Pillar Strength and Design Methodology for Stone Mines 

Gabriel S. Esterhuizen, Sr. Research Engineer 

Dennis R. Dolinar, Lead Research Engineer
 

John L. Ellenberger, Lead Research Scientist 

NIOSH-Pittsburgh Research Laboratory 

Pittsburgh, PA
 

ABSTRACT 

Underground stone mines in the United States make use of the 

room-and-pillar method of mining. A prerequisite for a safe 

working environment is that the pillars should adequately support 

the overburden and the pillar ribs and rooms should remain stable 

during mining. At present pillar dimensions are either based on 

experience at neighboring mines or on strength equations that 

were developed for metal or non-metal mines. This paper 

presents a pillar design methodology that was developed from a 

study of pillar performance in operating stone mines. Data were 

collected that describe the rock mass quality, pillar conditions, 

mining dimensions and intact rock strength. The results showed 

that current mining practices have resulted in generally stable 

pillar layouts with no recent cases of extensive pillar collapses. 

However, a small number of single failed or failing pillars were 

observed in otherwise stable layouts. Numerical analyses were 

used to supplement the observations and develop a pillar strength 

equation that describes the stable and small number of failed 

pillars observed. The developed strength equation can be used to 

design stable pillar layouts provided the factor of safety is greater 

than 1.8 and the width-to-height ratio of the pillars is greater than 

0.8. The paper concludes with guidelines for applying the 

developed equation and selecting appropriate input parameters. 

INTRODUCTION 

Underground stone mines in the United States use the roomand-pillar method to extract sedimentary formations that are 

generally flat lying. Pillar stability is one of the prerequisites for 

safe working conditions in a room-and-pillar mine. Unstable 

pillars can result in rock sloughing from the pillar ribs and can 

lead to the collapse of the roof if one or more pillars should fail. 

Fall of ground injuries from the roof and pillar ribs accounted for 

about 15% of lost work days in underground stone mines from 

1997 to 2006 (1). About one third of these incidents were 

associated with rib instability. 

Current research at the National Institute for Occupational 

Safety and Health (NIOSH) has the objective to reduce ground 

fall accidents in stone mines through safe pillar design. Studies 

of pillar performance and strength in stone mines (2, 3) 

highlighted the need for pillar design guidelines that specifically 

address pillar stability in U.S. stone mines. At present, pillar 

dimensions are either based on experience at neighboring mines or 

on strength equations that were developed for metal or non-metal 

mines. This paper presents a pillar strength equation with 

associated design guidelines that will assist in the design of safe 

room-and-pillar workings in stone mines. 

BACKGROUND 

In a room-and-pillar mine, the pillars are required to provide 

global stability which can be defined as supporting the overlying 

strata up to the surface. In addition, local stability in the form of 

stable pillar ribs and roof are required to provide safe working 

conditions. Pillar design is typically carried out by estimating the 

pillar strength and the pillar stress, and then sizing the pillars so 

that an adequate margin exists between the expected pillar 

strength and stress. The factor of safety (FOS) relates the average 

pillar strength (S) to the average pillar stress (σp), as follows: 

FOS = σ 

S

p 

(1) 

When designing a system of pillars, the FOS must be selected 

with care, because it must compensate for the variability and 

uncertainty related to pillar strength and stress and mining 

inconsistencies. The selection of an appropriate safety factor can 

be based on a subjective assessment of pillar performance or 

statistical analysis of failed and stable cases (4, 5, 6, 7). As the 

FOS decreases, the probability of failure of the pillars can be 

expected to increase. In practical terms, if one or more pillars are 

observed to be failed in a layout, it is an indication that the pillar 

stress is approaching the average pillar strength, causing the 

weaker pillars to fail. The relationship between FOS and failure 

probability, however, depends on the uncertainty and variability 

of the system under consideration (8). 

Pillar Strength 

Pillar strength can be defined as the maximum resistance of a 

pillar to axial compression (9). In flat lying deposits, pillar 

compression is caused by the weight of the overlying rock mass. 

Empirical evidence suggests that pillar strength is related to both 

its volume and its shape (4, 6, 9). Numerous equations have been 

developed that can be used to estimate the strength of pillars in 
 

 

 

 

 

coal and hard rock mines, and have been reviewed and 

summarized in the literature (10, 11, 12, 13). These equations are 

generally empirically developed and are only applicable for 

conditions similar to those under which they were developed. 

More recently, numerical model analyses combined with 

laboratory testing and field monitoring have contributed to the 

understanding of failure mechanisms and pillar strength (2, 6, 12, 

14, 15, 16, 17, 18). 

A pillar strength equation that captures both the pillar shape 

and volume effect is a power equation of the following form, 

where k is a parameter related to the rock strength, w and h are the 

pillar width and height and α and β are parameters related to the 

geomechanical conditions of the rock mass. 

wα (2) S = k × β

h 

Table. 1 Uniaxial compressive strength of limestone rocks collected at mine sites. 

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

Lower Strength 12,800 6,400 – 20,800 50 Burlington, Salem, Galena-Platteville 

Medium Strength 19,600 11,900 – 30,000 100 Camp Nelson, Monteagle, Plattin, Vanport, 

Upper Newman, Chickamauga 

High Strength 31,800 22,000 – 43,700 32 Loyalhanna, Tyrone 

Pillar Stress 

The average pillar stress (σp), in regular layouts of pillars, can 

be estimated by the tributary area method as follows, where γ is 

the specific weight of the overlying rocks, h is the depth of cover, 

w is the pillar width, l is the pillar length and C1 and C2 are the 

heading and crosscut center distances respectively. 

(C × C ) σ p = γ ×h × 1 2 

(w× l) (3) 

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. In conditions where the tributary 

area method is not valid, such as irregular pillars, limited extent of 

mining or variable depth of cover, numerical models such as 

Lamodel (19) can be used to estimate the average pillar stress. 

Pillar Failure 

Pillar failure occurs when a pillar is loaded beyond its peak 

resistance and load shedding, yielding, shearing or collapse occurs 

(9). Failure of a single pillar can result in hazardous pillar rib 

conditions, roof instability in the adjacent mining rooms and 

blockage of local access ways. Load redistribution caused by the 

failure of a single pillar can overload adjacent pillars, resulting in 

wide-area failure (20, 21). These wide-area failures can occur as a 

catastrophic collapse or gradual closure or “squeeze” over time. 

They can result in excessive convergence of the mine opening, 

surface subsidence and disruption of the overlying strata and may 

cause an air blast if the air is violently forced out of the collapsing 

area (21). Empirical evidence and theoretical studies suggest that 

as the width-to-height ratio of pillars is reduced, the potential for 

catastrophic failure increases as a result of the rapid decrease in 

strength of a slender pillar after it has reached its peak loadbearing capacity (20). 

Pillars can show signs of instability prior to failure. As the 

stress in a pillar increases, rock fracturing and spalling can occur 

at the pillar corners and perimeter. Pillars that are heavily loaded 

can exhibit an “hourglass” shape. Ultimately, the pillars develop 

open fractures and rib sloughing as the peak load-bearing capacity 

is exceeded (22, 23, 24, 25). These signs of failure can be used to 

visually assess the stability of pillars in underground workings. 

PILLAR PERFORMANCE IN STONE MINES 

The performance of pillars was assessed at 91 locations in 34 

different operating stone mines in the Eastern and Midwestern 

United States (3). 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 intended design dimensions were unknown, the 

measured dimensions were assumed to represent the design 

adequately. The approximate number of pillars in each layout was 

recorded and the depth of cover determined from surface 

topography and mine maps. The Lamodel software package (19) 

was used to estimate of the average pillar stress in cases where the 

tributary area method was considered inappropriate. Data from 

the Norton limestone mine in Ohio, which was not visited as part 

of this study, has been added to the records owing to its great 

depth and reported stable conditions (26). 

At each mine, rock samples were collected to determine the 

uniaxial compressive strength (UCS) and the rock mass was 

classified using the Rock Mass Rating (RMR) system (27, 28). 

The UCS results were grouped into three categories based on the 

average strength of the various formations and are shown in table 

1. It can be seen that there is considerable variation in the intact 

rock strength of the stone being mined. It was found that the 

RMR varied between 65 and 85 out of a possible 100, which 

indicates the relatively good rock mass conditions found in these 

mines. 

Successful Pillar Layouts 

The survey revealed that all of the 91 pillar layouts observed 

at the 34 different mines could be classified as successful in 

providing global stability by supporting the overburden load up to 

the ground surface. Table 2 summarizes the dimensions and cover 

depth of the pillar layouts that were investigated. Not all the 

pillar layouts were fully successful in providing local stability in 

the form of stable roof spans and pillar ribs. Single failed pillars 
 

 

 

 

 

 

 

 

 

 

 

 

 

were observed that were surrounded by stable pillars. These cases 

of individual pillar failure are evaluated below, and do not 

represent failure of the layout in providing global stability. 

Table 2. Summary of mining dimensions and cover depth of mines included in study. 

Dimension Average Minimum Maximum 

Pillar width (ft) 43.0 15.0 70.5 

Pillar height (ft) 36.5 15.8 124.6 

Width-to-height ratio 1.41 0.29 3.52 

Cover depth (ft) 385 75 2,200 

Individual Failed Pillars 

A total of eighteen cases of individual pillars that had failed in 

otherwise stable layouts were observed at five different mining 

operations. These failed pillars can represent a safety hazard 

because they are associated with unstable roof and ribs and 

typically require that the mining area be barricaded or abandoned. 

Each of the failed pillars was visually assessed and, where 

practical, was photographed to provide a record of the pillar 

conditions. The key parameters describing the failed pillars are 

summarized in table 3, together with notes regarding probable 

factors contributing to the failure. Factors contributing to pillar 

failure included the presence of through-going angular 

discontinuities, weak bedding bands, increased pillar height 

during bench mining and undersized pillars. The failed pillars 

exhibited one or more of the following characteristics: 

1) Collapse of entire pillar, as shown in figure 1. 

2) Rib spalling to a rounded hour-glass shape with open 

joints and fractures, shown in figures 2 and 3. 

3) Shearing along large angular discontinuities (dip 30 to 

70°) resulting in loss of pillar integrity, shown in figure 4. 

The failed pillars were typically surrounded by pillars that 

appeared to be stable, showing minimal signs of disturbance. The 

observations lead to the conclusion that the failed pillars represent 

the low end of the distribution of possible pillar strengths, and not 

the average pillar strength. The average FOS of the layouts 

containing these failed pillars can therefore be expected to be 

relatively high because of the low observed failure frequency. 

Table 3. Summary of failed pillar characteristics. 

Case 

Pillar 

width 

(ft) 

Pillar 

height 

(ft) 

Widthto-height 

ratio 

Average 

pillar stress 

(psi) 

UCS 

(psi) Factors contributing to pillar failure 

1 35 60 0.58 1,305 31,175 Partially benched pillar, contains angular discontinuities 

2 35 60 0.58 1,363 31,175 Partially benched pillar, contains angular discontinuities 

3 35 60 0.58 1,494 31,175 Partially benched pillar, contains angular discontinuities 

4 50 90 0.56 1,827 22,185 Pillar fully benched to 90-ft height causing reduced 

width-to-height ratio 

5 35 60 0.58 1,856 31,175 Benched pillar, contains angular discontinuities 

6 40 90 0.44 2,494 21,750 Partly benched pillar 

7 28 52 0.54 2,494 21,750 Large steep dipping discontinuity and elevated stress 

ahead of benching 

8 40 90 0.44 2,509 21,750 Partly benched pillar 

9 26 32 0.81 2,755 23,200 Thin weak beds in limestone, pillar undersized causing 

elevated stress 

10 42 24 1.73 2,525 23,200 Thin weak beds in pillar causing progressive spalling 

11 41 50 0.82 2,583 23,200 Thin weak beds in pillar and moist conditions, pillar 

collapsed 

12 20 40 0.49 2,755 23,200 Benched pillar is undersized causing elevated stresses 

13 22 40 0.54 2,900 23,200 Benched pillar is undersized causing elevated stresses 

14 12 28 0.43 3,495 31,175 Undersized pillar subject to elevated stresses 

15 27 30 0.90 3,625 23,200 Thin weak beds in pillar caused progressive slabbing 

16 18 24 0.75 3,915 23,200 Undersized pillar subject to elevated stresses 

17 40 52 0.77 1,220 23,900 Partially benched pillar, contains angular discontinuities 

18 40 52 0.77 1,100 23,900 Partially benched pillar, contains angular discontinuities 

The Impact of Large Angular Discontinuities 

Pillars that were associated with angular discontinuities were 

observed to have failed when the average pillar stress was only 

about 4-5% of the UCS. The potential weakening effect of a 

large angular discontinuity is clearly demonstrated in figure 4. 

These discontinuities are not always readily visible to production 

staff when developing a pillar, but only become apparent when 

the pillar becomes fully loaded or when bench mining is carried 

out around the pillars. Particularly hazardous conditions can 

result if large angular discontinuities cause unstable blocks to 

slide or topple from the pillar ribs. Of the eighteen failed pillars 
shown in table 3, seven were associated with large angular 

discontinuities. 

Large discontinuities were observed to be present in 22 of the 

34 stone mines surveyed. These large discontinuities can be 

widely spaced, extend from the roof to the floor of the workings 

and the strike extent can be several hundred feet. The spacing 

appears to follow a negative exponential distribution with 75% of 

the discontinuities less than 40 ft apart. The average dip was 81° 

with only 18% of the observations having a dip of less than 70°. 

The Impact of Weak Bedding Bands within a Pillar 

The presence of near-horizontal, thin weak bands within a 

pillar was observed to be a contributing factor in four of the failed 

pillar cases. A study of the effect of weak bands on pillar strength 

(29) showed that extrusion of the weak material can induce 

tension in the surrounding rock, which promotes failure by 

progressive spalling at much lower stress than the compressive 

strength of the rock mass. A similar mechanism is described by 

Brady and Brown (9). The pillar shown in figure 2, listed as case 

10 in table 3, is an example of a pillar that the authors believe 

failed by this mechanism. At present there is insufficient 

information to draw general conclusions regarding the conditions 

that might lead to this type of failure. It is not clear, for example, 

why only the single pillar shown in figure 2 failed while the rest 

of the pillars in the area did not show signs of distress, in spite of 

appearing to have similar weak bands. 

 

 

 

 

 

 

 

 

 

 

Figure 1. Remaining stump of a collapsed pillar in an 

abandoned area. Thin weak beds in the pillar and moist 

conditions are thought to have contributed to the failure. The 

width-to-height ratio was 0.82 and average pillar stress about 

11% of the UCS.ws 

Figure 2. Pillar that has an original width-to-height ratio of 

1.7 failed by progressive spalling. Thin weak beds are thought 

to have contributed to the failure. The average pillar stress 

was about 11% of the UCS prior to failure. 

Figure 3. Partially benched pillar failing under elevated 

stresses at the edge of bench mining. Typical hourglass 

formation indicating overloaded pillar. Width-to-height ratio 

is 0.44 based on full benching height. The average pillar stress 

was about 12% of the UCS. 

Figure 4. Partially benched pillar that failed along two angular 

discontinuities. Width-to-height ratio is 0.58 based on full 

benching height. The average pillar stress was about 4% of the 

UCS. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Wide-Area Failures 

Two cases of wide-area pillar failure have been reported in 

stone mines that are no longer operating (30). The first case was a 

reported collapse of a small stone mining operation that may have 

been the result of a sudden collapse of the pillars. The pillar 

dimensions were variable and insufficient information exists to 

evaluate this event for estimating pillar strength. 

The second case was a reported failure in which an area of 

about 20 pillars was alleged to have failed (30). An investigation 

of this reported failure revealed that the pillars had not actually 

failed, but moisture-related yield of the weak floor may have 

occurred and caused the roof to collapse around the pillars (31). 

The pillars were seen to be intact within the collapsed area. 

Consequently, this case has been discarded for estimating pillar 

strength because the pillars had not failed. However, it does 

highlight the fact that the potential for floor yield should be 

evaluated when designing a stone mine pillar layout, e.g. by 

drilling into the floor during exploration. 

These two case histories, while not directly useful for 

evaluating stone pillar strength, do emphasize the fact that widearea failures can occur. 

Pillar Rib Stability 

The provision of stable pillar ribs is necessary for safe mining 

operations. Rib instability can be caused by unfavorable jointing 

in the rock mass, by rock fracturing under elevated stresses caused 

by undersized pillars or by poor blasting practice. Figure 5 shows 

an example of stress-related rib spalling at approximately 900 ft of 

cover. It was found in some cases that stress-related rib spalling 

can occur when the average pillar stress exceeds about 11-12% of 

the UCS. Rib supports, such as screen and bolts, are sometimes 

used to secure the pillar ribs. 

Figure 5. Example of rib slabbing and resulting concave 

pillar ribs that can initiate when average pillar stress 

exceeds about 11% of the UCS. 

Summary of Stone Mine Pillar Observations 

The pillar layouts that were surveyed by NIOSH are presented 

in figure 6 which shows the pillar stress against the width-toheight ratio. The pillar stress is normalized by the average UCS 

of each formation, obtained from table 3. The chart also includes 

data points representing the 18 failed pillars, while failures 

associated with the presence of large angular discontinuities are 

indicated separately. Information on the approximate number of 

pillars in each pillar layout and layouts that are no longer in use 

are indicated. Layouts that are disused may have been abandoned 

because of stability concerns, depletion of reserves or changes in 

operating procedures. A bounding curve was drawn around the 

case histories, which represents the limit of current experience 

with stone pillar performance. 

For the purpose of preparing this chart, the width-to-height 

ratio of the pillars was based on the minimum pillar width. Where 

pillars were partially benched, the full height of benching was 

used to represent the pillar height. Actual underground 

measurements of room and pillar dimensions were used. 

All the pillar layouts shown in the chart can be considered to 

have been “successful” in the objective of providing global 

support. The results show that these “successful” pillar layouts 

contain many thousands of stable pillars while the single failed 

pillars represent only a very small part of the total population of 

pillars. The relatively low strength of the failed pillars that 

contained angular discontinuities is also clearly indicated. The 

chart can be used to compare a current or proposed pillar layout 

with past experience (32). 

A PILLAR STRENGTH EQUATION FOR STONE MINES 

The database on stone mine pillar performance contains 

information on many stable pillar systems but only eighteen 

individual failed pillars, which are likely to be the weakest 

members of the population of pillars. These data are therefore not 

representative of the average stone pillar strength and are not 

sufficient to develop a strength equation for stone mines. 

Consequently, information from other mining operations that are 

similar to stone mine room-and-pillar workings were sought to 

provide a basis for developing a strength equation. 

Records of stable and failed pillars in the lead mines of the 

Viburnum Trend in Southeastern Missouri were considered to be 

the most appropriate for developing a strength equation for stone 

mines. The workings are flat-lying and room-and-pillar 

operations have been conducted with mostly square pillars (33) 

since the 1960’s. The host rock is dolomitized limestone with 

similar strength characteristics to limestone. The average UCS of 

the rock is approximately 22,000 psi (34), which falls within the 

upper range of limestone formation strengths. The rock mass 

quality was assessed at several different underground locations by 

the authors and found to fall within the range found in stone mines. 

It is recognized that the presence of mineralization within the host 

rock can affect the rock strength and post failure behavior. 

However, the stages of failure development observed underground 

and reported by Lane et al. (35) are very similar to those seen in 

stone pillars. Importantly, a wide-area pillar collapse occurred at 

one of these mine operations during the 1980’s, which provides 

valuable data on the ultimate pillar strength (30). 

A well-documented pillar design procedure has been 

developed for these mines based on the observation of failed and 

stable pillars (35, 36). The design technique makes use of 

numerical models to estimate pillar loading while pillar strength is 

estimated by a set of strength relationships which are based on the 

confinement principle, modified after Lunder and Pakalnis (12). 

Direct observations of hundreds of pillars, which included both 
stable and failed case histories, have been used to refine the 

strength relationships. 

In principle, the pillar strength is determined by viewing each 

pillar in plan and subdividing it into 8 ft x 8 ft elements. Each 

element is considered either an “outer” or “inner” element. The 

outer elements have lower strength than the inner elements owing 

to the lack of confinement. The strength is also affected by the 

pillar height, according to relationships presented in Roberts et al. 

(36). For example, a 16-ft square pillar will consist of four 8-ft 

“outer” elements and will be weaker than a 24-ft square pillar that 

has eight “outer” and one “inner” element. The method therefore 

takes into consideration both the pillar shape and pillar volume for 

estimating pillar strength. 

In order to express the pillar strength relationships in the form 

of a power equation, a series of strength curves were developed 

for 24-ft, 32-ft and 40-ft wide pillars using the “inner” and “outer” 

element approach, shown in figure 7. The corresponding 

parameters for the power equation were then determined by least 

squares curve fitting to these results. The following equation was 

obtained, where w and h are the pillar width and height in feet. 

0.3 w S = k × 0.59 (4) h 

The strength parameter k was found to be 20,240 psi. The value 

of k can be expressed in terms of the UCS as follows based on the 

average UCS value of 22,000 psi for the formation. 

k = 0.92 ×UCS (5) 

Note that for pillar dimensions in meters, the k parameter becomes 

0.65 × UCS because of a dimensional imbalance in the equation. 

The strength results obtained by equation 4 are also shown in 

figure 7 for comparison. The difference between the two methods 

was less than 7% for width-to-height ratios between 0.5 and 2.0. 

Tens of pillars

Hundreds of pillars

>Thousand pillars

Current pillar layouts

Tens of pillars

Hundreds of pillars

>Thousand pillars

Disused pillar layouts Failed pillars

Failed pillar intersected by 

large angular discontinuity

Failed pillar (not disturbed by 

large angular discontinuity)

Tens of pillars

Hundreds of pillars

>Thousand pillars

Current pillar layouts

Tens of pillars

Hundreds of pillars

>Thousand pillars

Disused pillar layouts Failed pillars

Failed pillar intersected by 

large angular discontinuity

Failed pillar (not disturbed by 

large angular discontinuity)

0.30 

0.00

0.05

0.10

0.15

0.20

0.25

Pillar stress/UCS

Boundary of current experience

0.00 

0.05 

0.10 

0.15 

0.20 

0.25 

Pillar stress/UCS 

Boundary of current experience 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Width-to-hdth-to-height ratio 

Tens of pillars 

Hundreds of pillars 

>Thousand pillars 

Current pillar layouts 

Tens of pillars 

Hundreds of pillars 

>Thousand pillars 

Disused pillar layouts Failed pillars 

Failed pillar intersected by 

large angular discontinuity 

Failed pillar (not disturbed by 

large angular discontinuity) 

 

 

 

 

 

 

 

 

Figure 6. Summary of observed pillar layouts and single failed pillars. 

Application to Case Histories 

Equation 4 was used to calculate the average pillar strength 

and safety factors for the stone mine dataset to determine whether 

reasonable results would be obtained. In this calculation, the 

minimum width and the tallest edge of the pillars were used in the 

equation. This implies that the strength of long rectangular pillars 

and partly benched pillars may have been under-estimated. In 

addition, the seven 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 treated separately. 
 

 

 

 

 

 

 

24 ft-wide pillar

40 ft-wide pillar

32-ft-wide pillar

Roberts et al. (36)

0.10 Equation 3

0.40 

0.50 

0.60 

0.00

-

-

0.00 

24-ft-wide pillar 

40-ft-wide pillar 

32-ft-wide pillar 

Roberts et al. (36) 

Equation 3 

Pillar stress/UCS 

0.30 

0.20 

0.00 0.50 1.00 1.50 2.00 

Width-to-hdth-to-height ratioeight ratio 

Figure 7. Comparison of predicted pillar strength using the 

method of Roberts et al. (36) and equation 4. 

The results should show that the FOS values of the stable 

layouts are greater than 1.0 and greater than that of the failed 

cases. In addition, the failed cases can be expected to have a 

relatively lower average FOS. However, the average calculated 

FOS of the failed pillars can be expected to be somewhat greater 

than 1.0 because the calculation was carried out using average 

strength values while the failed pillars are likely to have been 

weaker, which contributed to their failure. 

The results are presented in figure 8 which shows the 

distribution of the FOS for the stable pillar layouts and the 

individual failed pillars. 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 

to 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 in cases where the average FOS 

approaches 1.0. 

Stable Pillar Layouts

Failed Single Pillars

25 

20 

Stable Pillar Layouts 

Failed Single Pillars 

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

Factor of Safety 

Frequency 15 

10 

5 

0
 

0
 

Figure 8. Distribution of factors of safety of successful 

pillar systems and single failed pillars using equation 4. 

Failed pillars that were intersected by large angular 

discontinuities were excluded from the chart. 

The failed pillars can be seen to fall in the FOS range of 1.0 to 

4.0 with an average FOS of 2.35. The relatively high value of 

FOS is not unexpected, since the calculation used best estimates 

of rock strength and loading, which did not necessarily include 

local strength variations or geological structures that may have 

caused the individual pillars to fail at these apparently high FOS 

values. 

It was concluded that equation 4 provides a reasonable 

agreement with the observed stable and failed pillar performance 

in stone mines. However, the observations showed that large 

angular discontinuities can have a significant impact on pillar 

stability and should be incorporated explicitly in the pillar strength 

equation. In addition, several of the stone 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 equation 4 are presented below 

that will allow these two parameters to be included. 

Adjustment for the Presence of Large 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 (37) 

program was used to assist in investigating the potential effect of a 

single large discontinuity on the strength of pillars with width-toheight ratios of 0.5 to 1.5. In these models the discontinuities 

were assumed to be smooth and planar, having a friction angle of 

30 deg with little or no cohesion. 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. Table 4 provides a summary of the results, expressed as a 

reduction factor that relates the strength of a pillar intersected by a 

single large discontinuity to the undisturbed pillar strength, and is 

referred to as the discontinuity dip factor (DDF). The table shows 

that discontinuities can have a significant impact on pillar strength 

and the impact is exacerbated as the width-to-height ratio 

decreases. 

When considering the stability of a single pillar that is 

intersected by a large discontinuity, the DDF values shown in 

table 4 are applicable. However, these factors would be 

conservative for assessing a layout of many pillars, because the 

large discontinuities can be widely spaced and may not 

necessarily intersect every pillar. A further adjustment, called the 

frequency factor (FF), is suggested. The average impact of large 

discontinuities on the strength of pillars in a layout can then be 

estimated as follows, and is called the large discontinuity factor 

(LDF), where DDF is the discontinuity dip factor shown in table 4, 

and FF is the frequency factor related to the frequency of large 

discontinuities per pillar shown in table 5. 

LDF = 1 – DDF FF (6) 

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 

frequency of large discontinuities per pillar can easily be 

estimated by dividing the pillar width by the average spacing of 

the large discontinuities. For example, if we are designing pillars 

that are 30 ft wide at a width-to-height ratio of 1.0 and we want to 

know the impact of large discontinuities that are spaced 100 ft 

apart, dipping at 50 degrees, we can proceed as follows: 
 

 

 

 

 

 

 

1)	 Calculate the expected frequency of large discontinuities 

per pillar (30/100 = 0.3), 

2)	 In table 5 we find that FF = 0.26 

3)	 Look up the discontinuity dip factor (DDF) in table 4, 

which is 0.37. 

4)	 Calculate the LDF for the pillar layout using equation 6, 

which is 0.90. 

This represents a 10% reduction in the average strength of pillars 

in the layout. 

The field observations did not include any cases where every 

pillar in a layout was intersected by one or more angular (30 to 70 

deg) discontinuities. Therefore, the validity of the LDF under 

such conditions could 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 30 and 70 deg. 

Table 4. Discontinuity dip factor (DDF) representing the strength reduction caused by a single discontinuity 

intersecting a pillar at or near its center, used in equation 6. 

Discontinuity 

dip (deg) 

Pillar width-to-height ratio 

≤0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3-2.0 

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 

Table 5. Frequency factor (FF) used in equation 6 to 

account for large discontinuities. 

Average frequency of large 

discontinuities per pillar Frequency factor (FF)

 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 

>3.0 1.00 

Adjustment for Rectangular Pillars 

Strength adjustments to account for the increased strength of 

rectangular pillars have been suggested by several researchers 

(7, 38, 39). A numerical model study that simulated brittle rock 

failure in stone pillars (40) indicated that slender pillars do not 

benefit as much from a length increase as wider pillars. This 

effect is caused by the lack of confinement in the slender pillars. 

The numerical model results indicate that the length benefit ratio 

(LBR) is likely to be zero when a pillar has a width-to-height ratio 

of 0.5 and it gradually increases to 1.0 as the width-to-height ratio 

approaches 1.4. 

The “equivalent width method”, proposed by Wagner (38) 

was selected as a basis for calculating the length benefit of 

rectangular pillars in stone mines. According to this 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. The modified form of Wagner’s equivalent 

width equation is proposed as follows, 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. 

⎛ 4 A ⎞ we = w + ⎜ − w ⎟× LBR (7)

⎝ C ⎠

Table 6 shows the suggested relationship between width-to-height 

ratio and the value of LBR. For square pillars, we will equal the 

pillar width w. 

Table 6. Values of the length benefit ratio (LBR) for 

rectangular pillars with various width-to-height ratios. 

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 

Adjustments for Other Geotechnical Conditions 

Observations showed that the presence of thin weak bedding 

bands appears to have contributed to the failure of several of the 

pillars presented in table 3. The data and understanding of this 

failure mode are not sufficient to account for weak bands in the 

pillar strength equation. Similarly, the presence of weak floor 

strata and its impact on stone pillar performance is not well 

described. The authors suggest that the services of a rock 

engineering specialist should be sought when these conditions 

exist, so that a detailed program of investigation can be conducted. 
 

Pillar Strength Equation for Stone Mines 

The strength equation for stone mine pillars with the large 

discontinuity and rectangular pillar adjustments can be rewritten 

as follows, where UCS is the uniaxial compressive strength of the 

intact rock, LDF is the large discontinuity factor, w and h are the 

pillar width and height in feet. For rectangular pillars, w is 

replaced by the equivalent width we, which can be calculated 

using equation 7. The value of LDF can be determined from 

equation 6. If no large discontinuities are present, LDF will equal 

1.0. 

w 0.3

S = 0.92 ×UCS × LDF × (8)

h 0.59

Pillar Stability Results from Modified Equation 

Equation 8 was used to calculate the adjusted strength and of 

all the pillars in the stone mine database. The completed FOS 

results are presented in figure 9, which displays the FOS against 

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

currently operating and disused layouts, failed pillars and the 

approximate number of pillars in the various layouts. Scaling for 

the FOS axis was limited to 10.0, causing thirteen cases with FOS 

values greater than 10.0 not to be displayed. It can be seen that 

only one of the current pillar layouts has an FOS of between 1.0 

and 1.8. This point represents a local area in a mine where rib 

instability occurred as a result of large discontinuities. 

Tens of pillars

Hundreds of pillars

>Thousand pillars

Current pillar layouts

Tens of pillars

Hundreds of pillars

>Thousand pillars

Disused pillar layouts Failed pillars

Failed pillar intersected by 

large angular discontinuity

Failed pillar (not disturbed by 

large angular discontinuity)

Width to height ratio = 0.8

Factor of safety = 1.8

Width to height ratio = 0.8 

 

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

- -

0.0 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

7.0 

8.0 

9.0 

10.0 

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 

Width-to-height ratio 

Tens of pillars 

Hundreds of pillars 

>Thousand pillars 

Current pillar layouts 

Tens of pillars 

Hundreds of pillars 

>Thousand pillars 

Disused pillar layouts Failed pillars 

Failed pillar intersected by 

large angular discontinuity 

Failed pillar (not disturbed by 

large angular discontinuity) 

4.0 

Figure 9. Chart showing the factor of safety against width-to-height ratio using equation 8. Current and disused pillar 

layouts are shown as well as single failed pillars. The recommended area for pillar design is shaded. 

DESIGN GUIDELINES AND RESTRICTIONS 

The results shown in figure 9 formed the basis for developing 

the design guidelines that follow. The guidelines are empirically 

based, their validity is therefore restricted to rock conditions, 

mining dimensions and pillar stresses that are similar to those 

included in this study. Therefore, they should be applicable to the 

majority of stone mines in the Midwestern and Eastern United 

States. The guidelines for designing stable pillar layouts are as 

follows: 
1)	 Equation 8 can be used to estimate the average strength of 

pillars in flat-lying room-and-pillar stone mines. The 

equation specifically excludes cases where the floor may 

yield or low strength bedding bands exist that can extrude 

and weaken the pillars. 

2)	 The presence of widely spaced large discontinuities that 

extend from the roof to the floor of the workings can be 

accommodated in the equation, however further investigation will be required if more than about 30% of the pillars 

in a layout are likely to be intersected by one or more large 

discontinuities that dip between dip 30 and 70 deg. 

3)	 The final pillar height should be used when calculating 

pillar strength. Where floor benching will be conducted, 

the final benched height should therefore be used. For 

rectangular pillars, the pillar strength should be calculated 

by using the equivalent width (equation 7 and table 6). 

4)	 The average pillar stress can be estimated by using the 

tributary area method (equation 3) or by using appropriate 

numerical models, as discussed earlier. 

5)	 The recommended factor of safety against pillar failure is 

1.8 and can be seen in figure 9 to represent the lower 

bound of current experience. Since the effect of large 

discontinuities is accounted for in the pillar strength 

equation, this safety factor is equally valid for pillars that 

are intersected by large discontinuities. 

6)	 Pillars having a width-to-height ratio of less than 0.8 

should be avoided. Figure 9 shows that there has been a 

natural tendency for mines to avoid these slender pillars. 

Nine of the layouts that had width-to-height ratios of less 

than 0.8 are no longer in use for various reasons, while 

only four mines are currently operating with these slender 

pillars. In addition, this study and other investigations 

have shown that slender pillars are more severely affected 

by the presence of discontinuities than wider pillars (15, 

41). 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 (11, 15, 42). The 

confining stress can be further reduced if low friction 

contacts exist between the pillar and the surrounding rock. 

7)	 Inspection of figure 9 reveals that a number of stable 

layouts exist that have large safety factors (>3.0) and 

width-to-height ratios that are less than 0.8. These layouts 

are mostly at very shallow depths of cover, typically less 

than 200 ft, and it has been possible to mine with very 

slender pillars. These pillars were found to be either very 

narrow, as little as 15 ft, or very tall, up to 125 ft. The 

strength and loading of narrow pillars are both sensitive to 

small variations in the overbreak, blast damage and pillar 

spacing. Large tall pillars, on the other hand, have high 

ribs which can represent a safety hazard and the roof 

becomes inaccessible and poorly visible with increased 

severity of potential rock fall impacts. The strength of 

these slender pillars is also more adversely impacted by the 

presence of unfavorable discontinuities than wider pillars. 

It is therefore not advisable to design layouts with such 

slender pillars, even with high calculated factors of safety. 

8)	 Equation 8 should only be used up to average pillar stress 

values that are 25% of the UCS to remain within the limits 

of past experience, as seen in figure 6. Since the design 

approach is entirely based on past experience, no 

comments can be made about pillar stability when pillars 

are loaded beyond 25% of the UCS. 

9)	 Rib spalling can initiate when the pillar stress exceeds 

 

 

 

 

 

 

 

about 11% of the UCS, which presents a potential safety 

hazard. Mitigation in the form of scaling or rib support 

may be required. Excessive scaling should be avoided, 

since it will reduce the pillar size and can result in 

increased pillar stress, exacerbating the situation. 

10)	 Pillar design cannot be carried out without considering 

roof stability. Roof spans directly impact the pillar stresses, 

because wider roof spans imply higher stresses in the 

pillars. As part of the pillar design, an evaluation of roof 

span stability and likely maximum stable spans should be 

conducted. Summaries of the experience with roof span 

stability in stone mines and layout considerations are 

provided in Esterhuizen et al. (43) and Iannacchione et al. 

(44). 

The shaded zone in figure 9 indicates the area in which 

equation 8 is likely to produce stable pillar layouts and coincides 

with the current experience in stone mines and observations of 

failed pillars. Since the design recommendations are based 

entirely on the observed performance of stone pillars, pillars that 

plot outside the shaded area in figure 9 are beyond the validity of 

these design guidelines. 

DESIGN INPUT 

Obtaining appropriate input to calculate pillar strength can 

sometimes be a challenge. Two design scenarios are presented 

below and the data requirements discussed. The first is called a 

scoping study in which no, or limited, geotechnical data might be 

available and the objective is simply to assess various pillar layout 

options. The second scenario is a final pillar design where 

accurate, representative geotechnical data is required. 

Data for a Scoping Study 

A scoping study should be carried out using conservative 

estimates of any unknown parameters. If the UCS is unknown, 

and the name of the stone formation is listed in table 1, the lower 

value of the range of strengths should be used. In other cases, a 

conservative estimate of the UCS at 6,400 psi is suggested. If the 

presence and frequency of large discontinuities is unknown, a 

conservative value of the discontinuity frequency of 1 per pillar 

should be used and a discontinuity dip of 60 deg. The average 

pillar stress can be calculated using the tributary area method 

(equation 3), since it provides a conservative upper limit result. 

The maximum depth of cover would typically be used in such a 

calculation. 

Data for a Final Design 

Equation 8 can be used to assist in developing a final pillar 

design if sufficient rock strength and rock mass data are available. 

Data required will include: 

1.	 A reliable determination of the average UCS of the pillar 

material. 

2.	 A geotechnical description of the rock mass and 

determination of the RMR value. This is required to 

verify that the rock mass quality, as expressed by the 

RMR, falls within the range of 65 to 85. 

3.	 Identification of the presence or absence of large angular 

discontinuities that can impact pillar strength. The 

average dip of the discontinuities and their average 

spacing should be determined. 
 

 

 

 

 

 

 

 

4.	 Verification that weak bedding bands do not exist in the 

proposed pillars and the floor strata are competent. 

5.	 The average pillar stress which can be calculated using the 

tributary area method (equation 3) if appropriate, or 

numerical models. 

The required data can be obtained through standard 

geotechnical procedures which can include geotechnical logging 

of core, strength testing of rock samples in a rock mechanics 

laboratory, and mapping of rock exposures in outcrops and 

underground in the mine. 

CONCLUSIONS 

This study has shown that the pillars in underground stone 

mines are expected to perform under a wide variety of depths of 

cover, rock strengths and geotechnical conditions. Pillar designs 

should be adapted to account for these variations in conditions. 

This paper provides a pillar strength equation that can be used to 

design stable pillar layouts provided the factor of safety is greater 

than 1.8 and the width-to-height ratio of the pillars are greater 

than 0.8. Recommendations are provided for selecting 

appropriate input parameters when geotechnical data is limited. 

The study has further shown that: 

•	 Two cases of wide-area roof failure occurred in stone 

mines in the past that may have been related to pillar and 

floor instability. These two cases could not be used to 

evaluate the pillar strength. 

•	 A total of eighteen individual failed pillars and several 

thousand stable pillars were observed. The failed pillars 

resulted in rib and roof stability hazards. 

•	 Large angular discontinuities were observed to cause a 

significant reduction in pillar strength and were a 

contributing factor in the failure of seven of the eighteen 

failed pillars. The suggested pillar strength equation 

accounts for the weakening effect of large angular 

discontinuities. 

•	 Fourteen of the eighteen failed pillars had width-to-height 

ratios of less than 0.8. The pillar design guidelines suggest 

that pillars with width-to-height ratios of less than 0.8 

should be avoided. 

•	 The upper limit of the average pillar stress of the layouts 

included in the study is about 25% of the UCS. The design 

guidelines suggest that the strength equation should not be 

used to design pillars when the average pillar stress will 

exceed 25% of the UCS. 

•	 Rib spalling can create a rock fall hazard. In some mines, 

rib spalling was observed to initiate when average pillar 

stress exceeded about 11% of the uniaxial compressive 

strength of the rock. 

•	 Rectangular pillars are increasingly used to assist in stone 

mine roof control and ventilation control. The suggested 

strength equation accounts for the strengthening effect of a 

rectangular pillar shape on its strength. 

The design guidelines presented in this paper were developed 

from direct observation of actual pillar performance in US stone 

mines, and therefore should only be applied in rock conditions and 

mining geometries that are similar to those included in the study. 

Disclaimer 

The findings and conclusions in this paper have not been 

formally disseminated by the National Institute for Occupational 

Safety and Health and should not be construed to represent any 

agency determination or policy. 

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