1 

AN EVALUATION OF THE STRENGTH OF SLENDER PILLARS 

G. S. Esterhuizen, NIOSH, Pittsburgh, PA 

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 widthto-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, which illustrate failure 

initiation by brittle spalling and the sensitivity of slender pillars to the 

presence of discontinuities. 

Introduction 

 The National Institute of Occupational Safety and Health (NIOSH), 

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 [1] showed that the room and pillar method was used 

in 69 of the 70 mines surveyed. The average depth of cover was 80 

m, 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-to-height (w:h) ratio 

was 1.73 but 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, all of 

them at width-to-height ratios of less or equal to 1.5. Since 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 [2], or if the layout is more complex, estimates of 

average pillar loading can be obtained from numerical models [3]. 

 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 [2, 4]. Several empirically based pillar strength 

equations have since been developed for hard rock mines [5, 7, 9]. 

 Analytical methods to estimate pillar strength have been 

developed, such as Wilson’s confined core model [10] and a similar 

model by Barron [11]. 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 [12]. For example, Hoek and Brown [12] used the results of 

elastic models to estimate the strength of pillars in various rock mass 

classes. Martin and Maybee [14] used elastic models to evaluate the 

effect of brittle failure on hard rock pillars, while Lunder and Pakalnis 

[9] 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 [1, 16, 15]. Models that 

simulate rock fracture and elemental particulate behavior show 

promise in developing more realistic simulations of pillar failure 

mechanics [17, 20]. 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 

semi-empirical hazard prediction system for pillars and stopes in a 

deep Canadian mine [20]. 

 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 

which 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 will be 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, since 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 [5, 

6, 7, 8] are summarized in Figure 1, which represents the pillar 

strength as a function of the width-to-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. 

 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 the cases, and pillar loads were 

estimated by the tributary area method [2] 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. 
 2 

Variability of Failure Strength of Case Histories 

)222.0778.0(

h

w S =σ p + 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 development pillar design in South Africa, 

achieving a failure probability of less than 0.5% [23]. 

 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: 

1. Power equations: the Hedley and Grant [5] equation is an 

example of a power equation used in hard rock pillar design: 

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 [2]. 

2. Linear equations: such as the equation originally proposed by 

Obert and Duvall [24] based on laboratory tests on rock 

samples: 

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

of 1.0. 

 3. An equation based on pillar confinement developed by Lunder 

and Pakalnis [9]: 

 where κ (kappa) is a pillar friction term, C1 and C2 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 κ can be determined as: 

where Cpav is the average pillar confinement, which can be 

found by: 

 where Wp is the pillar width and h 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 UCS 

in the Hedley–Grant [5] equation and similarly setting the value of κ

in the Obert-Duval [24] equation. The result is shown in Figure 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. Interestingly, 

the Lunder-Pakalnis equation predicts constant pillar strength below 

w:h ratios of 0.4. The Obert-Duval 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 UCS for a pillar with a w:h ratio of 0.5, while the Hedley-Grant 

75.0

5.0

h

w = kS

))(*( = 1 + κ CCUCSKS 2

⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

− = −

Cpav

Cpav

1

1

costan 1 κ

hW p p

h

W

Cpav /

4.1

75.0log46.0 ⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ = +

Figure 1. Pillar stability graph showing published case

histories of failed pillars from hard rock metal mines 

0

0.1

0.2

0.3

0.4

0.5

0 0.511.522.5

Width to height ratio

Pillar strength/UC

0.6

0.7

0.8

S

Hedley&Grant[5]

Potvin et al.[6]

von Kimmelman et al.[7]

Lunder[8]

Width to height ratio

0.0 0.5 1.0 1.5 2.0

Pillar strength/UCS 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Lunder-Pakalnis

Hedley-Grant

Obert-Duval

Figure 2. Comparison of pillar strength equations. 
 3 

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 [1]. The first category is failure of the rock mass, in which 

spalling or crushing occurs through the intact rock as well as shearing 

along natural joint planes in the rock. This failure mode is progressive 

and can be described in the following stages, after Krauland and 

Soder [21]: 1) Slight spalling of pillar corners and walls; 2) Severe 

spalling; 3) Appearance of fractures in the central part of the pillar; 4) 

Occurrence of rock falls from the pillar, emergence of an hour-glass 

shape; 5) Disintegration of the pillar, or, alternatively, the formation of 

a well developed hour glass 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 [26, 17]. 

 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 which 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 [25] 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 [27, 28, 17]. It has been found 

that the onset of brittle spalling typically occurs at 0.3-0.5 the uniaxial 

compressive strength of the rock, which is the stress level required for 

crack initiation. Stacey [29] presented a number of cases in which 

brittle spalling occurred below 0.2 times the UCS 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. [28], 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 0.3-0.5 times the UCS, followed 

by friction hardening at higher confinement, increasing up to the 

strength predicted by the Hoek-Brown [22] 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 [25]. This model can include ubiquitous joints, which 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 UCS, and the 

transition from brittle to frictional strength occurs at 1/ 3 = 20.0. For the 

brittle section of the strength curve, the friction value was set to zero, 

after Martin et al. [27]. The parameters for the fully developed 

frictional rock mass strength were based on the Hoek-Brown [12] 

criterion by approximating the predicted rock mass strength with 

appropriate Mohr-Coulomb parameters. 

 Figure 3 shows the Hoek-Brown [12] strength curve and the 

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

70, a UCS of 120 MPa (17,400 psi), and Hoek-Brown [12] mparameter of 12.0, to simulate a good quality rock mass. 

σ3(MPa)

0 5 10 15 20

σ1 (MPa) 0

20

40

60

80

100

120

140

Bilinear

Hoek-Brown

Spalling limit 

Figure 3. Bilinear and Hoek-Brown[13] rock strength plots for a 

Rock Mass Rating of 70.0 

 The strain softening parameters for the models were determined as 

part of the model calibration process because they are affected by 

model element size [25]. All the models were run using identical 

element sizes. 

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 Figure 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 one half of the width of the rooms was 

included in the models. The floor of the models was fixed in the 
 4

vertical direction. The top surface of the model was subject to an 

applied downward velocity which 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 [25]. 

Applied velocity under 

servo control

 Symmetry planes 

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 Mohr-Coulomb 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 mid height 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 [9], 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 [9] to develop the strength equation. 

Details of the input data for this model are presented in Table 1. 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 [9] curve. 

 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 [12] strength criterion. The spalling 

limit was maintained at 30% of the UCS in all the models. The results 

are presented in figure 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. 

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. 

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 

Width to height ratio

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Pillar strength/UCS 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Lunder-Pakalnis

Model results

Figure 5. Results of model calibration against the LunderPakalnis [9] empirically derived pillar strength equation. 
 5

This explains the flattening of the pillar strength curve to the brittle rock 

strength seen in figure 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 figure 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 [29]. 

 Slender pillars with w:h ratios of 0.8 and below 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 commence 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 so far 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 50° to 70° 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 [31]. The 

strength parameters were Cohesion = 1.2 MPa (170 psi) and Friction 

angle = 42o. 

Width to height ratio

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Pillar strength/UCS 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

RMR=80

RMR=75

RMR=70

RMR=60

 The results are summarized in Figure 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 70o 

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%. 

Figure 6. Effect of Rock Mass Rating (RMR) on pillar 

strength determined from numerical model results. 

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 which presents brittle 

spalling at low stress, and the other which shows the effect of throughgoing 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 between about 12.2 and 15.2 m (40-50 ft) 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 

Figure 7.

 Sections through the center of pillars with d

width to height ratios showing the extent of brittle and shear 

failure of the rock mass predicted by numerical modeling. 

ifferent 

W:H=0.5

W:H=2.0

W:H=1.0

Brittle failure

Shear failure

W idth to height ratio

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Pillar strength/UCS 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Isotropic rock mass

Joint dip 50 degrees

Joint dip 60 degrees

Joint dip 70 degrees

Average joint effect

Figure 8. Numerical model results of pillar strength versus width 

to height ratio, showing the effect of discontinuities inclined at 

50 to 70 degrees on pillar strength. 
 6 

width was measured to be 16.4 m (53 ft), and the depth of cover is 140 

m (464 ft). 

 The limestone is a strong rock mass with a UCS of 150 MPa 

(22,000 psi). Jointing is near vertical with an average spacing of about 

50 cm (1.6 ft). 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 the current hourglass shape, as shown in 

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

pillars have failed. Inspection of the pillars revealed that open vertical 

fractures or joints could be seen in the pillar ribs. Columnar fragments 

of rock about 2m 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 UCS 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, 

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

massive, fine to medium grained with cross bedding. Jointing is 

spaced at 0.4 to 2.0 m (1.3-6.6 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 UCS 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 ft) wide, and 8.2 m (27 ft) 

high on development. Room width was 13.4-14.6 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 91 m (300 ft). 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 that failed along two prominent 

discontinuities. The photograph was taken from the upper mining 

bench and does not show the full height of the 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 [9] equation for pillar strength, and a conservative 

value of the UCS 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. 

Figure 10. Partially benched pillar that failed along two prominent 

discontinuities dipping at approximately 60°. 

Conclusions 

 This evaluation of the strength of slender pillars has revealed the 

following: 

Figure 9. A pillar with a width to height ratio of 0.77 showing the 

effect of brittle failure and spalling. Open vertical f

are visible in the pillar. 

1. 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. ractures/joints 

2. Pillar strength equations developed from empirical studies can 

predict significantly different strengths for slender pillars, even if 

identical rock strength values are used. 

3. 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. 

4. 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. 

5. 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 
 7 

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. 

6. 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. 

Acknowledgements 

 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. 

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