Effects of Weak Bands on Pillar Stability in Stone Mines: Field 

Observations and Numerical Model Assessment 

Gabriel S. Esterhuizen, Senior Research Engineer 

John L. Ellenberger, Lead Research Scientist

NIOSH-Pittsburgh Research Laboratory 

Pittsburgh, PA, USA 

ABSTRACT 

Observations of pillar conditions in limestone mines showed 

that the presence of weak bands in the limestone can result in pillar 

damage at stresses that are lower than one would otherwise expect. 

The objective of this National Institute for Occupational Safety and 

Health study was to investigate the mechanism of failure caused by 

the presence of weak bands by using a series of numerical models 

based on realistic physical properties. The effect of the weak bands 

on pillar strength was also investigated by developing a series of 

pillar models with different width-to-height ratios and determining 

their strength by simulating a servo-controlled loading condition. 

The material properties of the bands and the limestone were varied 

and their effects on pillar strength were determined. 

Model results demonstrated failure development similar to the 

pillars observed in the field. The weak bands were seen to develop 

tensile stresses in the limestone as they extrude under increasing 

stress. The limestone fails in tension at a much lower stress than its 

expected uniaxial compressive strength. The results showed the 

load bearing capacity of the pillars can be reduced significantly by 

the presence of multiple thin weak bands. The degree of strength 

reduction is largely dependent on the compressive strength, the 

frictional resistance and the thickness of the weak bands. Single 

weak bands do not affect pillars as severely as multiple bands. 

However, the model results show that a single thick weak band, 

which comprises more than about 2 percent of the pillar height, can 

cause a significant reduction in pillar strength. The weakening 

effect of the bands become less severe as the pillar width-to-height 

ratio is increased. 

Field observation of spalling and failure associated with weak 

bands in limestone pillars show that the failure mechanism in the 

models resembles the observed failure. The model results provide 

insight into the important factors affecting pillar strength in the 

presence of weak banded materials. 

INTRODUCTION 

The National Institute for Occupational Safety and Health 

(NIOSH) has collected data at 34 underground stone mines in a 

study of factors that influence pillar and roof span stability. During 

field data collection, it was unusual to see significant pillar damage 

but in several instances damage was associated with thin weak 

bands in the pillars (Esterhuizen et al., 2006). These observations 

prompted an effort to understand the mechanism that produces 

damage near and between the weak bands and its effect on pillar 

strength. 

Field observations of pillar damage have been discussed in 

earlier works where pillars exhibited damage in excess of what one 

would intuitively expect from normal loading (Alber and Heiland, 

2001a, 2001b). It has been recognized that rock failure can be the 

result of purely tensile damage under low confinement. In some 

cases, tensile crack damage dominated over shear cracks by more 

than 50 to 1 (Diederichs, 2002). Studies in granite recognized that 

when confinement is low, extension fracturing is the dominant 

mode of failure. (Hajiabdolmajid et al., 2002). It has also been 

concluded that the absence of confinement in slender pillars can 

result in a brittle failure mode, while wider pillars also combine 

shearing (Esterhuizen, 2006). In this paper the term “brittle 

failure” will be used to describe the failure mode of extension 

fracturing parallel to the major principle stress, as opposed to shear 

failure where conjugate shear planes develop oblique to the 

direction of the major principal stress. 

The effect of end constraints on the strength of rock samples 

tested in the laboratory has received much attention in the literature, 

Jaeger and Cook (1979) give a good overview of the subject. A 

theoretical analysis of stresses within cylindrical specimens (Peng, 

1971) showed that significant tension can be induced if soft 

extruding end pieces are used in compression testing. 

Experimental results showed that the strength of granite samples 

can reduce from as high as 207 to 96 MPa (30,000 to 14,000 psi) 

by using various end materials in compression testing. 

The models described in this paper were constructed and run 

using the FLAC-2D (Cundall et al., 2005) finite difference program. 

In addition to the design of a series of basic models with a range of 

different width-to-height (W/H) ratios, physical properties of the 

limestone and the weak bands that could contribute to the pillar 

behavior were addressed. This study utilized the results of 

laboratory strength tests and the RocLab software to produce 

realistic properties for the models (Hoek et al., 2002). The effects 

of parametric variation are also presented and discussed. 
FIELD OBSERVATION 

During the course of visits to 34 limestone mines in the Eastern 

and Midwestern U.S. a number of pillars were observed exhibiting 

varying degrees of failure in association with one or more thin 

bands of weak material. The failure can manifest itself as a ledge 

or overhang either above or below a thin bed of material that was 

notably weaker than the encasing strata, shown in figure 1. The 

development of overhangs, ledges, and concave zones associated 

with thin planes of softer material was observed frequently, with 

the pillar presenting the appearance of being much more heavily 

loaded than one would expect as a result of simple tributary area 

pillar loading. 

Figure 1. Overhang caused by a weak band in a limestone pillar. 

A second failure mode that seemed to occur at higher pillar 

stress was vertical fracturing and spalling of thin slabs of limestone 

between the weak bands, shown in figure 2. In this case, the 

average pillar stress ranged between approximately 15 and 20 MPa 

(2,700 to 2,900 psi), which was only about 10-15 percent of the 

uniaxial compressive strength of the limestone beds at this mine. 

The material comprising the weak bands observed in the field 

included carbonaceous bedding planes, calcite fillings on bedding 

planes, and indurated clays or seat earths that had characteristics 

more closely resembling soils than rocks. Figure 3 shows an 

example of a weak band infilling that was easily penetrated by a 

geologist’s pick. 

The field observation of pillars showing failure controlled by 

discontinuities, more specifically bedding, at relatively shallow 

depths has been documented in other cases (Hoek et al., 1995). 

FAILURE MECHANISM 

Based on the field observations, it appeared that extrusion of the 

weak bands contributed to the failure of the stronger limestone. At 

lower loads, the soft material would extrude and release blocks of 

limestone defined by pre-existing joints. This causes the overhangs 

observed in figure 1. At higher vertical loads, the intact limestone 

Figure 2. Pillar damage observed in rock containing thin weak 

bands. Note spalling of the intact rock material between the weak 

bands. 

Figure 3. Low strength weak band showing geologists pick 

embedded in material. 

appears to fracture into thin vertical slabs. It was speculated that 

this failure is related to horizontal tensile stresses that develop as 

the weaker material extrudes under the elevated loads. Numerical 

models were developed to investigate the postulated mechanism of 

failure. 

Model Design 

The model to investigate the failure mechanisms simulated a 

single strong rock slab encased between two weak bands, shown in 

figure 4, representing a portion of a bedded rock mass. The 
Free face 

Applied loading 

Line for 

strength 

Weak bands 

determination 

Figure 4. Model layout to investigate failure mechanisms in 

strong rock encased between two weak bands. 

thickness of the weak bands was initially set to 10 percent of the 

strong rock thickness. The upper and lower boundaries of the 

model were constrained in the vertical direction. A symmetry 

plane was defined at the left boundary while the right boundary was 

a free surface representing an excavation surface, i.e. a pillar rib in 

this case. 

After constructing the models, the physical properties for the 

limestone and the weak bands were developed. The limestone 

represented a rock mass with uniaxial compressive strength of 

100 MPa (14,000 psi) and Geologic Strength Index (Hoek et al., 

2002) of 80. The Roclab software (Hoek et al., 2002) was used to 

develop the equivalent Coulomb parameters selected for FLAC 

modeling. The resulting limestone cohesion was 8.83 MPa 

(1,280 psi) with a friction angle of 38 degrees and tensile strength 

of 1.0 MPa (145 psi) . The weak band properties were selected 

from the database of properties in FLAC, and were generally 

representative of a strong, clayey soil. The initial strength 

parameters assigned to the weak bands were cohesion of 6.0 kPa 

(0.87 psi) with a friction angle of 24 degrees. The elastic modulus 

of the weak bands was initially set to 5% of the strong bed modulus. 

It has been recognized that continuum models with traditional 

failure criteria (e.g. Hoek-Brown or Mohr-Coulomb) based on 

simultaneous mobilization of cohesive and frictional strength 

components have not been successful in predicting the extent and 

depth of brittle failure, often seen as rock-slabbing parallel to the 

major principal stress. A bilinear failure criterion was therefore 

used to simulate brittle failure of the rock at low confinement, after 

Kaiser et al. (2000). At low confinement, the compressive strength 

of the rock material was set at one third uniaxial compressive 

strength, but with a zero friction angle to simulate the delayed 

mobilization of friction associated with brittle failure. When the 

confining stress exceeds 5 percent of the maximum stress, the 

friction and cohesion values revert to those developed from the 

Hoek-Brown failure criterion. Details of the procedure followed 

are presented in Esterhuizen (2006). 

The model was loaded by gradually moving the upper boundary 

downwards while the lower boundary remained fixed. The 

response of the model in the vicinity of the free face was monitored 

by observing stress and failure development as the loading 

increased. The strength of the rock was determined by averaging 

the vertical stress at mid-height of the slab near the free face, 

shown in Figure 4. Being near the free face, the average stress in 

this region approximates the uniaxial compressive strength of the 

rock mass consisting of the combined weak bands and strong rock. 

The reason for modeling a wide slab of rock, rather than say a cube, 

was to observe how failure propagates from the free face towards 

the interior of the slab. 

Model Results 

The effect of the weak bands on the strength of the slab model 

was assessed by determining the strength of the slab for various 

scenarios. A uniform rock slab model, consisting of only the 

stronger rock material, had a strength of 35 MPa (5,000 psi), which 

is approximately equal to the brittle strength of the material. When 

weak bands are added, the strength is reduced by 80 percent to 

6.8 MPa (1,000 psi). 

Inspection of the model output showed that failure of the 

layered rock mass occurs through an extrusion-tension mechanism. 

As the vertical load is increased, failure first occurs in the weak 

bands, because they are essentially cohesionless. As the load 

continues to increase, a zone of tension develops within the 

stronger slab (figure 5a), which is caused by the extrusion of the 

failed weak bed material. As the loading increases, the tensile 

stresses increase and tensile failure develops in the stronger slab 

(figure 5b), which relieves the initial zone of tension (Figure 5c). 

As the vertical loading continues to increase, tensile stresses are 

induced on either side of the initial tensile failure zone and tensile 

failure continues to occur. The process repeats until the entire slab 

has failed or the extrusion mechanism is inhibited by frictional 

resistance between the weak bands and the rock slab. If the tensile 

failure process is inhibited, the remainder of the slab fails by 

Coulomb shearing. The extrusion-tensile failure mechanism can 

explain the observed progressive spalling of intact rock at relatively 

low stress, shown in figure 2. 

The sensitivity of the rock mass to parameters such as elastic 

modulus of the weak bands, the strength of the weak bands and the 

tensile strength of the stronger material was tested. Variation of 

these parameters showed that the extrusion-tensile failure mode 

occurred for most scenarios. However, as the weak bands become 

stronger and stiffer, the role of tensile failure is diminished and 

Coulomb shear failure of the strong rock slab becomes more 

prevalent. On the contrary, when weak bands are thick and weak, 

extrusion without tensile failure tends to occur. 

PILLAR MODELING 

The simple slab model described above does not resemble a 

mine pillar, and the effect of weak bands is expected to be different 

when they occur within a mine pillar. Failure development in a 

mine pillar is affected by the end constraints provided by the roof 

and floor contacts, the pillar shape and by the initial stress state. 

Therefore, a series of models were developed to evaluate the effect 

of weak bands on typical pillars found in limestone mines. 

Model Design 

A series of models were developed to simulate a pillar and the 

surrounding roof and floor rocks. The models were designed so the 

roof, pillar, floor and weak bands can be independently assigned 

physical properties. The weak bands were evenly distributed 

vertically throughout the pillar. The roof and floor contacts were 

also assigned as weak bands. Initially the thickness of the weak 

bands was 3 percent of the thickness of the limestone beds. The 

models ranged in width-to-height ratio from 0.3 to 1.5 to bracket 

the typical range of width-to-height ratios observed in the field. 

The range of width-to-height ratios modeled was achieved by 

maintaining a constant model pillar width while changing the 

height. The weak bands were generated with three rows of 

elements each in order to permit distortion of the weak material. 
Weak bands

a)

b)

c)

d)

Failure in weakbands

Tensile stresscontours

Tensile failure

e)

f)

g)

Figure 5. Stages of failure development in a beam of strong rock encased between two weak bands. 

Roof 

Weak bands 

Limestone 

Floor 

Figure 6. Part of a pillar model showing level of detail and 

variation of element sizes to accommodate thin weak bands. 

Pillar width-to-height ratio is 1.0.

Figure 6 shows the element discretization used in a pillar with 

width-to-height of 1.0. 

The initial strength parameters were the same as described 

above for the simple slab models. The strength properties of the 

weak bands and the limestone were varied to investigate their effect 

on pillar strength. The thickness of the weak bands and number of 

bands was also varied. 

The pillar model was loaded by slowly moving the upper 

boundary of the model downwards, using the servo-control 

function available in FLAC. The function controls the velocity of 

the upper boundary by considering the unbalanced forces in the 

model. If failure occurs the rate of loading is slowed or stopped 

until the excessive loads have been dissipated, after which the 

velocity is increased again to continue loading the model. The 

average stress at mid height of the pillars was computed and 

recorded during the loading process. The pillar strength was 

defined as the peak value of the recorded average stress. The 

models were loaded until the recorded average stress had dropped 

20 percent below the peak value. 

Validation of Un iform Pillar Models 

Validation of the model results was carried out by comparing 

the model-predicted pillar strength for pillars without weak bands 

to the pillar strength equation proposed by Obert and Duvall (1967): 

S =σ p (0.778 + 0.222 w)

h

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

1.0, w is the pillar width and h is the pillar height. Model results 

for a uniform rock mass, without any weak bands, are shown 

against the equation in Figure 7. The value of σp in the equation 

was set to 26.4 MPa (3,800 psi), so that it would be normalized to 

the numerical model results. It can be seen that the model 

predicted strength follows a similar trend as that predicted by the 

equation. It was concluded that the modeling method provides a 

realistic representation of pillar strength over the range of width-toheight ratios shown. The strength and elastic parameters of the 

limestone were kept unchanged in all the model runs, except when 

sensitivity to the limestone tensile strength was investigated. 
5 

10 

Pillar strength (M

P

a)

35 

30 

25 

20 

15 

Model results 

Equation 

the weak-banded models initiated as tensile splitting at the outer 

extremities of the pillars, similar to the mechanism observed in the 

field and identified in the slab models (figure 8b). As the load 

increased, the tensile splitting would continue and, in the case of 

slender pillars, can extend right through the pillar. In the wider 

pillars, the tensile splitting process is inhibited by the frictional 

resistance provided by the roof and floor rocks. In these cases, the 

pillar loses some of its effective width to the tensile-extrusion 

failure mode, and ultimately fails by shearing of the pillar core 

shown in figure 8d. 

Figure 9 shows the change in pillar strength for width-to-height 

ratios of 0.3 to 1.5 due to the presence of thin and thick weak bands 

in the models. The thin and thick weak bands were 7% and 13% of 

0 

0.0 0.5 1.0 1.5 2.0 2.5 

Pillar width-to-height ratio 

Figure 7. Graph showing modeled pillar strength for a uniform 

rock mass compared to the Obert-Duval (1967) equation for 

pillar strength. 

PILLAR MODELING RESULTS 

the limestone bed thickness, respectively. It can be seen that the 

reduction in pillar strength is affected by the pillar width-to-height 

ratio and the weak bed thickness. For example, the models with a 

0.5 width-to-height ratio and thin weak bands showed a drop in 

peak strength to about 25 percent of the uniform pillar strength. 

Significantly, the pillars with width-to-height ratios of greater than 

0.8 did not show such a large drop in strength. A greater thickness 

of the weak bands results can result in a very significant loss of 

pillar strength. 

1.2 

The Effect of Adding Weak Bands 

Weak bands with cohesion of 6 kPa (0.87 psi) and friction angle 

of 24 degrees were added to the models using the initial strength 

parameters and were loaded until the peak pillar strength was 

exceeded. The presence of the weak bands considerably reduced 

the pillar strength. An examination of the development of failure in 

the models showed that a uniform rock mass initially fails in a 

brittle-compressive mode and shear failure occurs through the pillar 

core during the final stages of failure (figure 8a and 8c). Failure in 

Shear failure Brittle spalling Tensile spalling 

A B 

Weak bands 

Ratio of banded to unifor

m pillar strength 1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 

W/H ratio 

Thin weak bands 

Thick weak bands 

Figure 9. Model results showing strength reduction caused by the 

presence of multiple weak bands in pillars. Thin weak bands are 

3% and thick weak bands 7% of the limestone bed thickness 

respectively. Friction angle is 24 degrees with 6 kPa (0.87 psi) 

cohesion. 

Shear failure Shear failure 

C D

Figure 8. Model results showing rock failure in pillars without 

It appears, therefore, that the strength of slender pillars 

containing weak bands can be significantly reduced by the 

extrusion-tensile failure mode. However, wider pillars are 

compromised to a lesser extent. 

The Effect of Weak Band Strength 

The initial pillar models all had weak bands with strength 

equivalent to that of a weak clayey soil, which had a compressive 

strength of only 0.1% of the strength of the limestone beds. A 

series of additional models were run to determine how the pillar 

and with weak bands. Pillar (a) consists of uniform rock and (b) 

contains weak bands. Both are loaded to 80 percent of their peak 

strength. Pillars (c and d) are uniform and banded respectively, 

and both have been loaded beyond their peak strength. 

strength would be affected by bands that were not as weak. The 

strength of the weak bands was increased in stages up to 38 percent 

of the strength of the limestone that comprised the remainder of the 

pillar. 
1.2 

The results for the pillars containing thin weak bands are 

presented in figure 10. The results show that the effect of the weak 

bands on pillar strength is again dependent to the width-to-height 

ratio of the pillars. When the weak bands are relatively strong, they 

have little effect on the wider pillars, in some cases the models 

showed an increase in strength when the bands are strong, this is 

thought to be a secondary effect of the loading rate and interaction 

between extrusion and compressive failure which produces variable 

model behavior. The weak bands reduce the strength of the more 

slender pillars to a greater extent than the wider pillars. 

results, slender pillars experience a greater decrease in strength 

compared to the wider pillars. However, low friction values, such 

as 18 degrees, can also result in large reductions in the strength of 

the wider pillars. 

In practical situations, the presence of moisture can further 

reduce the friction between the weak bands and the surrounding 

rock, either by lubricating the contact planes or by an increase in 

the pore-pressure within the weak bands as pillar load is increased. 

In such cases, the frictional resistance can be reduced to almost 

zero and pillar strength is likely to be further reduced. 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 

W/H ratio 

38% 

10% 

1% 

0.10% 

Weak band 

strength ratio 

Ratio of banded to uniform pillar strength 0.6 

0.4 

0.2 

0.0 

Figure 10. Model results showing the effect of the strength of 

multiple weak bands on pillar strength. Weak band strength is 

expressed as a percentage of the limestone strength. Weak band 

thickness was 3% of the limestone bed thickness, with 24 degrees 

friction angle and 6 kPa (0.87 psi) cohesion. 

The Effect of Weak Band Friction 

The sensitivity of the pillar strength to weak-band friction was 

tested by conducting a further set of runs using the model with thin 

weak bands. For these analyses the weak-band friction was varied 

from 18 to 35 degrees. The resulting pillar strengths are presented 

in figure 11. The results show that as the weak-bed friction 

decreases, the pillar strength is decreased. Similar to the previous 

Ratio of banded to unifor

m pillar strength

1.0 

0.8 

0.6 

Effect of Limestone Tensile Strength 1.0 

The relationship between pillar strength and tensile strength of 0.8 the limestone was also evaluated. The base case material properties 

friction angle but the thickness of the weak bed was varied. It can 

included a tensile strength of the limestone of 1 MPa (145 psi). 

The influence of a lower tensile strength of 0.5 MPa (72 psi) was 

tested on both uniform pillars and pillars containing thin weak 

bands. A lower tensile strength resulted in little change in the peak 

strength of the uniform pillars regardless of the width-to-height 

ratio. This result is expected, since tensile failure does not play a 

role in the failure of uniform pillars. However, the pillars with 

width-to-height ratios of 0.5 or less, that contain weak bands, 

showed strength reduction of about one half the strength shown in 

figure 9. Wider pillars did not show a material reduction in 

strength. 

Effect of Elastic Modulus Ratio 

Models were also modified to establish the impact of various 

elastic moduli of the weak bands. It was argued that the elastic 

extrusion of the weak bands should have an effect on the induced 

tension in the stronger limestone, and therefore reduce the pillar 

strength. The elastic modulus of the weak bands was varied from 

0.1 through 10 percent of the limestone modulus using a 1.0 widthto-height model with weak bands having a 24 degree friction angle. 

It was determined that only when the weak band modulus dropped 

below 1.0 percent of the limestone modulus did it have a significant 

effect on the pillar strength. 

Single Weak Bands 

weak bands in the pillars. Models of pillars containing single weak 

bands showed that the strength is not affected as severely as when 

multiple weak bands are present. Figure 12 shows how the 

presence of a single weak band at mid-height of a pillar can affect 

the strength of a pillar with width-to-height of 1.0. In these models 

the weak bands all had 6 kPa (0.87 psi) cohesion and 24 degree 

1.2 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 

W/H ratio 

30 deg 

24 deg 

21 deg 

18 deg 

Weak band 

friction 

The analyses reported so far were all conducted with multiple 

be seen that a single thin weak band has an insignificant effect on 

the pillar strength, however, as the thickness is increased, its effect 0.4 

on pillar strength becomes significant. When the single weak band 

was 13 percent of the pillar height, it was found that failure is 

0.2 confined to the weak band only and the stronger limestone does not 

fail in tension. Under these conditions, the failure mechanism is 

simply the extrusion of the weak band and the pillar strength is 

determined by the weak band properties. 

0.0 

During the field studies, pillars were observed that contained 

Figure 11. Model results showing the effect of the friction angle single weak bands up to 15 cm (6 in) wide. Most of these pillars of multiple weak bands on pillar strength. Weak band thickness 

was 3% of the limestone bed thickness. Weak band cohesion 

was 6 kPa (0.87 psi) in all the runs. 

did not show any signs of tensile spalling of the intact rock, but 

joint defined blocks were ejected on occasion. A review of the data 
Ratio of strength of pillar containing weak

band to unifor

m pillar strength 1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0.1% 1.0% 10.0% 100.0% 

Weak band thickness/pillar height 

Figure 12. Model results showing the effect of the thickness of a 

single weak band on the strength of a pillar with width-to-height 

of 1.0. Weak band friction angle was 24 degrees and cohesion 

6 kPa (0.87 psi) in all the runs. 

showed that the pillar stresses were in the range of 5 to 11 MPa 

(700 to 1,600 psi), which did not initiate the extrusion-spalling 

mechanism. 

SUMMARY AND CONCLUSIONS 

The model studies have shown that weak bands can 

significantly reduce the strength of pillars. The observed pillar 

instabilities associated with weak bands can be explained by a 

mechanism of extrusion of the weak bands followed by tensile 

failure of the strong rock layers. The models are a simplification of 

reality, but are useful for understanding the mechanisms of failure 

and the sensitivity of pillar strength to the relevant parameters. The 

study has shown that: 

1. The extrusion-tensile failure mode typically initiates at 

the perimeter of a pillar and progresses inwards, reducing 

the effective width of the pillar. 

2. Observations in operating mines show that weak bands 

can cause rib failure to initiate when the average pillar 

stress is only about 10 percent of the limestone strength. 

3. Slender pillars (width-to-height<1.0) are more severely 

affected by the presence of weak bands than wider pillars. 

4. Pillar strength is adversely affected as the thickness of 

the weak bands increase. 

5. Model results show that single weak bands can have a 

significant effect on pillar strength if their thickness 

exceeds about 2 percent of the pillar height. 

The failure mechanism in a pillar with weak bands is 

predominantly caused by extrusion of the weak bands, which 

induces tension in the stronger rock slabs. The strong rock fails in 

tension, which is manifested as rib spalling in underground 

limestone mines. At lower stresses the extrusion process can 

release blocks defined by joints or blasting fractures. 

 The extrusion-tension failure mechanism and model results 

appear to adequately explain the observed failure of pillars 

containing weak bands at relatively low stress. The results have 

helped to develop an understanding of the failure mechanism 

associated with weak bands in limestone pillars and have shown 

which parameters are important to consider when designing pillars 

that may contain weak bands. The information can be used to 

design safer pillar layouts in limestone mines. 

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