Evaluation of the Effects of Length on Strength of Slender Pillars in 

Limestone Mines Using Numerical Modeling 

Dennis R. Dolinar, Lead Research Engineer
 

Gabriel S. Esterhuizen, Senior Research Engineer 

NIOSH-Pittsburgh Research Laboratory 

Pittsburgh, PA
 

ABSTRACT 

In this National Institute for Occupational Safety and Health 

(NIOSH) investigation, numerical modeling is used to evaluate the 

degree the pillar strength is increased with a rectangular pillar over 

a square pillar. Three-dimensional models are developed with 

pillar sizes and geometries typical of underground limestone mines 

in the United States. Physical properties for the models are based 

on geotechnical surveys of nearly 30 underground limestone mines. 

To examine the effects of pillar length on pillar strength with the 

modeling, a parametric study was conducted where the pillar width 

was held constant while the length and height were varied resulting 

in width-to-height ratios ranging from 0.5 to 1.5 and width to 

length ratios ranging from 1 to 6. 

There are two aspects to pillar strength and failure. At 

relatively low stresses, pillars can fail as a result of geologic 

structure. At greater depths and sufficient extraction ratios, the 

pillars can also fail as a result of the vertical stresses exceeding the 

pillar strength. The effects of increasing pillar length regarding 

both strength aspects are presented in this study. 

Equations were developed for the increase in pillar strength 

with both height and length based on the model results. The results 

from these equations were compared to those from previously 

developed empirical equations for rectangular pillars. For the more 

slender limestone pillars with a width-to-height ratio below one, 

there is little increase in pillar strength for length-to-width ratios 

above 2 and almost no increase with length for a width-to-height 

ratio of 0.5. Only with the more squat pillars is there a significant 

improvement in strength with length. The gain in pillar strength 

with length for the pillars is explained by the change in the ratio of 

the area of the pillar affected by the ends to the area of the pillar. 

For the squattest pillar, the change in strength is also affected by an 

increase in pillar confinement along the central length of the pillar 

as the pillar transitions from a square to a long rectangular pillar. 

With geologic structure, added pillar length can significantly 

improve the pillar strength, depending on the strike of that structure 

with respect to the long pillar axis. 

INTRODUCTION 

This investigation was conducted as part of an effort to 

improve safety in underground limestone mines by evaluating and 

improving the performance of the pillars. Highly stressed pillars 

that are approaching failure have an increased potential for spalling 

and fall of material from the ribs; thus increasing the hazard of 

ground fall injuries to workers while pillar failure can impact 

operations. In this investigation, the effects on pillar strength of the 

pillar length are evaluated. There are two aspects to pillar strength 

and failure. At relatively low stresses, pillars can fail as a result of 

geologic structure (Esterhuizen et al., 2006; Iannacchione and 

Coyle, 2002). A detailed analysis of the strength reduction of 

square limestone pillars with geological weaknesses has previously 

been conducted (Iannacchione, 1999). The analysis did not include 

the effects of pillar length. However the following comment was 

made about rectangular pillars “The orientation of a discontinuity is 

important when the pillars are rectangular in that the strength will 

be affected most if the discontinuity is aligned with the long axis of 

the pillar.” At greater depths and sufficient extraction ratios, the 

pillars can also fail as a result of the vertical stresses exceeding the 

pillar strength. The effects of increasing pillar length regarding 

both strength aspects will be presented in this study. 

Rectangular pillars have been used in situations where 

horizontal stress is an issue. Through the use of rectangular pillars, 

the roof exposed to failure from the maximum horizontal stress can 

be minimized (Iannacchione et al, 2003). Longer pillars can also 

increase the efficiency of ventilation in the limestone mines (Grau 

et al., 2002 and 2006). With longer pillars, the number of 

ventilation stoppings can be reduced. However, this study is 

designed to determine to what degree the pillar strength can be 

increased by using rectangular pillars over the more standard 

square pillars. The approach in this study is to use numerical 

modeling to evaluate the effects of pillar length on the pillar 

strength. There have been a number of equations developed to 

predict the increase in strength from a square to a rectangular pillar. 

Many of these equations have not been substantiated, or have been 

used for coal pillars which are much squatter pillars than used in 

limestone. The pillar sizes and geometries used in the models are 

typical of underground limestone mines in the United States. 
 

 

 

 

 

 

 

 

 

 

 

BACKGROUND 

There have been a number of equations proposed for the 

increase in pillar strength with length. These include the 

Bauschinger-Johnson, Mark-Bieniawski, Grobbelaar and Wagner 

equations (Babcock, 1994; Mark and Chase, 1997; Mark 1999; 

Salamon, 1983). These equations were developed using either an 

intuitive, empirical, or analytical approach or a combination of 

these approaches. Figure 1 shows the strength increase over a 

square pillar from these equations for two fairly small width-toheight ratios. The equations obviously give somewhat different 

results, although all the equations indicate that the rate of increase 

in pillar strength diminishes as the pillar length increases. 

The Bauschinger-Johnson equation was developed to a large 

extent from laboratory testing. This experimental work formed the 

basis of the effects of the width-to-height ratio that other researches 

have developed for pillar design in both coal and hard rock mines. 

However, based on figure 1, the results of this equation are not 

relevant for a width to height ratio of 0.5, a ratio that is seen in 

benched areas of limestone mines. This ratio appears to fall outside 

the range of the experimental data used to develop the equation. 

Wagner’s approach develops an effective width for rectangular 

Strength Ratio Strength Ratio

1.4 

1.3 

1.2 

1.1 

1.0 

0.9 

1.4 

1.3 

1.2 

1.1 

1.0 

0.9 

0 2 4 6 8 10 12

 Length-to-Width Ratio

Bauschinger-Johnson 

Mark-Bieniawski 

Wagner 

Grobbelaar 

W/H=1.0 

Square pillar 

0 2 4 6 8 10 12 

Length-to-Width Ratio 

Wagner 

Mark-Bieniawski 

Bauschinger-Johnson 

Grobbelaar 

W/H=0.5 

Square pillar 

Figure 1. The ratio of the compressive strength of various 

rectangular geometries compared to a square pillar from four 

different pillar equations (Babcock, 1994; Mark and Chase, 

1997; Mark 1999; Salamon, 1983). (Top charts shows pillar 

width-to-height ratio equal to 1.0. Bottom charts shows pillar 

width-to-height ratio equal to 0.5). 

pillars based on the pillar area and perimeter that can then be input 

into equations for pillar strengths that use a width-to-height ratio. 

For an infinitely long pillar, the effective width is twice that of a 

square pillar. Grobbelaar developed a rather complex analytical 

approach to pillar design. However, the equations can be 

simplified to develop an equation with the effects of length 

included. The Grobbelaar and Wagner equations or approaches 

have not been proven or substantiated (Roberts et al., 2005) 

The Mark-Bieniawski equation is based on the Bieniawski 

formula for square pillars with an analytical analysis for load 

distribution in the pillar at peak load used to develop an equation 

for rectangular pillars. However, the constants and parameters in 

this equation are related to squat pillars in coal. Both equations 

form the basis of determining the pillar strength in both ALPS 

(Analysis of Longwall Pillar Stability) and ARMPS (Analysis of 

Retreat Mining Pillar Stability), which have been used in coal pillar 

design and are empirically developed from actual case studies of 

the performance of pillars (Mark, 1990; Mark and Chase, 1997; 

Mark, 1999). 

For underground limestone, the pillars are very slender 

compared to those used in underground coal mines. On 

development, the pillars may be 8 m high by 12 m wide resulting in 

a width-to-height ratio of 1.5. With first bench mining, the height 

can be 12 m resulting in a width-to-height ratio reaching one. With 

multiple benches, the pillar can reach 24 m in height or greater, 

resulting in a width-to-height ratio of 0.5 or less. 

Another consideration for limestone mines is the type of failure 

that can occur in these high openings with slender pillars. It has 

been noted in hard rock mines that pillar spalling and failure occur 

at stress levels well below the expected rock and pillar strength 

(Stacey and Yathavan, 2003). This has been attributed to brittle 

failure that can develop where the confining stresses are low and 

the geometry does not constrain crack growth. The application of 

brittle failure considerations to rectangular pillar design and 

performance will be discussed. 

MODEL FAILURE CRITERIA 

For modeling, an appropriate failure criteria must be selected 

along with the compatible properties that can produce realistic 

results. Pillars in hard rock mines, including limestone, will 

typically slab and split at stresses that are only 0.04 to 0.35 of the 

uniaxial compressive strength of the intact rock (Stacey and 

Yathavan, 2003). This is thought to be caused by brittle failure 

(Hajiabdolmajid et al., 2002a; Diederichs, 2002). With brittle 

failure, the mobilization of the frictional strength is delayed until a 

significant amount of the cohesive strength has been lost as a result 

of extension cracks. The slabbing and splitting is the result of the 

initiation and uninhibited growth and propagation of extension 

cracks into near vertical fractures at low confinement (Diederichs, 

2002). At low confining stresses near the surface of an excavation, 

the cracks can freely propagate and interact upon crack initiation 

because the frictional strength is not mobilized (Hajiabdolmajid et 

al., 2002b). This results in the yield strength of the rock reducing 

to the crack initiation stress (Diederichs, 2002; Hajiabdolmajid et 

al., 2003). Essentially, once a crack initiates, it is free to grow and 

propagate in such an environment. In a more confined environment, 

crack propagation and growth would be restricted and yield would 

only occur as the amount of cracking and coalescence of the cracks 

was sufficient to allow for interaction between the cracks. Along 

the side of a limestone pillar, the confining stresses are low and the 
 

 

 

 

 

 

 

 

 

 

geometry should not restrict the propagation of cracks that do 

initiate. Also from observations in a number of limestone mines, 

pillar spalling has initiated at stress levels that are about 10 percent 

of the uniaxial compressive strength of the rock (Esterhuizen et al 

2006). Therefore, since the brittle failure criterion appears to be 

applicable to limestone pillars, this criterion is used in the models 

to determine failure. Further, brittle failure involves the failure of 

the intact rock and is not related to the more general rock mass 

strength. Therefore, the brittle failure is associated with the intact 

rock strength. 

In this study for brittle failure, the rock strength is based on 

crack initiation during a uniaxial compression test. This can be 

about 1/3 of the uniaxial compressive strength of the intact rock. 

To model the brittle failure, a bilinear Mohr-Coulomb criterion is 

used. Brittle failure can be considered a special case of the MohrCoulomb criterion where the friction angle is zero. The second 

portion of the envelope at higher confining pressures is based on 

the cohesion and friction angle for the rock mass. From the 

intersection of these two envelops in the maximum-minimum 

(sigma 1-sigma 3) stress space, the confining pressure where this 

transition will take place in the model can be calculated. For 

comparison, the models were also run using a linear MohrCoulomb criterion with the cohesion and friction angle based on 

the rock mass properties. 

The effects of increasing pillar length when geologic structures 

weaken a pillar are also examined. For these models, the Coulomb 

criterion of failure with a plane of weakness is utilized to determine 

failure (Horino and Ellickson,1970; Jaeger and Cook, 1976). In 

this case, the dip of the weakness and the angle of friction and 

cohesion along the planes of weakness are the controlling strength 

parameters. 

MODEL PARAMETERS 

A three-dimensional finite difference code was used to model 

the pillars. This is the FLAC3D code version 3.0 (Itasca, 2005) 

The model layout and input parameters are discussed below. 

Model 

The basic layout and geometry used for the models is shown in 

figure 2. The model consisted of a main and immediate roof and a 

main and immediate floor above and below the pillar. The total 

thickness of the roof is 56 m and of the floor 64 m. The crosscuts 

and entries surrounding the pillar were 6 m wide (one half a 12-m 

roof span). For the pillar, the elements were 1 m by 1 m by 1 m 

cubes. 

In the models, the roof and floor were fixed along the sides in 

the x and y directions with the floor fixed along the bottom in the 

vertical direction. A force (through the use of grid point velocities) 

was applied to the top of the model in the vertical direction, thus 

loading the pillar at a controlled rate until failure. The average 

stress across the pillar was then calculated during model loading 

with the peak pillar stress being captured. This average peak pillar 

stress is the maximum pillar strength and is used to evaluate the 

effects of the changes in pillar geometry. 

Model Properties 

The rock mass properties used in the model are based on the 

rock mass rating developed from 30 underground limestone mines 

(Esterhuizen et al., 2006). The average RMR for these mines is 

approximately 75, and this value was used to calculate the "s" and 

Pillar

Main Roof

Main Floor

Floor

Roof

Pillar 

Main Roof 

Main Floor 

Floor 

Roof 

Figure 2. Basic model configuration used in the numerical 

modeling. 

"m" parameters for the Hoek-Brown failure criteria (Hoek and 

Brown, 1980; Brady and Brown, 1993). These values were then 

converted to a friction angle and cohesion that could be used in the 

Mohr-Coulomb criteria (Hoek and Brown, 1997). Table 1 gives 

the properties used in the models for both the brittle and the MohrCoulomb criteria. The failure strength parameters are at zero 

percent plastic strain, while the values at the other levels of plastic 

strain are the post failure properties used in the models at that strain 

level. Strain softening is used to simulate the post failure 

performance of the elements in the pillar. The percent of plastic 

strain is the amount of strain after failure has occurred. These post 

failure properties control how fast the material strength decreases to 

the residual level after failure. In the models, a bulk modulus of 40 

GPa and a shear modulus of 24 GPa were used for the pillar, roof 

and floor. Further, the roof and floor elements were not allowed to 

fail. The roof and floor are in this case composed of the same 

material as the pillar. Even if the roof and floor elements were 

allowed to fail, the elements are more confined than the pillar 

elements and the pillar would fail first thus reducing the stress in 

the roof and floor resulting in the roof and floor elements never 

reaching the failure stress. Not allowing the roof and floor elements 

to fail reduces the model run times. 

For the brittle failure, an intact uniaxial compressive for the 

limestone of 150 MPa was used. The brittle pillar compressive 

strength was then taken as 1/3 of this value or 50 MPa. Again this 

value is based on the assumed stress level required for crack 

initiation. However, for the modeling, a friction angle and 

cohesion must be calculated. These are calculated for the brittle 

failure using the Mohr-Coulomb criteria, the brittle failure uniaxial 

compressive strength and a friction angle of zero. With the brittle 

failure, above a certain confining stress, the Mohr-Coulomb 

criterion based on the rock mass properties is used. This 

necessitates a bilinear model for the brittle failure. Figure 3 shows 
 

 

 

 

 

 

 

 

 

 

 

- - - 

- - 

- - - - - 

- - - 

Table 1. Peak and post failure strength properties used in models for the different failure criteria. Peak properties are at zero 

percent plastic strain. 

Plastic Strain 

Property 0 0.001 0.002 0.005 0.01 0.02 

Brittle Failure 

Cohesion, MPa 25 - - 5 1.25 -

Tension, MPa 2.7 0 - - - -

Friction Angle, deg 0 - - - - -

Dilation, deg 30 - - - 0 -

Mohr-Coulomb Failure 

Cohesion, Mpa 8.1 

Tension, Mpa 2.7 

Friction Angle, deg 47.6 

Dilaton, deg 30 

Jcohesion, MPa 1 

Jtension, MPa 0.4 

Jfriction Angle, deg 42 

Jdilation, deg 0 

0 - -

Ubiquitous Joints 

0.2 - - - 0 

0 - - - -

- 30 - - -

- - - - -

Sigma 1, MPa 80 

70 

60 

50 

40 

30 

20 

10 

0 

Mohr-Coulomb 

Brittle 

Mohr-Coulomb (nonbrittle) 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 

Sigma 3, MPa 

Figure 3. Mohr-Coulomb and bilinear brittle failure criteria as 

used in the numerical models. 

the bilinear brittle Mohr-Coulomb criterion and the Mohr-Coulomb 

criterion for the rock mass. 

Geologic structures were modeled using the ubiquitous joint 

feature in FLAC3D. The properties of the ubiquitous joints are 

also given in table 1. In these models, the failure may occur along 

the joints or through the solid rock or the rock mass. Where failure 

occurs in the rock or rock mass the bilinear brittle failure criteria is 

used. The dip angle for the planes of weakness was taken as 60 

degrees from the horizontal. Again, strain softening is used to 

develop the post failure behavior of the elements in the pillars. 

PARAMETRIC DESIGN 

In this investigation both the pillar length and model failure 

criteria were varied. The criteria included brittle, Mohr-Coulomb 

or a ubiquitous joint failure. The ubiquitous joint models were 

used to evaluate the effects of geologic structure that could 

2.43 0.8 

0 -

significantly reduce the pillar strength and be a factor at relatively 

shallow depths. The brittle and Mohr-Coulomb failure criteria 

were used to evaluate the conditions where the pillar stress is 

sufficient to cause pillar failure. 

For each of these failure criteria, the pillar width-to-length ratio 

was varied as was the pillar width-to-height ratio. In all cases, the 

pillar width was 12 m. Table 2 shows the pillar heights and lengths 

used in the models for a given failure criteria. The base case in 

each series is a square pillar.

 Table 2. Experimental matrix for modeling of limestone pillar 

strengths. All pillars had a width of 12 m. 

Length-to­ Width-to-height ratio 

width ratio 0.5 0.66 1 1.2 1.5 

Brittle Failure 

1 x x x x x 

2 x x x x x 

3 x x x x x 

4 x x x x x 

5 x 

6 x 

Mohr-Coulomb Failure 

1 x x x x x 

2 x x x x x 

3 x x x x x 

4 x x x x x 

5 x x x 

6 x x x 

Ubiquitous Joint Failure, joint strike 90 °

 to long axis of pillar 

1 x - x - x 

2 x - x - x 

3 x - x - x 

Ubiquitous Joint Failure, joint strike 0 °

 to long axis of pillar 

1 x - x - x 

2 x - x - x 

3 x - x - x 
 

 

 

0 

20 

40 

60 

80 

100 

120 

140 

160 

180 

Strength, MPa 

W/H = 1.5 

W/H = 1.2 

W/H = 1.0 

W/H = 0.66 

W/H = 0.5 

Mohr-Coulomb Failure Criteria 

MODEL RESULTS 

The parameter used to determine the effects of pillar length on 

the pillar strength was the average vertical stress on the pillar at 

failure. This stress is the maximum pillar strength for the 

conditions. The strength of the square pillar at a width of 12 m is 

then used as a baseline to evaluate the increase in pillar strength 

with length. Table 3 shows the maximum pillar strength of the 

model pillars for brittle and Mohr-Coulomb failure criteria and for 

both joint orientations for the ubiquitous joint models. 

Table 3. Pillar strengths determined from models for different 

pillar geometries and failure criteria. 

Width-to Height Ratio Width-to- 180 

Brittle Failure Criteria 0.5 0.66 1 1.2 1.5 length ratios 160 

Pillar Strength, MPa 140

Brittle Failure 

1 49.6 51.6 74.1 89.4 105.8 120 Strength, MPa 

2 49.4 58 78.4 100.5 121.1 

3 49.3 59.6 78.5 111 126.3 

4 49.1 60.6 79 112 139.1 

100 

80 

60 

W/H = 1.5 

W/H = 1.2 

5 - - - - 156.4 

6 - - - - 157.5 40 

W/H = 1.0 

Mohr-Coulomb Failure 20 W/H = 0.66 

W/H = 0.5 

1 43.7 52.4 80.5 101.3 127.6 0 

0 10 20 30 40 50 60 70 80 

2 45.7 59.2 101 124.1 147.3 Length, m 

3 46.1 61.2 107.9 133.5 155.6 

4 46.3 62.3 111.9 137.7 163.3 Figure 4. The strength increase with pillar length for various 

pillar width-to-height ratios for the Mohr-Coulomb failure 5 - - 114.5 140.3 168.4 criteria (top) and for the brittle failure criteria (bottom). 6 - - 116 142.1 170.8 

Ubiquitous Joint Failure, 90 ° to long axis of pillar 120 

1 4.7 - 39 - 62.8 

2 24.8 - 59.8 - 91.2 100 

3 34.5 - 69.3 - 103 80 

Ubiquitous Joint Failure, 0 ° to long axis of pillar 

1 4.8 - 39 - 62.8 

Strength, MPa 

90 degrees 

0 degrees 

W/H = 1.5 

W/H = 1.0 

W/H = 0.5 

60 

2 4.8 - 40.6 - 68 

3 4.7 - 40.6 - 70.6 40 

Figure 4 shows the maximum pillar strength versus the pillar 20 

length for both the brittle and Mohr-Coulomb failure criteria for 

each pillar height. The maximum pillar strengths occur with the 0 

For the ubiquitous joint models, the direction of the strike of 

the planes of weakness was either 0 or 90 degrees to the long axis 

of the pillars. For a 0-degree orientation, the strike of the joints 

was parallel to the long axis of the pillar and for a 90-degree 

orientation, the strike of the joints was perpendicular to the long 

axis of the pillar. 

Coulomb failure. Further, with increasing length, the rate at which 

the pillars gain strength decreases for both failure criteria. 

Figure 5 shows the pillar strength versus the pillar length for 

each pillar height for the ubiquitous joint models for both joint 

orientations. When the strike of the joint system is perpendicular to 

pillars with the largest width-to-height ratios and the longest 

lengths for both failure criteria. In general, the pillar strength does 

increase with pillar length, although this depends on the pillar 

width-to-height ratio. For the slenderest pillars with a width-toheight ratio of 0.5, there is no increase in strength with length with 

brittle failure and only a small increase in strength with the Mohr12 24 36 

Length, m 

Figure 5. The strength increase with pillar length for various 

pillar heights with ubiquitous joints either perpendicular (90 

degrees) or parallel (0 degrees) to the long pillar axis. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

the long axis of the pillar, there is clearly an increase in the pillar 

strength with pillar length. Again, the rate of increase in pillar 

strength decreases with the length. However, when the structures 

strike is parallel with the long axis of the pillar there is a much 

smaller increase in strength with length for the squat pillars and 

little or no increase in pillar strength with length for the slender 

pillars. 

PILLAR STRENGTH EQUATIONS BASED ON 

GEOMETRY 

Clearly, there is interaction between the pillar width-to-height 

and width-to-length ratios for both the brittle and Mohr-Coulomb 

failure with the dominant parameter being the width-to-height ratio 

(table 3, figure 4). Further, the contribution of the length in 

increasing the strength diminishes with increasing length. 

Therefore, any equations that are developed for the pillar strength 

must reflect this interaction and the diminishing effects of length. 

The following equations were developed for these two failure 

criteria based on a multiple regression analysis of the data. The 

dependent variable in the regression analysis was the ratio of the 

pillar strength to the pillar strength for a square pillar with a widthto-height ratio of one. 

For the brittle failure criteria the following equation was developed

 CRbf = CS1(1.51(W/H) +0.46(W/L) -0.78(W/H)(W/L)-0.18) (1) 

where: CRbf = compressive strength with brittle failure for a 

given pillar width and length, 

CS1 = compressive strength for a pillar with a width-toheight to length ratio of 1.0, 

W = pillar width,

 H = pillar height, and

 L = pillar length. 

The multiple correlation coefficient squared for this regression is 

0.96. 

For the Mohr-Coulomb failure criteria, the following equation 

was developed 

CRmc = CS1(1.64(W/H)+0.15(W/L)-0.58(W/H)(W/L)-0.18) (2) 

where: CRmc = compressive strength with Mohr-Coulomb for a 

given pillar width and length. 

The multiple correlation coefficient squared for this regression 

is 0.99. The correlation coefficients of both these equations are 

very high. However, the equations are based on modeled pillars 

with the range of width-to-height ratios from 0.5 to 1.5 and width 

to length ratios from 1.0 to 6.0. Therefore, the application of these 

equations should be limited to those ranges. 

Figure 6 shows a graph of the ratio of the rectangular to square 

pillar strength versus the pillar length-to-width ratio for different 

width-to-height ratios developed from both these equations. For 

the brittle failure criteria, there is a large difference in the gain in 

pillar strength with length depending on the width-to-height ratio of 

the pillar. An increase in strength of over 40 percent occurs for the 

squattest pillar. For a width-to-height ratio of 0.6, the strength ratio 

is nearly one for length-to-width ratios from one to six. Essentially 

there is no increase in pillar strength with length for width-toheight ratios less than 0.6. Therefore, the brittle failure equation is 

not valid for ratios that are below 0.6 since there is no change in 

Strength Ratio 

1.5 

1.4 

1.3 

1.2 

1.1 

1.0 

0.9 

0 1 2 3 4 5 6 7 

Length-to-Width Ratio 

Brittle Failure 

Mohr Coulomb Failure 1.5 

1.0 

1.5 

1.0 

0.5 

0.66 

0.66 

0.6 

Figure 6. The ratio of a rectangular to a square pillar strength 

versus the pillar length-to-width ratio for various pillar widthto-height ratios from equations developed from both the brittle 

failure and Mohr-Coulomb failure criteria. 

strength for more slender pillars. Based on the models results, the 

width-to-height ratio where there is no further increase in strength 

with length occurs between 0.5 and 0.66. Therefore, for the slender 

pillars with width-to-height ratios of about 0.66 and below, there is 

little increase in the pillar strength with length. 

For the Mohr-Coulomb failure criteria, the change in the 

strength ratio for a similar range of width-to-height ratios is much 

less than with brittle failure. This results in part from a much lower 

coefficient for the independent width to length term in the equation. 

With the squattest pillar, the strength ratio is just under that for the 

brittle failure. There is also an increase in pillar strength with 

length with the most slender pillar as compared to no strength 

increase when the same pillar is subject to brittle failure. Even 

though there is some increase in pillar strength with length for 

pillars with a width-to-height ratio of 0.5, the amount of the 

strength increase is small because of the reduced pillar strength for 

these slender pillars. 

For the more slender pillars with width-to-height ratios from 1 

to 0.66 (12- and 18-m-high pillars) with brittle failure and with a 

width-to-height ratio of 0.66 for the Mohr-Coulomb failure, there is 

some strength gain from a square pillar to a rectangular pillar with 

a length-to-width ratio of two, but little increase in strength for a 

pillar with a length-to-width ratio of three. Essentially, for pillars 

with these width-to-height ratios there is no benefit of increased 

pillar strength beyond a pillar length-to-width ratio of two. In 

general for both failure criterion, with width-to-height ratios below 

one, ratios that will occur with benching, the benefits of length on 

pillar strength diminish rapidly especially beyond a length-to-width 

ratio of two. With the benching, as the pillar width-to-height ratio 

reaches 0.5, all gains in strength from the pillar length are 

essentially lost. 

Of these two criteria used in the modeling, the brittle failure 

criterion is probably more relevant to use in the underground 

limestone mines because of the type of failure that could be 

expected. Therefore, the brittle failure equation will probably give 

more representative pillar strengths when rectangular pillars are 

used in underground limestone mines. The brittle failure equation 

can also account for the lack of strength increase with length in the 

most slender pillars, while the Mohr-Coulomb equation does not. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.12 

1.08 

1.04 

1.00 

0.96 

Brittle Failure Model 

Grobbelaar 

Bauschinger-Johnson 

Mark-Bieniawski 

Wagner 

W/H ratio = 0.6 

Strength Ratio Strength Ratio 1.50 

1.40 

1.30 

1.20 

1.10 

1.00 

W/H ratio = 1.5 

Bauschinger-Johnson 

Brittle Failure Model 

Mark-Bieniawski 

Wagner 

Grobbelaar 

0 1 2 3 4 5 6 7 8 

Length-to-Width Ratio 

Figure 7. The ratio of a rectangular to square pillar strength 

versus the length-to-width ratio for the pillar comparing the 

brittle failure pillar equation to other pillar equations used to 

estimate the strength of rectangular pillars(Babcock, 1994; Mark 

and Chase, 1997; Mark 1999; Salamon, 1983) . 

The results of the equation for the brittle failure can be 

compared to the four published equations discussed in the 

background section. Figure 7 shows a comparison of these 

equations for two pillar width-to-height ratios. For the squat pillar, 

the brittle failure equation predicts larger strength increases with 

pillar length than the other equations. However, for the slender 

pillar, the brittle failure equation is the only equation that predicts 

little or no increase in pillar strength with pillar length. 

BASIS FOR INCREASE IN PILLAR STRENGTH 

WITH LENGTH 

The increases in the pillar strength with the change in length 

can be evaluated in terms of the pillar geometry. In this case the 

geometry of interest is the relationship between the areas affected 

by the pillar ends to the total area of the pillar. Figure 8 shows a 

pillar separated into the areas affected by the ends of the pillar and 

a central portion of the pillar length not affected by the ends. 

Essentially, there is central portion of the pillar length that behaves 

as if there were no pillar ends. 

Based on figure 8 the following equation can be developed for 

the relationship between the strength of a square and rectangular 

pillar; 

CR = CS + Fc (1 – Ae/Ap ) (3) 

Lp 

Pillar End Pillar End Central Pillar Core Central Pillar Core Pillar End

Le Lc Le 

Wp 

Plan View 

Figure 8. Relationship of the dimensions of a pillar, the central 

pillar core and the pillar ends where Lp = length of pillar, Wp = 

width of pillar, Lc = length of central pillar core and Le = length 

of pillar ends. 

where: CR = compressive strength of a rectangular pillar, 

CS = compressive strength of a square pillar, 

Fc = confining stress factor, 

Ae = area of the ends of the pillar, and 

Ap = area of the pillar. 

Essentially, there is a central pillar core that is more confined than 

the ends of the pillars, and will therefore carry more stress at peak 

pillar strength than the ends. A higher percent of the pillar in the 

central region results in higher pillar strengths. 

For the Mark-Bieniawski rectangular pillar equation, from 

geometric considerations, the pillar stress distribution is calculated 

based on end effects that are one half the pillar width (Mark and 

Chase 1997). If the length of the pillar in an end area is assumed to 

be equal to one half of the pillar width (6 m), the ratio of the pillar 

end area to the pillar area for the different lengths of modeled 

pillars are 1.0 for the 12 m pillar, 0.5 for the 24 m pillar, 0.33 for 

the 36 m pillar, 0.25 for the 48 m pillar, 0.2 for the 60 m pillar, and 

0.17 for the 72 m pillar. These ratios can then be used to calculate 

the confining stress factor for each of the modeled pillar results 

using equation 3 since the compressive strengths for the square and 

rectangular pillars are known for a given geometry (table 3). 

Table 4 shows the calculated confining stress factors. For pillars 

that are 12 m or higher, the stress factor for either failure criteria 

does not vary significantly with pillar length. For these slender 

pillars, the change in the ratio of the area of the ends to the pillar 

area can explain the increase in pillar strength. 

This is not the case for the squattest pillar. The confining 

stress factor increases significantly above a pillar length of 36 m for 

both types of failure. Not only is ratio of the area of the ends to the 

total pillar area of importance, but also that the ends are 

significantly reducing the confinement along a portion of the length 

of the pillar. This may be occurring in the more slender pillars but 

can not be observed in the pillar strength because the pillar 

performance is dominated by the pillar height. However, the 

increase in the confining stress is limited to a transition from a 

square to a rectangular pillar that appears to be completed with a 

60-m pillar length. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. Confining stress factor calculated for each failure 

criteria. 

Width-to-Length Width-to-Height Ratio 

Ratio 0.5 0.66 1 1.5 

Brittle Failure -Confining Stress, MPa 

2 0 8.6 12.8 30.6 

3 0 6.6 11.9 30.5 

4 0 6.5 12 44.4 

5 - - - 63.3 

6 - - - 62.3 

Mohr-Coulomb Failure-Confining Stress, MPa 

2 4 13.6 41 39.4 

3 3.6 13.1 40.9 41.8 

4 3.5 13.2 41.9 47.6 

5 - - 42.5 51 

6 - - - 52 

The reduction in confinement along a portion of the length of 

the squattest pillar can be seen with the brittle failure with the depth 

of brittle failure depending on confinement. The depth of brittle 

failure will be less with higher confinement. For the square, 8-mhigh pillar, the depth of brittle failure is 2 m around the entire pillar. 

For the 24-m-long pillar, the depth of brittle failure is only one 

meter along the middle third of the pillar and 2 m deep along each 

third of the pillar at the ends. For the 60-m-long pillar, the depth of 

failure is only 1 meter along the middle two thirds of the pillar and 

2 m deep along the remaining pillar ends. 

For the most slender pillars with a width-to-height ratio of 0.5, 

there is little or no change in strength with increasing length. For 

the brittle failure, the yield stress is the peak pillar strength with a 

rapid collapse in pillar strength after yield. Figure 9 shows stressstrain curves for a 24 m high pillar with a 36 m length for both 

brittle and Mohr-Coulomb. Essentially, with brittle failure there is 

no confinement that develops in the pillar at peak pillar strength. 

With the Mohr-Coulomb failure, the confinement is sufficiently 

small such that any decrease in the end effects with length makes 

only a small difference in the pillar strength. Essentially pillar 

height dominates the pillar behavior. Stress, MPa 60 

50 

40 

30 

20 

10 

0 

Brittle Failure 

Mohr-Coulomb Failure 

0 0.05 0.1 0.15 0.2 0.25 

Strain, pct 

Figure 9. Stress-strain curves for a 24 m high pillar with a 36 m 

length for both brittle and Mohr-Coulomb failures. 

GEOLOGIC STRUCTURE 

For geologic structure in the form of ubiquitous joints, there is 

a significant reduction in pillar strength compared to non-jointed 

pillars with the same geometry. The amount of this reduction 

depends on the pillar geometry as well as the orientation of the 

joints. However, the largest reduction in strength occurs with the 

most slender pillars. 

For the jointed pillars, the pillar geometry and orientation of 

the geologic structure also have a significant effect on the jointed 

pillar strength. A longer pillar significantly increases the pillar 

strength when the joint strike is orientated perpendicular to the long 

axis of the pillar. Essentially, the joints are daylighted at the ends 

thus weakening the ends of the pillar but leaving a central core that 

depends on the general rock mass or intact rock strength. With 

increasing pillar length, the size of this central core increases and 

thus increasing the strength. Therefore, to improve the pillar 

strength in the presence of geologic structure, the amount of the 

pillar that is not affected by a daylighted structure must be 

sufficient. It should be noted that joint dip, in this case 30 degrees 

from vertical will affect the amount of the central pillar core that 

can develop. For the most slender pillar with a width-to-height 

ratio of 0.5, a very large increase in strength occurs from a square 

pillar to a rectangular pillar with a length-to-width ratio of 2. The 

joints are daylighted across the entire square pillar, completely 

controlling the pillar strength resulting in a very weak pillar. Thus, 

when a central pillar core develops with an increase in length, there 

is a significant gain in strength. 

If the strike of the joints or structure is parallel to the long 

pillar axis, significant improvement in the pillar strength only 

occurs with the squattest pillars. For a slender pillar with a widthto-height ratio of 0.5, there is no increase in strength as a result of 

the structure daylighting through both the square and rectangular 

pillars. However, significant gains in the pillar strength can be 

achieved where joints or structure are orientated perpendicular, 

rather than parallel, to the long axis of the pillar. 

CONCLUSIONS 

The effects of increased pillar length on pillar strength for 

pillars in limestone mines was investigated using numerical 

modeling. Based on the results of the modeling, several 

conclusions can be developed. 

For the failure of the rock and rock mass without specific 

geologic structures, two failure criteria were examined in the 

models: a Mohr-Coulomb and a brittle failure criterion. Thus, two 

different pillar design equations involving the pillar length were 

developed from a regression analysis of the modeling results. 

However, because of the nature of limestone mine and pillar 

geometry and the type of expected pillar failure, the equation based 

on the brittle failure criteria is the one most relevant to estimating 

rectangular limestone pillar strengths. 

For the pillar strength equation involving brittle failure, the 

effects of length vary with the width-to-height ratio of the pillar. 

Below a width-to-height ratio of about 0.66, there is little or no 

increase in strength with length. At these slenderness ratios, there 

is little or no confinement across the pillar at peak strength. Only 

with the more squat pillars is there a significant gain in strength 

with pillar length. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

With brittle failure and width-to-height ratios of one or less, 

there is little or no increase in the pillar strength with length 

especially beyond a length-to-width ratio of two. As a result, if 

rectangular pillars are used with benching, as the width-to-height 

ratio drops below one, the benefits of increased pillar strength from 

pillar length will be lost. 

The effects of pillar length on strength as determined by the 

equation using the Mohr-Coulomb failure criteria is not as 

dependent on the pillar width-to-height ratio as with the brittle 

failure. However, with very slender pillars there is only a limited 

amount of confinement; therefore the gain in strength is also small. 

Based on either equation, there is little benefit to be gained in pillar 

strength with increased length for slender pillars. 

The gain in pillar strength with length for the more slender 

pillars can be explained by the geometry of the pillar. The 

geometry that determines the strength increase is the ratio of the 

area of the pillar affected by the ends to the area of the pillar 

resulting in the rate of increase in strength diminishing with length. 

For the squattest pillar, the change in strength is also affected by an 

increase in pillar confinement along the central length of the pillar 

as the pillar transitions from a square to a long rectangular pillar. 

Where geologic structure can weaken a pillar and the structure 

is daylighted, additional pillar length can significantly increase the 

pillar strength. In general, however, the greatest gain in pillar 

strength is obtained when the strike of the geologic structure is 

orientated 90-degrees to the long axis of the pillar. Essentially, 

with length, the core area of the pillar not affected by the structure 

increases, thus resulting in a gain in pillar strength. 

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