Pillar strength in underground stone mines in the United States

G.S. Esterhuizen n,

n Corresponding author. Tel.: +1 412 386 5207; fax: +1 412 386 6891.
 

E-mail address: eee5@cdc.gov (G.S. Esterhuizen).
 

D.R. Dolinar, J.L. Ellenberger

National Institute for Occupational Safety and Health, PO Box 18070, Pittsburgh, PA 15236, USA

abstract

Stone mines in the Eastern and Midwestern United States make use of the room-and-pillar method of

mining to extract relatively flat-laying sedimentary formations. A survey of pillar performance was

carried out to identify potential modes of instability. Pillars were found to have been successful in

providing support to the overburden, but a small number of individual failed pillars were observed.

Failure of the pillars was observed to be related to spalling of the hard brittle rocks, shearing along preexisting angular discontinuities or progressive extrusion of soft infill materials on bedding planes. A

method of estimating the pillar strength and selecting a safety factor for design was developed based on

observations of stable and failed pillars, supplemented by numerical models. The developed pillar

strength equation can be used to design stable stone mine pillars provided the rock conditions are

similar to those included in the study.

1. Introduction

There are more than 120 operating underground stone mines

in the United States that use the room-and-pillar method of

mining. Mining is typically conducted in flat laying or gently

undulating deposits ranging from highly siliceous limestone to

chemical-grade dolomite and limestone. Large mechanized

equipment, such as 50-ton dump trucks and 5-m3 wheel loaders,

are used to achieve the required economy of scale. The mining

equipment requires adequately sized openings for effective

operation. Where the thickness of the deposit allows it, second

pass bench-mining of the floor may be carried out.

The function of pillars in room-and-pillar mines is to provide

both local and global stability. Local stability is defined as the

provision of stable ribs (pillar walls) and stable roof conditions,

allowing safe access to working areas. Falls of ground from the

roof and pillar ribs account for about 15% of all injuries in

underground stone mines [1] and are related to local instabilities.

Regional stability is defined as the need to support the weight of

the overburden rocks up to the ground surface. Inadequate

regional stability can result in the collapse of multiple pillars

over large sections of a mine, which can result in significant safety

hazards [2,3]. A review of the literature revealed that two cases of

multiple pillar failures have occurred in stone mines in the United

States [3], both without injury or fatalities. One of the cases was

assessed to be related to punching of the pillars into a weak floor

stratum, while the other case appears to have been the collapse of

irregular sized pillars [4]. Insufficient data are available to

estimate the pillar stress at the time of failure of the latter case.

The design of slender pillars in hard brittle rocks has received

considerable attention in recent years [5–8]; however, a widely

accepted design method does not exist for pillars in stone mines.

The research described in this paper had the objective to reduce

the rock fall hazard and the potential for multiple pillar collapse

in stone mines by providing a method for designing pillars. The

research was carried out by first conducting a detailed survey of

mining practices and pillar performance in operating stone mines

in the eastern and midwestern United States. Data were collected

on the mining dimensions, pillar and roof stability conditions,

spacing and orientation of discontinuities and the rock mass

conditions using the rock mass rating (RMR) system of Bieniawski

[9]. Rock samples were collected for laboratory strength testing.

The field data together with numerical modeling results were

used to develop a method for estimating the pillar strength and

selecting an appropriate safety factor for design.

2. Geological setting

The stone mines included in this study are concentrated in the

Interior Plains and the Appalachian Highlands physiographic

regions [10]. Stone deposits located in the Interior Plains region

are generally flat laying or only gently dipping and include rocks

ranging across most of the Paleozoic Era, from the Ordovician Age

to the Pennsylvanian. Overall, the rocks encountered in the

Appalachian Highlands region are similar in age to those found in

the Interior Plains region. They differ in that they have been

transformed through mountain building processes to consist of
elongated belts of folded and faulted sedimentary rocks. Mines

located in strata dipping greater than 101 were excluded from the

study.

3. Rock mass characteristics

The characteristics of the rock mass were evaluated at ninetytwo investigation sites in the thirty-four different operating

mines. Rock samples were collected for strength testing at the

National Institute for Occupational Safety and Health (NIOSH)

laboratory in Pittsburgh, Pennsylvania. Uniaxial compressive

strength (UCS) tests were conducted on nominally 50-mmdiameter cores drilled from the rock samples. Cores were

prepared and tested in accordance with ASTM standards [11].

The results are summarized in Table 1, grouped into three

strength categories.

Table 1

Uniaxial compressive strength of stone mine rocks grouped by strength.

Group Average (MPa) Range (MPa) Samples tested Representative limestone formations

Lower strength

Medium strength

High strength

88

135

220

44–144

82–207

152–301

50

100

32

Burlington, Salem, Galena-Plattesville

Camp Nelson, Monteagle, Plattin, Vanport, Upper Newman, Chickamauga

Loyalhanna, Tyrone

It can be seen that there is considerable

variation in the intact rock strength of the stone formations being

mined, the average strength ranging from 44 MPa for the weakest

formations to 301 MPa in the strongest.

Discontinuities in the stone formations were categorized into

bedding-related discontinuities, which are nearly horizontal, and

joints which are typically steeper-dipping discontinuities. The

bedding discontinuities are the most prominent feature in most of

the mines. The average bedding discontinuity spacing was 0.98 m

with 42% of the observed strike lengths falling in the range of 3–

10 m. In some cases, persistent beds could be traced over more

than 100 m. Bedding discontinuities were typically very rough

and tight or coated with thin carbonaceous materials. Occasionally, the bedding discontinuities contained weak infill materials

such as indurated clay minerals, calcite or carbonaceous matter.

Extrusion of weak materials in the beds was observed to

contribute to instability of the pillar ribs.

Two or more steeply-dipping joint sets were observed to be

present at all the sites investigated. The majority of the joints are

nearly vertical, tight with no infilling or weathering, and the

average spacing between joints is 0.41 m. 94% of the joint trace

lengths fall in the range of 15 cm–3.0 m, with an upper limit of

about 10 m. Joints are typically tight, rough, planar with no

infilling or weathering. Isolated cases were observed where joints

were weathered and contained soft calcitic or clayey infill, usually

at locations near the surface outcrop. The steeper joints did not

appear to have an impact on pillar stability, while angular joints

were observed to cause block release from pillar ribs.

Large, widely spaced discontinuities that extend from the roof

to the floor or across the width of an excavation were observed in

about 40% of the locations visited. These large discontinuities

were logged separately because of their infrequent occurrence

and their potential impact on excavation stability. The average

dip was 821 and the average spacing was 7.9 m, with 72% of the

spacings less than 12 m. The distribution of measured spacings

follows the shape of a negative exponential distribution. The

discontinuity surfaces are rough, planar and can have thin soft

infill materials. These large angular discontinuities can cause a

safety hazard where they intersect the roof or the pillar ribs. Also,

they can have a significant impact on the strength of a pillar,

particularly if they bisect a tall, slender pillar, such as the one

shown in Fig. 1.

Fig. 1. Example of a pillar that is bisected by a large angular discontinuity.

Groundwater conditions were generally dry with damp or

dripping conditions observed only in six out of the 92 locations

mapped. Damp conditions are thought to have contributed to

pillar instability where weak bedding infill materials were

softened by the presence of moisture.

Rock mass classification was carried out at each underground

data collection site. Two perpendicular rock faces, typically two

sides of a pillar, were classified using the 1989 version of the rock

mass rating (RMR) system [9]. The RMR makes use of the rock

quality designation (RQD) [12] which is best obtained from drill

core logging. Since drill core was not available at the mine sites,

the combined RQD and joint spacing rating was determined using

the fracture frequency approach proposed by Laubscher [13]. The

mean and standard deviation of the various classification

parameters and their ratings are summarized in Table 2.

The combined discontinuity spacing and RQD rating is shown as

the ‘‘discontinuity frequency’’ rating out of a possible 40 points.

Table 2

Summary of rock mass component ratings.

Parameter Rating mean Rating Maximum

standard possible

deviation rating

UCS (MPa) 11.6 1.79 15

Discontinuity frequency 23.0 3.91 40

Discontinuity length 4.0 1.23 6

Separation 5.5 1.14 6

Roughness 4.7 1.27 6

Infilling (gouge) 5.7 0.82 6

Weathering 6.0 0.1 6

Groundwater 14.4 2.1 15

Total rating 74.9 – 100
The table shows that the calculated mean RMR is 74.9, which is

classified as ‘‘Good’’ rock according to the RMR system [9]. The

distribution of RMR values is shown in Fig. 2 which shows that the

RMR varied between a value of 60 and 85 out of a possible 100.

40 

Frequency 

35 

30 

25 

20 

15 

10 

5 

0 45-50 50-55 55-60 60-65 65-70 70-75 75-80 80-85 85-90 90-95 95-100 

Rock Mass Rating 

Fig. 2. Distribution of rock mass rating [9] values in underground stone mines.

The fact that only about half of the mines need to regularly install

roof reinforcement in the 10–15-m wide openings is evidence of

the quality of the rock mass being mined.

The rock classification data were collected within the mining

horizon and are therefore representative of the material forming

the pillars. The floor and overlying roof strata were not classified.

The overlying and underlying strata can be similar in strength and

composition to the strata being mined, or may consist of weaker

sedimentary rocks such as shale or siltstone that is strongly

bedded [14]. Although low strength roof or floor strata can have

an impact on pillar performance, it did not appear to have an

impact on pillar stability in any of the mines included in the

study. However, roof and floor strength should not be excluded

from geotechnical investigation because one of the reported pillar

collapses [4] was assessed to have resulted from pillars punching

into a weak floor stratum.

4. Observed pillar performance

A survey of stable and unstable pillars in underground stone

mines within the Eastern and Midwestern United States identified

the causes of pillar instability to provide data for estimating pillar

strength [15]. Mines that were likely to have unstable pillars

owing to their depth of working or size of pillars were identified

as targets for the survey. Data were collected that included both

the intended design dimensions and the actual pillar and room

dimensions in the underground workings. In older areas of mines,

where the original design dimensions were unknown, the

measured dimensions were assumed to adequately represent

the intended design. Table 3 summarizes the dimensions and

cover depth of the pillar layouts that were investigated.

Table 3

Summary of mining dimensions and cover depth of mines included in study.

Dimension Average Minimum Maximum

Pillar width (m) 13.1 4.6 21.5

Pillar height (m) 11.1 4.8 40.0

Width-to-height ratio 1.41 0.29 3.52

Room width (m) 13.5 9.1 16.1

Cover depth (m) 117 22.9 670

The

approximate number of pillars in each layout was recorded to the

nearest order of magnitude and the depth of cover determined

from surface topography and mine maps. The LaModel stress

analysis software [16] was used to estimate the average pillar

stress in cases where the tributary area method was considered

inappropriate. Data from one abandoned limestone mine, which

was not observed as part of this study, was added to the records

owing to its great depth and reported stable conditions [17].

All the pillar layouts surveyed were considered to be successful

in providing global stability because they all provided satisfactory

support of the overburden weight up to the ground surface.

However, not all the pillar layouts were fully successful in

providing local stability in the form of stable roof spans and

pillar ribs. A total of eighteen cases of individual unstable pillars

among otherwise stable pillars were observed. Table 4 presents a

summary of the relevant dimensions, estimated stress, rock

strength and notes related to the failed pillars.

Table 4

Summary characteristics of failed pillars.

Case Width (m) Height (m) W:H ratio Pillar stress (MPa) UCS (MPa) Notes

1 10.7 18.3 0.58 9.0 215 Partially benched pillar, contains angular discontinuities

2 10.7 18.3 0.58 9.4 215 Partially benched pillar, contains angular discontinuities

3 10.7 18.3 0.58 10.3 215 Partially benched pillar, contains angular discontinuities

4 15.2 27.4 0.56 12.6 153 Pillar fully benched to 27 m height causing reduced width-to-height ratio

5 10.7 18.3 0.58 12.8 215 Benched pillar, contains angular discontinuities

6 12.2 27.4 0.44 17.2 150 Partly benched pillar

7 8.5 15.8 0.54 17.2 150 Large steep dipping discontinuity and elevated stress ahead of benching

8 12.2 27.4 0.44 17.3 150 Partly benched pillar

9 7.9 9.8 0.81 19.0 160 Thin weak beds in limestone, pillar undersized causing elevated stress

10 12.8 7.3 1.73 17.4 160 Thin weak beds in pillar causing progressive spalling

11 12.5 15.2 0.82 17.8 160 Thin weak beds in pillar and moist conditions, pillar collapsed

12 6.1 12.2 0.49 19.0 160 Benched pillar is undersized causing elevated stresses

13 6.7 12.2 0.54 20.0 160 Benched pillar is undersized causing elevated stresses

14 3.7 8.5 0.43 24.1 215 Undersized pillar subject to elevated stress

15 8.2 9.1 0.9 25.0 160 Thin weak beds in pillar caused progressive slabbing

16 5.5 7.3 0.75 27.0 160 Undersized pillar subject to elevated stress

17 12.2 15.8 0.77 8.4 165 Partially benched pillar, contains angular discontinuities

18 12.2 15.8 0.77 7.6 165 Partially benched pillar, contains angular discontinuities

The instability

resulted in hazardous local conditions requiring the area to be

barricaded, and in some cases mining operations had to be halted

or re-directed to avoid the hazardous situations. Therefore, the

pillars can be considered to have failed in their function to provide

local stability. These failed pillars are a small percentage of the

more than 1000 pillars that were assessed.
The survey further showed that the following factors can

contribute to pillar failure or instability:

(a) Large angular discontinuities that typically extend from roofto-floor in a pillar. Sliding can occur along these discontinuities that can significantly weaken these slender pillars

[18]. Of the eighteen unstable pillars observed, seven were

affected by these large angular discontinuities. Fig. 3 shows a

pillar that is weakened by two angular discontinuities that

contributed to failure of the pillar at a relatively low pillar

stress.

Fig. 3. Partially benched pillar that failed along two angular discontinuities.

Width-to-height ratio is 0.58 based on full benching height and average pillar

stress is about 4% of the UCS.

(b) Weak	 bands within pillars that can extrude resulting in

progressive spalling of the pillar ribs [19]. Fig. 4 shows a pillar

that has been severely compromised by this mechanism of

failure.

Fig. 4. Pillar that had an original width-to-height ratio of 1.7 failed by progressive

spalling. Thin weak beds are thought to have contributed to the failure. Average

pillar stress is about 11% of the UCS.

It appears that moisture present in the weak beds was

a contributing factor in this failure. Pillars in the immediate

surrounding area appeared to be unaffected.

(c) High pillar	 stress caused by deep cover or high extraction

ratios can cause spalling of the pillar ribs [5,20–22]. It was

found that spalling can initiate when the average pillar stress

exceeds about 10% of the uniaxial compressive strength of the

pillar material. Pillars tend to take on an hourglass shape

when spalling initiates. Fig. 5 shows a pillar that has failed

and taken an hourglass shape due to rib spalling.

Fig. 5. Partially benched pillar failing under elevated stress at the edge of bench

mining. Typical hourglass formation indicating overloaded pillar.

Fig. 6 shows

stable pillars at one of the deeper stone mines in which minor

spalling has resulted in concave pillar ribs.

The failed pillars were typically surrounded by pillars that

appeared to be stable, showing minimal signs of disturbance.

These observations led to the conclusion that the failed pillars

represent the low end of the distribution of possible pillar

strengths, and not the average strength. As a result, the average

safety factor of the layouts containing the failed pillars can be

expected to be substantially higher than that of the individual

failed pillars. However, from a design and safety point of view, it is

important to identify these low strength conditions and take them

into account when designing a proposed pillar layout.

5.	 Pillar strength estimation

The stability of a pillar can be evaluated by calculating a factor

of safety (FOS), which is the ratio of the pillar strength to the

average stress in the pillar. The average stresses in pillars that are

of similar size and are located in a regular pattern can be

estimated with relative ease using the tributary area approach,

which assumes that the overburden weight is equally distributed

among the pillars. This provides an upper limit of the pillar stress

and does not consider the presence of barrier pillars or solid

abutments that can reduce the average pillar stress. The

conditions where the tributary area method is not valid, such as
irregular pillars, limited extent of mining or variable depth of

cover, numerical models can be used to estimate the average

pillar stress.

Fig. 6. Stable pillars in a limestone mine at a depth of cover of 275 m. Slightly

concave pillar ribs formed as a result of minor spalling of the hard, brittle rock.

Estimating pillar strength is more difficult and has been the

subject of much research in the mining industry [5–7,18–26].

Owing to the complexity of pillar mechanics, empirically based

pillar strength equations, which are based on the statistics of

failed and stable pillar systems, have found wide acceptance [27].

Very few pillar failures have occurred in stone mines, and

consequently the field data alone are inadequate to develop a

statistically based pillar strength equation. An empirically developed pillar strength estimation method that was originally

developed by Roberts et al. [7] was used as a starting point. The

method is based on the observation of a large number of collapsed

and stable pillars in the Missouri Lead Belt mines, where the lead

mineralization is hosted in dolomitic limestone rocks [20]. The

rock strength, rock mass rating and mining dimensions are similar

to those found in stone mines [15]. The method predicts the pillar

strength that can be expressed as a power equation incorporating

the UCS of the rock, the pillar width (w) and the pillar height (h) as

follows:

w0:30

S 1⁄4 0:65 x UCS x h0:59 ð1Þ

Eq. (1) was used to calculate the average pillar strength and

safety factors for the limestone mine dataset to determine

whether reasonable results would be obtained. In this calculation,

the minimum width and the tallest face of the pillars were used as

the width and height in the calculations. This implies that the

strength of long rectangular pillars and partly benched pillars

might have been under-estimated. The six cases of failed pillars

that were weakened by large angular discontinuities were

excluded from the calculation because their strength is dominated

by the properties of the discontinuities and should be evaluated

separately.

The results are presented in Fig. 7, which shows the

distribution of the FOS for the successful pillar layouts and

the individual failed pillars.

25 

20 

15 

10 

5 

0 

Factor of Safety 

Stable Pillar Systems 

Failed Single Pillars 

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 

Frequency 

Fig. 7. Distribution of factors of safety of successful pillar systems and failed single

pillars using Eq. (1). Failed pillars that were intersected by large angular

discontinuities were excluded from the chart.

It can be seen that the FOS values of

the successful pillar layouts are all greater than 1.0, as expected,

with the largest concentration of FOS values falling in the range of

4.0–5.0. The minimum FOS of the successful cases was 1.5, which

is considered to be reasonable, since no instances were observed

in which large numbers of pillars showed signs of distress and

overloading, as one might expect if the average FOS approaches

1.0. When calculating the FOS of the failed pillars, the actual pillar

dimensions and local mine layouts were simulated using the

LaModel [16] program to obtain improved estimates of the pillar

stress. However, the average rock strength for each mine site was

used in the calculations. The FOS of the failed pillars can be seen

to fall in the range of 1.0–4.0 with an average value of 2.35. The

relatively high value of FOS is not entirely unexpected, since the

failed pillars represent the weakest members in the array.

It was concluded that Eq. (1) provides reasonable agreement

with the observed stable and failed pillar performance in limestone mines. However, the observations showed that large

angular discontinuities and weak bedding bands can have a

significant impact on pillar stability and should be incorporated

explicitly in the pillar strength equation. In addition, several of the

limestone mines are making use of rectangular pillars to assist

with roof control and ventilation control. These rectangular pillars

can be expected to be stronger than square pillars and should also

be accommodated in the pillar design equation.

Adjustments to Eq. (1) are presented that take into consideration the presence of large angular discontinuities and the effect of

rectangular pillar shapes. The impact of weak bedding bands was

not included as a modification because the rare occurrence of this

failure mode made it difficult to generalize regarding the

particular conditions leading to failure and it would be difficult

to verify a suggested adjustment. Should weak bedding bands be

identified at a proposed mine site, a detailed geotechnical study of

the character of the weak beds and their likely impact on pillar

strength is recommended.

6. Adjustment for large angular discontinuities

An adjustment for the presence of large discontinuities in

pillars should account for both their inclination and spacing. Large

discontinuities can be widely spaced and do not necessarily

intersect each pillar in a layout. The two-dimensional UDEC [28]

software for geotechnical analysis was used to assist in investigating the potential effect of a single large discontinuity on the
strength of pillars with width-to-height ratios of 0.5–1.5. In these

models, the discontinuities were assumed to be smooth and

planar, having a friction angle of 301 and negligible cohesion. The

intact rock was modeled to simulate a typical limestone

formation displaying brittle spalling at low confinement. The

two-dimensional models simulated rib-pillars in which the strike

of the discontinuities were parallel to the pillar edges, and the

discontinuities were assumed to pass through the centers of the

pillars, producing conservative results.

Various analyses were carried out in which the dip of the

discontinuity was varied from 301 to 901 and the strength of the

pillar was determined by simulating the gradual compression of

the pillar until it reached its peak resistance and started to shed

load. A series of curves were fitted to the peak resistance of each

modeled pillar and are shown in Fig. 8. Strength reduction (%) 

0% 

10% 

20% 

30% 

40% 

50% 

60% 

70% 

80% 

90% 

100% 

20 30 40 50 60 70 80 90 

W:H = 3.0 

W:H = 2.5 

W:H = 1.5 

W:H = 2.0 

W:H = 1.0

W:H = 1.2 

W:H = 0.8 

0.7 

W:H = 0.5

W:H = 

Dip of large discontinuity (degrees) 

Fig. 8. Chart showing the impact of large angular discontinuities on the strength

of pillars, based on numerical model results.

The results show firstly

that as the discontinuity dip increases from 301 to about 601, its

impact on the pillar strength increases. When the discontinuity

dip is greater than 701, the effect starts to diminish. A vertical

joint through the center of a pillar is seen to have a relatively

small impact on pillar strength. These trends in the relationship

between pillar strength and discontinuity dip are similar to the

results obtained when testing laboratory specimens with inclined

planes of weakness [24].

The width-to-height ratio is also shown to be a significant

factor affecting the impact of large discontinuities. The chart

shows, for example, that a pillar with a width-to-height ratio of

0.5 will suffer a 95% reduction in strength if it is intersected by a

601 dipping joint while a pillar with a width-to-height ratio of 1.0

would only suffer a 34% reduction in strength. The observation

that slender pillars that are intersected by large angular

discontinuities can fail when the average stress is only about 5%

of the intact rock strength confirms that these large strength

reductions do occur in the field and are similar to those predicted

by the numerical models.

Table 5 summarizes the effect of the dip of large angular

discontinuities on pillar strength, based on the numerical model

results, expressed as a factor that relates the strength of a pillar

intersected by a single large discontinuity to the undisturbed

pillar strength.

Table 5

Discontinuity dip factor (DDF) representing the strength reduction caused by a

single discontinuity intersecting a pillar at or near its center, used in Eq. (3).

Dip (1) Pillar width-to-height ratio

r0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 41.2

30 0.15 0.15 0.15 0.15 0.16 0.16 0.16 0.16 0.16

40 0.23 0.26 0.27 0.27 0.25 0.24 0.23 0.23 0.22

50 0.61 0.65 0.61 0.53 0.44 0.37 0.33 0.30 0.28

60 0.94 0.86 0.72 0.56 0.43 0.34 0.29 0.26 0.24

70 0.83 0.68 0.52 0.39 0.30 0.24 0.21 0.20 0.18

80 0.53 0.41 0.31 0.25 0.20 0.18 0.17 0.16 0.16

90 0.31 0.25 0.21 0.18 0.17 0.16 0.16 0.15 0.15

This factor is called the discontinuity dip factor

(DDF) which can be used by designers to estimate the strength of

pillars that are intersected by these large angular discontinuities.

The DDF values in Table 5 are applicable for a single pillar

intersected by a single large discontinuity. However, these factors

would be conservative for assessing the strength of pillars in a

layout because the large discontinuities are typically widely

spaced and may not necessarily intersect every pillar. Therefore, a

further adjustment is required to account for the fact that only a

few of the pillars may be intersected if the large discontinuities

are widely spaced relative to the pillar dimensions. Based on the

observation that the spacing between large discontinuities

appears to follow a negative exponential distribution, the likelihood that any individual pillar will be intersected by a large

discontinuity can be calculated as follows:

P 1⁄4 1-e-f ð2Þ

where f is the expected frequency of large discontinuities per

pillar and the simplifying assumption is made that the pillar

width is perpendicular to the strike of the discontinuities. The

intersection probabilities for a likely range of discontinuity

frequencies per pillar has been compiled in Table 6 and is called

the frequency factor (FF).

Table 6

Frequency factor (FF) used in Eq. (3) to account for the spacing of large

discontinuities.

Average frequency of large Frequency factor (FF)

discontinuities per pillar

0.0 0.00

0.1 0.10

0.2 0.18

0.3 0.26

0.5 0.39

1.0 0.63

2.0 0.86

3.0 0.95

43.0 1.00

The DDF and FF can be combined as

follows to estimate the average impact of large discontinuities on

the strength of pillars in a layout and is called the large

discontinuity factor (LDF):

LDF 1⁄4 1-DDF x FF ð3Þ

If there are no large discontinuities present, FF is equal to zero

and LDF will equal 1.0, having no effect on pillar strength. The

expected frequency of large discontinuities per pillar can easily be

estimated by dividing the pillar width by the average spacing of

the large discontinuities.

The field observations did not reveal any cases where every

pillar in a layout was intersected by one or more angular

discontinuities. The validity of the LDF under such conditions

could therefore not be verified. It is recommended that a detailed

rock engineering investigation and pillar strength assessment

should be carried out if more than about 30% of the pillars are

expected to be intersected by large discontinuities that dip

between 301 and 701.
The final form of the pillar strength equation for stone mines is

as follows:

w0:30

S 1⁄4 0:65 x UCS x LDF x h0:59 ð4Þ

where UCS is the uniaxial compressive strength of the intact rock,

LDF is the large discontinuity factor shown in Eq. (3), and w and h

are the pillar width and height, respectively, in meters. Note that

if the units of distance are in feet, the 0.65 factor becomes 0.96

because of the volume effect inherent in the equation.

Eq. (4) was used to calculate the strength of a 14-m-wide pillar

with varying width-to-height ratios. The results are presented in

Fig. 9, which shows the strength curve for a pillar that is

unaffected by large angular discontinuities and the results for a

pillar that is intersected by a large discontinuity dipping at 601. Strength/UCS 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

Pillar without angular discontinuities 

Pillar intersected by angular 

discontinuity dipping at 60° 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Width-to-height ratio 

Fig. 9. Chart showing the predicted strength of a 14-m-wide pillar with varying

width-to-height ratios, using Eq. (4). The effect of a large angular discontinuity

dipping at 601 on the strength is also shown.

It

can be seen that the discontinuity has caused a much greater

reduction in pillar strength as the width-to-height ratio drops

below 1.0.

7. Adjustment for rectangular pillars

Rectangular pillars are used in stone mines to provide

ventilation control and to assist with roof control. Rectangular

pillars can be expected to be stronger than square pillars of the

same width. Strength adjustments to account for the increased

strength of rectangular pillars have been suggested by several

researchers [29–31]. These methods all assume that the pillar

strength will increase if the length is increased, regardless of the

width-to-height ratio. However, the stone mines make use of very

slender pillars which may not have been considered in the

methods. A numerical model study that simulated brittle rock

failure in stone mine pillars [32], indicated that the benefit of an

increased length is likely to be zero when a pillar has a width-toheight ratio of 0.5 and it gradually increases as the width-toheight ratio approaches 1.4, upon which the length benefit

becomes fully mobilized. The absence of a length benefit at the

small width-to-height ratios is caused by the low confinement,

which is insufficient to mobilize the frictional resistance of the

brittle material [33].

The so called ‘‘equivalent-width method’’, proposed by Wagner

[30] was selected as a basis for calculating the length benefit of

rectangular pillars in limestone mines. According to Wagner’s

method, the length benefit is expressed as an equivalent increase

in pillar width, which then replaces the true pillar width in the

pillar strength equation. An additional parameter was added to

account for the absence of a length-related strength increase in

the very slender pillars used in stone mines. The parameter is

called the length benefit ratio (LBR), which is a factor that

increases from zero to 1.0 as the width-to-height ratio increases

from 0.5 to 1.4. The modified form of Wagner’s equivalent-width

equation is proposed as follows:

( ) 4A we 1⁄4 wþ -w x LBR C ð5Þ

where w is the minimum width of the pillar, A is the pillar plan

area, C is the circumference of the pillar and LBR is the length

benefit ratio. Table 7 shows the suggested values of LBR based on

numerical model results.

Table 7

Values of the length benefit ratio (LBR) for rectangular pillars with various widthto-height ratios, used in Eq. (5).

Width-to-height ratio Length benefit ratio (LBR)

0.5 0.00

0.6 0.06

0.7 0.22

0.8 0.50

0.9 0.76

1.0 0.89

1.1 0.96

1.2 0.98

1.3 0.99

1.4 1.00

The calculated value of we is used in the

pillar strength equation instead of the true width. When pillars

are square, we will be equal to the pillar width w.

8. Factor of safety for design

An appropriate factor of safety for designing pillars can be

selected from failure statistics if a sufficient number of failed and

stable case histories have been observed [25,34]. In the case of

stone mines, the few individual pillars that have failed are not

sufficient to conduct this kind of study. A more pragmatic

approach was followed in which the FOS of all the layouts and

failed pillars was recalculated using Eq. (4) and the results were

assessed by comparing the FOS values of the failed pillars and the

successful pillar layouts, the FOS of the abandoned layouts, the

numbers of pillars in the various layouts, the average pillar stress

magnitude and the limits of past experience. The recalculated FOS

values are presented in Fig. 10, which displays the FOS against the

width-to-height ratio. Various symbols were used to indicate

currently operating and abandoned layouts, failed pillars and the

approximate number of pillars in the various layouts. Layouts

might have been abandoned because of stability concerns or

changes in operating procedures. The axis displaying the FOS is

limited to 10.0, causing thirteen cases with FOS values greater

than 10.0 not to be displayed.

The results show that the calculated average FOS of all the

failed pillars is 2.0, which includes the cases that were intersected

by large angular discontinuities. The average FOS of those pillars

that are intersected by large discontinuities is 1.5. The minimum

FOS for the stable layouts is 1.27, which is one of the disused

layouts. It can also be seen that only one of the current pillar

layouts has a FOS of between 1.0 and 1.8.
Based on the available data, it would appear there is little

experience with layouts having FOS values of less than 1.8.

Layouts that approach the FOS1⁄41.8 line in Fig. 10 are typically

deep layouts (4180 m) and are subject to relatively high average

pillar stresses where rib spalling can become an issue.

Failed pillars 

Failed pillar intersected by

large angular discontinuity 

Failed pillar (not disturbed by

large angular discontinuity) 

Width-to-height ratio 

Factor of Safety 

0.0 

1.0

2.0 

3.0 

4.0 

5.0 

6.0 

7.0 

8.0 

9.0 

10.0 

1.0 

Width-to-height ratio = 0.8 

Factor of safety = 1.8 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

Tens of pillars 

Hundreds of pillars 

>Thousand pillars 

Current pillar layouts 

Tens of pillars 

Hundreds of pillars 

>Thousand pillars 

Disused pillar layouts 

Fig. 10. Chart showing the factor of safety against width-to-height ratio using Eq. (4). Current and abandoned pillar layouts are shown as well as single failed pillars.

The recommended area for pillar design is shaded.

Therefore,

it is concluded that it would be prudent to design pillars that have

safety factors of at least 1.8 to remain within the range of known

successful layouts.

9. Pillar width-to-height ratio considerations

Fig. 10 shows that there has been a natural tendency for mines

to avoid slender pillars. Nine of the layouts that had width-toheight ratios of less than 0.8 are no longer in use for various

reasons, while only four mines included in the study are currently

operating with these slender pillars. In addition, this study has

shown that slender pillars are more severely affected by the

presence of discontinuities than wider pillars. Studies have also

shown that as the width-to-height ratio decreases below 0.8,

the confining stresses within a pillar approach zero and brittle

fracturing can occur throughout the unconfined pillar core

[5,6,18]. The confining stress can further be reduced if low

friction contact surfaces exist between the pillar and the

surrounding rock. The fracturing and spalling failure mechanism

is poorly understood and it seems prudent to avoid designing

pillars that might fail in this manner.

Inspection of Fig. 10 reveals that a number of stable layouts

exist that have large safety factors (43.0) and the width-toheight ratios less than 0.8. These layouts are mostly located at

very shallow depths of cover, typically less than 60 m. These

pillars were found to be either very narrow, as little as 4.5 m wide,

or very tall, up to 38 m high. The strength and loading of narrow

pillars are both sensitive to small variations in the over-break,

blast damage and pillar spacing. Large tall pillars have high ribs,

which can represent a safety hazard, the high roof becomes

difficult to inspect because it is poorly visible, and the severity of

potential rock fall impacts increases. Therefore, it is not advisable

to design layouts with such slender pillars, even if the calculated

factors of safety are high.

10. Conclusions

The slender pillars used in stone mines in the United States

present a unique challenge to mine designers. This study has

shown that isolated pillar failures have occurred, which have

resulted in safety and production concerns. An equation has been

developed which can be used to estimate the pillar strength.

The equation takes into account the intact rock strength and the

potential impact of large angular discontinuities on pillar

strength. An adjustment procedure is proposed to account for

the increased strength of rectangular pillars.

The presence of weak, clay-filled or softening-bedding bands

was seen to have a significant impact on pillar strength and was a

contributing factor in the only collapsed pillar observed by the

authors. Weak bedding bands are not accounted for in the design

procedure and should be the subject of a detailed geotechnical

investigation if they are encountered in a proposed mining area.

Pillars having a width-to-height ratio of less than 0.8 should be

avoided because of their sensitivity to the presence of angular

discontinuities, variations in dimensions and the absence of

confinement in the core of the pillar.

A pillar factor of safety of at least 1.8 is recommended when

using Eq. (4) for design. This will ensure that pillars remain within

the limits of previously successful layouts.

The pillar strength equation and design recommendations are

based on the empirical evidence from underground stone mines

in the United States and should only be used to design pillar

layouts in similar conditions.
Disclaimer

The findings and conclusions in this report are those of the

authors, and do not necessarily represent the views of the

National Institute for Occupational Safety and Health.

Acknowledgements

The authors wish to thank the stone mine personnel who were

willing to share their experiences and failures with the research

team and who accompanied us to the various underground mine

sites. Without their cooperation this research would not have

been possible.

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