceptance as design tools in the mining industry. 

More recently, numerical models have found increasing use in 

pillar design (Mark, 1999). For example, Hoek and Brown (1980) 

used the results of elastic models to estimate the strength of 

pillars in various rock-mass classes. Martin and 
 

2.5

Maybee (2000) used elastic models to evaluate the effect of brittle 

failure on hard-rock pillars, while Lunder and Pakalnis (1997) 

used numerical model results to assist in developing an 

empirically based pillar strength equation for hard-rock mines. 

Numerical models have been used to assess geological effects 

such as through-going joints and weak floor on pillar strength 

(Gale, 1999; Iannacchione, 1999; Esterhuizen, 2000). Models that 

simulate rock fracture and elemental particulate behavior show 

promise in developing more realistic simulations of pillar failure 

mechanics (Diederichs, 2002; Elmo et al., 2005). The role of 

numerical modeling in pillar design is now well established and 

has assisted in developing new approaches to pillar stability 

assessment and design, such as the development of a semiempirical hazard-prediction system for pillars and stopes in a 

deep Canadian mine (Diederichs et al., 2002). 

Owing to the limited cases of pillar failure in underground 

limestone mines, purely empirical methods that rely on the study 

of pillar failures have limited application. NIOSH is, therefore, 

following an approach that combines empirical observations and 

numerical models to develop a pillar design methodology for 

underground limestone mines. This paper presents the results of 

an evaluation of slender pillar strength through a review of 

empirical pillar design methods for hard rock mines, observation 

of pillar performance in limestone mines and numerical model 

analysis of slender pillars. Two examples of slender pillar 

instability in limestone mines are presented and discussed. 

For the purpose of this paper, pillars with width-to-height 

ratios of less than 1.0 are called slender pillars. Pillar strength is 

defined as the peak load-bearing capacity per unit area of a pillar. 

A pillar is considered to be failed if it is compressed beyond its 

strength and sheds load. During underground observations it can 

be difficult to visually assess whether a pillar has failed or not, 

because rock failure might be observed around the perimeter of 

the pillar, but the pillar as a whole may not have reached its peak 

resistance. 

Slender pillars in empirical studies 

Case histories of pillar failure from a number of empirical' studies 

(Hedley and Grant, 1972; van Kimmelman et al., 1984; Potvin et 

al., 1989; Lunder, 1994) are summarized in Fig. 1, which 

represents the pillar strength as a function of the widthto-height 

ratio. The pillar strength is normalized by the uniaxial 

compressive strength (UCS) of the rock material. The graph also 

shows the upper and lower bounds of the failed cases. 

Figure 2— 

0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0 0.5 1 1.5 2

Width to height ratio 

Pillar strength/UCS 

Hedley&Grant(1972) 

Potvin et al.(1989) 

von Kimmelman et al.(1984) 

Lunder(1994) 

Figure 1—Pillar stability graph showing published case 

histories of failed pillars from hard rock metal mines. 

The empirical studies were all carried out at metal mines with 

good- to very good-quality rock masses (RMR 60-85). Pillar 

failure was determined by visual inspection in all cases, and pillar 

loads were estimated by the tributary area method or through 

numerical modeling. None of the failed pillars were affected by 

major structures such as faults, so that the pillar stability was 

reflective of the general rock mass behavior. It can be seen that 

slender 'pillars are well represented by the case histories. 

Variability of failure strength of case histories. Figure 1 clearly 

shows that, for the presented case histories, the pillar strength 

becomes highly variable as the width-to-height ratio decreases. 

The standard deviation of the strengths of slender pillars is 

25.4%, while it is 7.8% for the wider pillars. The variability can 

be caused by several factors, which can include uncertainty of the 

actual rock strength, uncertainty of the pillar stress, variations in 

the degree and severity of jointing, a variation in the bedding 

characteristics and the presence of weak bands in the pillars. 

The uncertainty and variability of pillar strength and loading is 

accounted for in pillar design by selecting an appropriate safety 

factor. The safety factor is the ratio of the average pillar strength 

to average pillar load. If pillar strength or loads are highly 

variable, a larger safety factor is required to account for the 

increased variability. The objective when selecting a safety factor 

is to limit the failure probability of the pillars to some acceptable 

level. For example, a safety factor of 1.6 is commonly used for 

pillar design in South Africa, achieving a failure probability of 

less than 0.5% (Wagner, 1992). 

The high variability of slender pillar strength, seen in the case 

history database, implies that slender pillars require a higher 

safety factor than wider pillars. 

Empirical equations and slender pillar strength. A review of 

the empirically developed pillar strength equations for hard rock 

mines reveals that the equations can be placed into three groups: 

•	 Power equations: Power equations such as the Hedley and 

Grant (1972) equation used in hard rock pillar design 

0.5 

S = k w	 (1)

h0.75 

where 

k is the strength of a unit cube of the rock material 

forming the pillar, 

w is the pillar width and 

h is the height of the pillar. 

This equation follows the form of the coal pillar strength 

equation developed by Salamon and Munro (1967). 

•	 Linear equations: Linear equations, such as the equation 

originally proposed by Obert and Duvall (1967) based on 

laboratory tests on rock samples 

 w S =σ p (0.778 + 0.222 ) (2) 

h 

where 

σ'p is the strength of a pillar with a width-to-height ratio of 

1.0. 
•	 Equation based on pillar confinement: An equation based 

on pillar confinement was developed by Lunder and 

Pakalnis (1997) 

S=(K*UCS)(C1 +kCz) (3) 

where 

k is a pillar friction term, 

C 1 and C 2 are empirically derived constants 

determined to be 0.68 and 0.52, respectively, and 

K is the rock mass strength size factor, determined to 

be 0.44. 

The value of x can be determined by 

−1 ⎛ 1 − Cpav ⎞

κ = tan ⎢

⎡

cos ⎜

⎜ ⎟⎟⎥

⎤ (4)

1 + Cpav ⎣ ⎝ ⎠⎦

where 

Cpav is the average pillar confinement, which can be 

found by 

Wp / h ⎡ ⎛W ⎞⎤ p	 Cpav (5) = 0.46⎢log⎜

⎜ + 0.75⎟⎟⎥ h ⎣ ⎝ ⎠⎦

1.4 

where
 

Wp is the pillar width and 

h is the pillar height. 

These three forms of equations were compared by entering 

similar rock strength parameters in each. This was achieved by 

setting the large-scale strength of the rock mass (k) equal to 

0.42 times the DCS in the Hedley-Grant equation and 

similarly setting the value of 0"£ in the Obert-Duval equation. 

The result is shown in Fig. 2. Comparing the three curves 

shows that the Hedely-Grant and the Lunder-Pakalnis 

equations predict similar pillar strengths when the w:h ratio 

exceeds 0.6, but they diverge at lower w:h ratios. 

Pillar strength/UCS 0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

Lunder-Pakalnis 

Hedley-Grant 

Obert-Duval 

0.0 0.5 1.0 1.5 2.0 

Width to height ratio 

Figure 2—Comparison of pillar strength equations. 

Interestingly, the Lunder-Pakalnis equation predicts 

constant pillar strength below w:h ratios of 0.4. The ObertDuval equation is linear and predicts higher strength for 

slender pillars than the other two equations. For example, the 

Obert-Duval equation predicts a strength of 0.38 times the 

DCS for a pillar with a w:h ratio of 0.5, while the HedleyGrant equation predicts 0.30 and the Lunder-Pakalnis equation 

predicts 0.31. There is a difference of 26% between the 

highest and the lowest predictions. 

The review shows that the three forms of pillar strength 

equations considered will result in significantly different 

estimates of slender pillar strength for the same rock mass 

strength data. The equations also predict different trends in 

strength, especially at low width to height ratios. When 

designing slender pillars, the selection of a strength equation 

can, therefore, have a significant impact on the resulting 

dimensions of slender pillars. The numerical modeling 

discussed in this paper was carried out partly to evaluate pillar 

strength issues at low w:h ratios. 

Pillar failure in hard rock mines 

Pillar failure modes in hard rock mines can be divided into 

two categories (Iannacchione, 1999). The first category is 

failure of the rock mass, in which spalling or crushing occurs 

through the intact rock and shearing occurs along natural joint 

planes in the rock. This failure mode is progressive and can be 

described in the following stages, after Krauland and Soder 

(1987): 

•	 Stage 1: slight spalling of pillar corners and walls; 

•	 Stage 2: severe spalling; 

•	 Stage 3: the appearance of fractures in the central part of 

the pillar; 

•	 Stage 4: the occurrence of rock falls from the pillar, 

emergence of an hourglass shape; and 

•	 Stage 5: the disintegration of the pillar or, alternatively, 

the formation of a well-developed hourglass with the 

central parts completely crushed. 

In this category of failure, brittle spalling occurs initially 

through the intact rock, followed by shearing and crushing of 

the rock mass. The initial brittle failure appears to be independent of the natural joints and bedding planes in strong rock 

(Diederichs et al., 2002; Diederichs, 2002). 

The second category of failure is structure controlled, 

where shearing occurs along an individual geological structure 

such as a through-going joint or fault. Other modes of 

structural failure can occur when weak bedding layers or soft 

joint fill exists in a pillar that can extrude and destroy the 

pillar by inducing tension in the surrounding rock. Sliding 

along weak roof or floor contacts can induce similar failure 

modes in a pillar and is classified under the structural failure 

mode. 

Assessment of slender pillar strength using 

numerical models 

Numerical models were used to further investigate the 

strength of slender pillars and to address some of the issues 

related to the pillar strength equations. The FLAC3D (Anon., 

2002) finite difference software was used to conduct the 

modeling. The software has the capability to model elastic and 

strain softening behavior using an elasto-plastic constitutive 

law. Important to this project was for the models to replicate 

realistically the failure processes observed in hard rock pillars. 

Modeling brittle rock mass failure. It is important that the 

two-stage process of brittle spalling followed by shearing 

should be replicated in the numerical models. The phenomenon of brittle spalling has received much attention in the rock 

mechanics literature in recent years (Kaiser et al. 2000; Martin 

and Maybee, 2000, Rojat et al., 2003). It has been found that 

the onset of brittle spalling typically occurs at 0.3 to 0.5 the 

uniaxial compressive strength of the rock, which is the stress 

level required for crack initiation. Stacey and Yathavan (2003) 
 

presented a number of cases in which brittle spalling occurred 

below 0.2 times the DCS and possibly as low as 0.04 in one case. 

The brittle cracks typically extend and develop into fractures that 

are parallel to the major principal stress. According to Kaiser et al. 

(2000), at low confinement, stresses crack. dilation inhibits the 

mobilization of frictional resistance until the rock is sufficiently 

damaged. They proposed a bilinear strength envelope for rock 

around underground openings in which the strength at low 

confinement is independent of friction and is equal to 0.3 to 0.5 

times the DCS, followed by friction 

hardening at higher confinement, increasing up to the strength 

predicted by the Hoek-Brown (1997) or similar rock strength 

criteria. The change from brittle spalling to frictional resistance 

occurs at a ratio of the maximum to minimum principal stress of 

10 to 20, which depends on the heterogeneity and jointing in the 

rock mass. 

The FLAC3D software has a built-in constitutive model for 

bilinear rock strength based on the Mohr-Coulomb strength 

criterion, in which strain hardening or softening is a function of 

the deviatoric plastic strain (Anon., 2002). This model can 

include ubiquitous joints that can be used to evaluate the effect of 

through-going joint sets on rock mass strength. The bilinear 

model is well suited to simulate the brittle/frictional development 

of rock mass strength as a function of confining stress. The initial 

brittle strength was based on the assumptions that spalling 

initiates at 0.33 times the DCS, and the transition from brittle to 

frictional strength occurs when the ratio of maximum to 

minimum principal stress (σ1/σ3) is 20.0. For the brittle section of 

the strength curve, the friction value was set to zero, after Martin 

and Maybee (2000). The parameters for the fully developed 

frictional rock-mass strength were based on the Hoek-Brown 

(1997) criterion by approximating the predicted rock mass 

strength with appropriate Mohr-Coulomb parameters. 

Figure 3 shows the Hoek-Brown (1997) strength curve and the 

approximate bilinear strength curve for a rock mass rating (RMR) 

of 70, a DCS of 120 MPa (17,400 psi) and Hoek-Brown (l997) 

m-parameter of 12.0, which simulates a good-quality rock mass. 

The strain softening parameters for the models were determined as part of the model-calibration process because they are 

affected by model element size (Anon., 2002). All the models 

were run using identical element sizes. 

σ1 (MPa) 

140 

120 

100 

80 

60 

40 

20 

0 

Bilinear 

Hoek-Brown 

Spalling limit 

0 5 10 15 20 

σ3(MPa) 

Figure 3—Bilinear and Hoek-Brown (1980) rock-strength 

plots for a rock mass rating of 70.0. 

Modeling of structure controlled failure. The effect of 

through-going joints was modeled using the ubiquitous joint 

facility of the bilinear constitutive model in FLAC3D. The 

software allows joint sets to be defined in each model element 

having a specific orientation and Coulomb strength parameters. 

During the analysis of the effect of structure controlled failure, 

the rock mass maintained its brittle characteristics through the 

bilinear constitutive model. 

Model geometry and loading conditions. The models were 

set up to simulate a single pillar with the adjacent roof and floor 

rocks, as shown in Fig. 4. Both the pillar width and room width 

were set to 12 m, resulting in 75% extraction. The height of the 

pillar was varied to simulate different width-to-height ratios. 

Vertical symmetry planes were defined to coincide with the 

vertical sides of the model, simulating a repeating system of 

rooms and pillars. Owing to symmetry, only half of the width of 

the rooms was included in the models. The floor of the models 

was fixed in the vertical direction. The top surface of the model 

was subject to an applied downward velocity that simulated 

crushing of the pillar under increased compression. The applied 

velocity was subject to servo control to maintain the unbalanced 

forces in the model within acceptable levels (Anon., 2002). 

 Symmetry planes 

Applied velocity under 

servo control 

Pillar

 Fixed floor 

12m 

132m 

Figure 4—Model used to simulate a pillar in FLAC3D. 

The models were run to equilibrium under elastic conditions 

subject to a vertical field stress of 2.7 MPa (390 psi), simulating a 

mine at 100 m (328 ft) depth. The horizontal stress was also set at 

2.7 MPa (390 psi). After reaching equilibrium in the elastic state, 

the pillar material was changed from elastic to the bilinear MohrCoulomb material type. The model was then subject to increasing 

vertical loading by applying the servo-controlled 
velocities at the top of the model. The models were compressed 

until the pillar had completely failed and had reached a residual 

strength of less than 50% of the peak strength. 

During the simulations the average vertical stress at midheight of the pillar was calculated at regular intervals. The peak 

value of this stress was considered to represent the pillar strength. 

In addition, the closure between the top and bottom of the pillar 

was recorded, so that a pillar stress-strain curve could be 

developed. A routine was developed using the internal 

programming language available in FLAC3D, which recorded 

whether failure of an element occurred during the initial brittle 

stage or the shearing stage of the strength curve. 

Model calibration and testing. Model calibration was carried 

out by simulating pillars with w:h ratios of 0.3,0.4,0.66,0.8, 1.0, 

1.5 and 2.0 and comparing the results to the Lunder-Pakalnis 

(1997), empirically developed pillar strength equation. The 

models were all set up to simulate a good-quality rock mass with 

an RMR value of 70. This value of RMR is in the center of the 

range of RMR values of 60 to 80 reported for the case histories 

used by Lunder and Pakalnis (1997) to develop the strength 

equation. Details of the input data for this model are presented in 

Table l. 

Table 1—Input Parameters for the RMR=70 model 

Parameter Value 

Elastic modulus 70 GPa (1x107 psi) 

Poisson ratio 0.2 

Intact rock strength (UCS) 120 MPa (17,400 psi) 

First stage (brittle) cohesion 20 MPa (2,900 psi) 

First stage (brittle) friction angle 0o 

Second stage cohesion 6.5 MPa (940 psi) 

Second stage friction angle 42.7o 

Tensile strength 7 MPa (1,000 psi) 

Dilation angle 30o 

The calibration was carried out by varying the rate of cohesion 

softening in the models and keeping all the other parameters 

constant. Figure 5 shows the final result of the calibration runs. 

As shown, the model results predict a flattening of the strength 

curve at low w:h ratios similar to the Lunder-Pakalnis (1997) 

curve. 

0.7 

0.6 Lunder-Pakalnis 

Model results 

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 

Width to height ratio 

0.5 

Pillar strength/UCS 0.4 

0.3 

0.2 

0.1 

0.0 

Figure 5—Results of model calibration against the LunderPakalnis (1997) empirically derived pillar-strength equation. 

The sensitivity of the models to the rock strength parameters 

was tested by varying the rock-mass strength parameters to 

simulate RMR values of 60 to 80. This was achieved by modifying the uniaxial compressive strength as well as the cohesion 

and friction values in accordance with the Hoek - Brown (1997) 

strength criterion. The spalling limit was maintained at 30% of 

the DCS in all the models. The results are presented in Fig. 6, 

which shows that a reduction in the RMR to 60 does not have a 

significant effect on the pillar strength, while an increase to RMR 

= 80 results in a rapid increase in the strength of wider pillars. In 

all cases, the strength of pillars with w:h of 0.8 and less was equal 

to the brittle strength of the rock. All the model runs described 

below were carried out using the rock mass strength parameters 

for an RMR value of 70, as shown in Table 1. 

1.2 

1.1 

RMR=80 

RMR=75 

RMR=70 

RMR=60 

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 

Width to height ratio 

1.0 

0.9

Pillar strength/UCS 0.8 

0.7 

0.6 

0.5 

0.4 

0.3

0.2 

0.1 

0.0 

Figure 6—Effect of rock mass rating (RMR) on pillar 

strength determined from numerical model results. 

Pillar failure modes derived from model results. Inspection of the 

extent of brittle failure and shearing failure in the models showed 

that the pillars with w:h ratios of 0.8 and below fail in the brittle 

mode, owing to the absence of sufficient confinement in these 

pillars to mobilize the frictional component of the rock strength. 

This explains the flattening of the pillar strength curve to the 

brittle rock strength seen in Fig. 6. The extent of brittle and shear 

failure in pillars with w:h ratios of 0.5, 1.0 and 2.0 are presented 

in Fig. 7, illustrating the increasing role of brittle failure as the 

w:h ratio decreases. 

Failure of the wider model pillars initiates by brittle failure 

around the outside of the pillar, which commences when the 

stress in the outer skin of the pillar exceeds the brittle rock 

strength. The brittle failure process continues as the pillar load 

increases. As the pillar approaches its peak strength, shear failure 

starts to develop behind the brittle failure zone. The pillar load 

can start to decrease before shear failure has progressed to the 

pillar core. This type of behavior is similar to the results of 

compression tests on small coal pillars reported by Wagner 

(1974). 

Slender pillars with w: h ratios of 0.8 and less also start to fail 

by brittle spalling when the average pillar stress approaches the 

brittle rock strength. However, a small increase in load results in 

failure of the entire pillar followed by rapid load shedding. In 

these slender models, brittle failure did not always com- 
W:H=0.5 

Brittle failure 

Shear failure 

Figure 7—Sections through the center of pillars with different 

width to height ratios showing the extent of brittle and shear 

failure of the rock mass predicted by numerical modeling. 

W:H=2.0

W:H=1.0

W:H=2.0 

W:H=1.0 

mence at the outer skin of the pillar, but could start near the 

pillar center. According to the model results, pillars with w:h 

ratios of 0.8 or less will be at or near their ultimate strength 

when they start to show signs of brittle failure. 

Model results of the effect of inclined discontinuities. The 

strength parameters used in the models discussed above are 

based on the assumption that the rock-mass strength is 

isotropic, implying that the discontinuity orientations and 

spacings are also isotropic. In practice, one of the 

discontinuity sets can be dominant and will result in 

anisotropic strength in the rock mass. To investigate the effect 

of a single dominant discontinuity set on pillar strength, the 

ubiquitous joint facility in FLAC3D was used, A single 

discontinuity set, striking parallel to one of the pillar sides, 

was introduced into the pillar models. The discontinuity dip 

was varied from 500 to 700 in each model. The discontinuity 

strength was selected to simulate rough joints with unaltered 

joint walls that are continuous relative to the pillar 

dimensions. The Coulomb parameters used for these 

discontinuities were determined using the approach of Barton 

and Choubey (1997). The strength parameters were Cohesion 

= 1.2 MPa (170 psi) and friction angle = 42°. 

The results are summarized in Fig. 8, which shows that the 

presence of the inclined discontinuities can have a significant 

effect on the strength of slender pillars, while the wider pillars 

are affected to a much lesser degree. For example, 

discontinuities dipping at 70° reduce the strength of a pillar 

with w:h ratio of 2.0 by 13%, while the strength of a pillar 

with a w:h ratio of 0.5 is reduced by 62%. 

Pillar strength/UCS 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

Isotropic rock m ass 

Joint d ip 50 degrees 

Joint d ip 60 degrees 

Joint d ip 70 degrees 

A verage joint effect 

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 

W idth to height ratio 

Figure 8—Numerical model results of pillar strength vs. widthto-height ratio, showing the effect of discontinuities inclined at 

50° to 70° on pillar strength. 

Examples of slender pillar performance 

Observations of pillars in- underground limestone mines have 

revealed most of the characteristics of slender pillar failure 

described above. Presented below are two examples, one 

presents brittle spalling at low stress and the other shows the 

effect of through-going discontinuities. 

Example of brittle spalling. Brittle spalling and hourglass 

formation at relatively low stress was observed at a mine in 

northern Tennessee that uses the room-and-pillar method. In 

the area of concern, the pillars were square with side 

dimensions varying from 12.2 to 15.2 m (40 to 50 it) and were 

developed about 15.8 m (52 ft) high. Benching was partially 

carried out, which increased the pillar height to 21 m (70 ft). 

The room width was measured to be 16.4 m (53 ft), and the 

depth of cover was 140 m (464 it). 

The limestone is a strong rock mass with a DCS of 150 

MPa (22,000 psi). Jointing is near vertical with an average 

spacing of about 0.5 m (1.6 it). Joint surfaces are rough, and 

the joint continuity is less than 3 m (10 ft), Bedding joints are 

poorly developed and did not appear to affect the pillar 

stability. 

The pillars were about 15 years old and were reported to be 

progressively spalling to an hourglass shape, as shown in Fig. 

9. Based on visual observations, it is not certain whether these 

pillars had failed. Inspection of the pillars revealed that open 

vertical fractures or joints could be seen in the pillar ribs. 

Columnar fragments of rock about 2 m (6.6 ft) long were scattered about the pillars, as seen in the foreground. The average 

pillar stress, calculated by the tributary area method, is 15 

MPa (2,175 psi), which is only 10% of the DCS of the intact 

rock. This is at the lower end of the range of observed cases of 

brittle spalling. The presence of near-vertical open fractures 

and joints seems to confirm that a brittle failure process is taking place in these pillars. 

Example of the effect of inclined discontinuities. The 

second case is a limestone mine in western Pennsylvania that 

uses the room-and-pillar method of mining. The limestone is 

massive and is fine to medium grained with cross bedding. 

Jointing is spaced at 0,4 to 2.0 m (1.3 to 6.6 ft), and the joint 

trace length is seldom more than 3 m (10 ft). Joint surfaces are 

rough and do not contain any fill material. The bedding joints 

are poorly developed. Occasional prominent discontinuities 

with variable dip exist within the limestone formation. The 

DCS of this very strong limestone has been found to be up to 

265 MPa (38,420 psi). 

The pillars are square, 10.4 m (34 it) wide and 8.2 m (27 ft) 

high on development. Room width was 13.4 to 14.6 m (44 to 

48 ft). Benching was carried out, increasing the pillar height 

to 18.6 m (61 ft), which reduced the width-to-height ratio 

from 1.3 to 0.56. The depth of cover was approximately 90 m 

(300 it). Several of the benched and partially benched pillars 

in this layout failed, while the development pillars are in good 

condition. Figure 10 shows one of the failed pillars at the edge 

of the benching operation. The pillar failed along two 

prominent discontinuities. The photograph was taken from the 

upper mining bench and does not show the full height of the 
 

 

Figure 9—A pillar with a width to height ratio of 0.77 

showing the effect of brittle failure and spalling. Open vertical 

fractures/joints are visible in the pillar. 

Figure 10—Partially benched pillar that failed along two 

prominent discontinuities dipping at approximately 600

. 

benched side of the pillar. The stress at failure of this pillar is 

estimated to be 14.4 MPa (2,080 psi) based on tributary area 

loading. However, using the Lunder-Pakalnis (1997) equation for 

pillar strength and a conservative value of the DCS at 200 MPa 

(29,000 psi), the benched pillars are predicted to have a strength 

of 64 MPa (9,280 psi), and one would not expect failure to occur. 

The failure can, however, be explained by the weakening 

effect of the prominent discontinuities observed in the pillar. The 

discontinuities could have significantly reduced the strength of 

the benched pillar while having only a minor effect before 

benching, as predicted by the numerical models. This example 

demonstrates the importance of considering the potential effect of 

prominent discontinuities when designing slender pillars. 

Conclusions 

This evaluation of the strength of slender pillars has revealed the 

following: 

•	 Empirical studies show that the strength of slender pillars 

is more variable than the strength of wider pillars. The 

increased variability implies that higher safety factors are 

required when designing slender pillars to account for the 

variability. 

•	 Pillar-strength equations developed from empirical studies 

can predict significantly different strengths for slender 

pillars, even if identical rock strength values are used. 

•	 Numerical models revealed that the process of brittle 

spalling and failure at low confinement plays an important 

role in the strength of slender pillars. The absence of a 

confined core causes failure to occur at the relatively low 

brittle strength of the rock. 

•	 Numerical model results show that, for slender pillars, the 

difference between the pillar load at the onset of brittle 

spalling and the ultimate pillar strength can be small, 

implying that slender pillars are at or near the point of 

failure when they start to spall. This is not the case with 

wider pillars, where the ultimate strength can be much 

higher than the load required to initiate brittle spalling. 

•	 Slender pillars are more sensitive to the presence of 

inclined discontinuities than wider pillars. Numerical 

models showed that relatively strong, inclined discontinuities can reduce the strength of slender pillars by as 

much as 70%, while wider pillars are affected to a much 

lesser degree. This sensitivity can partly explain the large 

variability in slender pillar strength seen in the results of 

published empirical pillar strength studies. 

•	 The onset of brittle failure at relatively low stress and the 

significant reduction of slender pillar strength by prominent discontinuities have been observed in underground 

limestone mines. 

Acknowledgments 

The work presented in this paper forms part of the research 

program of the National Institute for Occupational Safety and 

Health in ground control and safety in underground mines. The 

contribution of Mr. John Ellenberger, who assisted in conducting 

the FLAC3D analyses, is gratefully acknowledged. 

References 

Anon .• 2002, "FLAC 3D, Fast Lagrangian Analysis of Continua," User's Manual Version 

2.1, Itasca Consulting Group, Minneapolis, Minnesota. 

Barron, K. 1986. "A new method for coal pillar design" in Conference on Ground 

Movement and Control Related To Coal Mining, N. Aziz, ed., Aus. I.M.M. 

Melbourne, pp. 118-124. 

Barton, N., and Choubey, v.. 1977. "The shear strength of rock joints in theory and 

practice," Rock Mechanics, Vol. 1 0, pp. 1-54. 

Brady, B.H.G .. and Brown, E.T., 1985, Rock Mechanics for Underground Mining, 

George Allen and Unwin, London, 527 pp. 

Bieniawski, Z.T., and van Heerden, W.L., 1975, "The significance of in-situ tests on large 

rock specimens," Int. J. Rock Mech. & Min. Sci. & Geomech., Abstr., Vol. 12, pp. 

101-103. 
Diederichs, M.S., 2002, "Stress induced damage accumulation and implications for 

hard rock engineering," NARMS-TAC 2002, Hammah et aI., eds., University of 

Toronto, pp. 3-12. 

Diederichs, M.S., Coulson,A, Falmagne, V., Rizkalla, N., and Simser, B., 2002, 

"Application of rock damage limits to pillar analysis at Brunswick Mine," in 

NARMS-TAC 2002, Hammah et aI., eds., University of Toronto, pp. 1325­

1332. 

Elmo, D., Coggan, J.S., and Pine, R.J., 2005, "Characterization of rock mass 

strength using a combination of discontinuity mapping and fracture mechanics 

modeling," in 40th US Rock Mechanics Symposium, American Rock 

Mechanics Association, Anchorage, Alaska, Paper 05-733. 

Esterhuizen, G.S., 2000, "Jointing effects on pillar strength," in 19th International 

Conf. on Ground Control in Mining, Morgantown, West Virginia, University of 

West Virginia, pp. 286-290. 

Gale, W.J., 1999, "Experience of field measurement and computer simulation 

methods of pillar design," Second International Workshop on Coal Pillar 

Mechanics and Design, National Institute for Occupational Safety and Health, 

IC 9448, pp. 49-61. 

Hedley, D.G.F., and Grant, F., 1972, "Stope and pillar design for the Elliot lake 

uranium mines," Can. Inst. Min. Metall. Bull,. Vol. 65, pp. 37-44. 

Hoek, E. and Brown, E.T., 1980, Underground Excavations in Rock, Inst. Min. 

Metall., London. 

Hoek, E., and Brown, E.T., 1997, "Practical estimates of rock mass strength," Int. J. 

Rock Mech. & Min. Sci. & Geomech., Abstr., Vol. 34, pp. 1165-1186. 

lannacchione, AT., 1999, "Analysis of pillar design practices and techniques for 

U.S. limestone mines," Trans. Instn. Min. Metall. (sect. A: Min. lndustry, 108, 

September-December pp. A 152-A 160. 

Kaiser, P.K., Diederichs, M.S., Martin, D.C., and Steiner, w., 2000, "Underground 

works in hard rock tunneling and mining," Keynote lecture, Geoeng2000, 

Melbourne, Australia, Technomic Publishing Co., pp. 841-926. 

Krauland, N., and Soder, P.E., 1987, "Determining pillar strength from pillar failure 

observations," Eng. Min. Journal, Vol. 8, pp. 34-40. 

Lunder, P.J., 1994, "Hard Rock Pillar Strength Estimation an Applied Approach," 

MASc. Thesis, Dept. Mining and Mineral Process Engineering, University of 

British Columbia. 

Lunder, P.J., and Pakalnis, R., 1997, "Determining the strength of hard rock mine 

pillars," Bull. Can. Inst. Min. Metall., Vol. 90, pp. 51-55. 

Mark, C., 1999, "Introduction to the proceedings," Second International Workshop 

on Coal Pillar Mechanics and Design, National Institute for Occupational 

Safety and Health, IC 9448, pp. 2-4. 

Martin, C.D., and Maybee, W.G., 2000, "The strength of hard rock pillars," Int. 

Jnl. Rock Mechanics and Min. Sci., Vol. 37, pp. 1239-1246. 

Obert, l., and Duvall, W.I., 1967, "Rock mechanics and the design of structures," in 

Rock, Wiley, New York. 

Potvin, Y, Hudyma, M., and Miller, H.D.S., 1989, "Rib pillar design in open stoping," 

Can. Inst. Min. Metall. Bull. Vol. 82, pp. 31-36. 

Rojat, F, Labouise, V., Descoeudre, F., and Kaiser, P.K., 2003, "Lötschberg base 

tunnel: brittle failure phenomena encountered during excavation of the steg 

lateral adit," in 70th ISRM Congress, South African Inst. Min Metall., pp. 927­

933. 

Salamon, M.D.G., and Munro, AH., 1967, "A study of the strength of coal pillars," J. 

South African Inst. Min. Metall., Vol. 68, pp. 55-67. 

Stacey, T.R., and Yathavan, K., 2003, "Examples of fracturing of rock at very low 

stress levels," ISRM 2003 Technology Roadmap for Rock Mechanics, S. Afr. 

Inst. Min. Metall., pp. 1155-1159. 

Von Kimmelman, M.R., Hyde, B., and Madgwick, R.J., 1984, "The use of computer 

applications at BCL Limited in planning pillar extraction and the design of 

mining layouts," In Design and Performance of Underground Excavations, Int. 

Soc. Rock Mech. Symposium, Brown and Hudson, eds., Brit. Geotech. Soc., 

London, pp. 53-63. 

Wagner, H., 1974, "Determination of the complete load-deformation characteristics 

of coal pillars," in 3rd Congress, Int. Soc. Rock Mech., Denver, Colorado, VII, 

Part B, pp. 1067-1081. 

Wagner, H., 1992, "Pillar design in South African collieries," in Workshop on Coal
 

Pillar Mechanics and Design, U.S. Bureau of Mines, IC 9315, pp. 283-301.
 

Wilson, AH., 1972, "An hypothesis concerning pillar stability," The Mining Engineer, 

Vol. 131, pp. 409-417.