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Stability Assessment in wide Underground Mine Openings by 
Mathews’ Stability Graph Method 
 
 
Choon Sunwoo1, Young-Bok Jung , U.M. Rao Karanam 2 
 
 
1 Korea Institute of Geoscience and Mineral Resources, Daejeon, Korea, 305-350 
2 Dept. of Mining Engineering, Indian Institute of Technology Kharapur, India 
 
 
 
ABSTRACT 
 
Stability of underground openings is a major concern for the safety and productivity of mining 
operations. Rock mass classifications methods form the basis of many empirical design methods as 
well as form a basis for numerical analysis. Of the many factors which influence the stability of 
openings, span of the opening for a given rock mass condition forms a single parameter of design. In 
this paper, the critical span curves proposed by Lang, the Mathews’ stability graph method, and the 
modified critical span curve obtained by the authors have been assessed. The modified critical span 
curve propose by the author has been successfully used to assess the stability of wide underground 
openings in several limestone mines. 
 
 
1. INTRODUCTION 
 
The condition of instability of underground openings is exhibited from the time of occurrence of rock 
falls from roof and wall of opening. Rock falls vary in size and amount depending upon the state of 
stress condition around opening, distribution of discontinuities, the strength and the condition of rock 
mass and finally on the dimensions of the opening. It is often noticed that very few metalliferous 
mines adopt systematic design procedures or methodologies. Consequently, unplanned operations 
usually end in local roof falls to major catastrophic events. Hence, a rock engineering design 
methodology for the design of stable underground openings is therefore essential for safety and 
productivity of mine operations. The design methods which are available for assessing the stability of 
underground openings have been broadly categorized as empirical methods, analytical methods, and 
observational methods. Rock mass classification systems constitute an integral part of empirical mine 
design. In more recent years, classification systems have often been used in tandem with analytical 
and observational methods. The primary objective of all classification systems is to quantify the 
intrinsic properties of the rock mass based on past experience. The second objective is to investigate 
how external loading conditions acting on a rock mass influence its stability. An understanding of 
these processes can lead to the successful prediction of rock mass behavior for different conditions. 
The RMR and Q system have evolved over time to better reflect the perceived influence of various 
rock mass factors on excavation stability. However, there have been certain modifications 
incorporated into these systems to enhance their applicability to the mining conditions. In the present 
investigations it is attempted to assess suitability of a rock mass classification system specific to the 
limestone mines. Six underground limestone mines have been identified for the present study and all 
the parameters required for estimating RMR and Q values have been measured from 140 locations of 
six underground limestone mines. A comparison is drawn between Rock Mass Rating(RMR), and 
Rock Mass Quality(Q) for the rock mass conditions prevailing at the mine sites. Since the main 
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objective of the study was to obtain a critical span design curve for the present set of underground 
limestone mines, the stability graph proposed by Mathews was applied 
 
 
2. STABILITY GRAPH METHOD 
 
The stability graph method for open stope design was initially proposed by Mathews et al.(1981) and 
subsequently modified by Potvin(1988) and Nickson(1992) to arrive at the modified stability graph 
method. In all instances, stability was qualitatively assessed as either being stable, potentially unstable 
or collapses. The modified stability Number N represents the ability of the rock mass to stand up 
without support under a given stress condition. 
The stability factor N, proposed by Mathews et al.(1981) is calculated as follows 
 
CBAQN ����= (1) 
 
Where, Q’ = The rock quality index after Barton et al.(1974) with the stress reduction factor(SRF) and 
the joint water reduction factor(JW) equal to one as they are accounted for separately within the 
analysis. 
Factor A - This value is designed to account for the influence of high stresses reducing the rock mass 
stability. The value of ‘A’ is determined by the ratio of the unconfined compressive strength to the 
intact rock divided by the maximum induced stress parallel to the opening surface. The value of A is 
set to 1.0 if the intact rock strength is 10 or more times the induced stress indicating that high stress is 
not a problem. The factor is set to 0.1 if the rock strength is 2 times the induced stress or less 
indicating that high stresses significantly reduce the opening stability. 
Factor B - This value looks at the influence of the orientation of discontinuities with respect to the 
surface analyzed. This factor states that joints oriented at 900 to a surface are not a problem to stability 
and a value of 1 is given to the value of B. Discontinuities dipping within 200 to the surface are the 
least stable representing structure which can topple within the stope. 
Factor C - This value considers the orientation of the surface being analyzed. A value of 8 is assigned 
for the design of vertical walls and a value of 2 is given for horizontal backs. This factor reflects the 
inherently more stable nature of a vertical wall compared to a horizontal back. 
Hydraulic radius accounts for the combined influence of shape and size in a more accurate way 
and it requires calculation. In short hydraulic radius or the shape factor(S) is defined as the ratio of the 
area to the perimeter of the opening, and is given as 
 
)(2
)(
HW
HW
SHR
+
�= (2) 
Where W is the width of the open stope / or the opening 
 H is the height of the open stope/ or the opening. 
 
The following relationship was used in converting the RMR values to Q. In the stability graph method 
the following relationship proposed by Bieniawski(1989) was employed to convert the RMR values to 
Q: 
 
44ln9 += QRMR (3) 
 
However, for mines considered in the present study the relationship between RMR and Q is as shown 
in the Figure 1, and it is given by equation (4). 
 
2.45ln3.6 += QRMRBasic (4) 
 3 
BRMR= 6.3Ln(Q) + 45.2
R2 = 0.51
0
20
40
60
80
100
0.001 0.01 0.1 1 10 100 1000
Rock Mass Quality (Q)
B
as
ic
 R
oc
k 
M
as
s 
R
at
in
g
 (B
R
M
R
)
Daesung-1
Pyunghae
Choongmu
Daesung-2
Chunglim
Samsung
 
Figure 1. Correlation between rock quality index Q and Basic Rock Mass Rating (B-RMR) 
 
 
3. CRITICAL SPAN CURVE 
 
The critical span curve is a simple and useful tool that aids in the design of underground openings. The 
critical span is defined as the diameter of the largest circle that can be drawn within the boundaries of 
the exposed back as viewed in plan. This exposed span is then related to the prevailing rock mass of 
the immediate back to arrive at a stability condition. The stability of an excavation, according to 
Pakalnis(1988) is classified into three categories(Table 1). 
 
Table 1. Three categories for the stability of an excavation 
Stable Excavations 
- No uncontrolled falls of ground 
- No noticeable movement in the back 
- No extraordinary support measure requirement 
Potentially Unstable excavations 
- Requirement of extra ground support to prevent potential falls 
of ground 
- Movement in the back of 1mm or more within 24 hours. 
- Increase in the frequency of poppingand cracking indicating 
ground movement 
Unstable Excavations - Area has collapsed - Support was not effective in maintaining stability 
 
 
Figure 2. Critical span curve for the mines sites 
 
Figure 3. Updated critical span curve 
 
Figure 2 shows the span curve(after Lang et al.,1994) obtained for 6 underground limestone mines. 
The database for this graph consisted of 140 points from the six mine sites with the RMR values 
ranging between 40 and 70. It is apparent from this curve that almost all the points fall within 
potentially unstable and unstable zone. The curve shown in Figure 2 has certain uncertainties since 
most of the data points are within the unstable zone. It is shown in the Figure 3 that around 45% of the 
openings of the mines considered in the present investigations can remain stable with good support 
system, while more than 50% is totally unstable. The modified critical span curve(Figure 4) suggested 
by Ouchi(2004) has also been drawn for the limestone mines considered in the present study. 
However, the ambiguity regarding the stable condition of the openings remained prevalent. 
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4. EXTENDED MATHEWS’ STABILITY GRAPH METHOD 
 
The Mathews’ stability graph encompasses a broad range of open-stoping experience and is essentially 
a self-validating model. Nevertheless, the validity and accuracy of the Mathews’ method depends on 
the quality and quantity of the stability data that it contains. The extended Mathews’ graph is a log-log 
plot with linear stable, failure and major failure zones rather than the traditional log-linear format with 
multiple curvilinear stability boundaries(Figure 4). The extension of the database has provided a wide-
ranging data-set to which statistical methods can be applied. The strength of the extended Mathews’ 
method lies in the improved stability graph with statistically determined stability zone boundaries and 
iso-probability contours. The first step is the use of Mathews’ stability graph is to determine the 
stability number N, and shape factor S, using the equation (1) and (2) respectively. The value of stress 
factor A is taken in the present case as 1.0 since the ratio of the compressive strength/induced strength 
is around 20 for all the case sets. Similarly, the value of joint orientation factor B is taken as 0.4 since 
the orientation of the joints with respect to the direction of the opening are between 600 and 450. 
Finally the value of surface orientation factor C is between 1 – 2 based on the orientation of roof 
surface. According to the modified Mathews’ stability graph method, certain locations of Samsung, 
Daesung-II, and Pyunghae mines are with the failure zone and the stability of these points can be 
improved following a suitable support system. A similar graph(Figure 5) has been obtained by 
modifying the critical span curve to suit to the limestone mines of present case history. A linear 
regression analysis was used to get statistically a best fit line for the data points obtained in the field. 
The dark line in the Figure 5 represents the best fit line and it satisfies the stability condition given by 
the equation (5). 
 
)(3.0 RMRBasicW = (5) 
 
Where W is the safe unsupported span for the rock mass conditions in the underground limestone 
mines of the case study. 
Although the stability zones can be defined statistically, a number of case histories in the present 
analysis have been placed in the unstable zone. This is to be expected, given the inherent variability of 
rock mass data that can be somewhat subjective. In order to include the locations of ambiguity, points 
within the 10% confidence interval from the mean have been considered. The dotted line, for the 
present set of data points, forms the upper-bound line of a stability zone. Any point outside the upper-
bound line is most unstable opening and the openings within the dotted lines though are stable with a 
suitable mechanical support system. 
 
 
Figure 4. Extended Mathews’ stability graph 
based on logistic regression 
 
Figure 5. Modified critical span curve for the 
limestone mines considered in the present study 
 
The shape and size of the open stope is one of the key factors that affect its stability. The 
application of various methods of structural analysis for the stability of the stopes, for a given rock 
quality, depends principally on its geometry. Hence, for a given rock quality, the stability of a stope is 
set by its dimensions(that is, height, width and the length of the stope). For the extraction of an ore-
body, the openings are determined by the ore-body shape and the chosen mining method. Most of the 
classification systems define stability with respect to a single value of the span. This is because the 
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database is essentially from tunnels where the long span can be assumed to be infinite and in which the 
short span (width) is critical. In mine openings the long dimensions are reduced by virtue of pillars and 
thus the stability increases. However, for a safe unsupported span of a stope, it is essential to estimate 
the limiting safe height of the opening. In order to ascertain the limiting height of an opening for a 
stable width, RMR has been correlated with the shape factor(S) of the underground locations (Figure 
6) and from the linear regression analysis the best fit line obtained is given by the following 
relationship. 
 
5.1)(02.0 += RMRS (6) 
 
The RMR values for the limestone mines of the case study are lying between the lowest of 40 and the 
highest of 70. From the graph it can be stated that the shape factor of an opening varies with the RMR 
values. For a lower RMR value of 40 the value of S is less than 2.5 and similarly for RMR value of 70, 
S is less than or equal to 3. For a measured value of RMR, using the equation (5) and (6) it is possible 
to estimate the limiting height of the stope for the limestone mines of the case study. Table 2 is an 
example of the measured and computed dimensions of locations in Samsung limestone mine 
 
Table 2. The measured and computed dimensions of locations in Samsung limestone mine (All the 
locations are assessed as unstable) 
Site Measured Width (m) 
Measured 
Height (m) RMR 
Shape 
Factor (S) 
Computed 
Width (m) 
Computed 
Height (m) 
P-62-1 15.5 7.6 53 2.56 15.9 7.55 
P-21 15.3 8.5 49 2.45 143.7 7.49 
P-82 15.6 7 45 2.40 13.5 7.45 
P-61 15 7.5 50 2.50 15.0 7.50 
P-131 15.7 7 53 2.56 15.9 7.55 
P-184 15 5.5 44 2.38 13.2 7.44 
 
The Lang’ s critical span curve, though is a very simple design curve, has not given any 
satisfactory assessment of stability of openings under present investigation. The data collected in the 
present investigation indicated that the lowest RMR value at which the roof fall has been noticed was 
40 and the measured width was 12m. Similarly, on the higher range of RMR that is at RMR 65 the 
unstable opening had a dimension of 18m. Therefore a line passing through these limiting points has 
established a modified critical span curve for the mines considered in the present analysis (Figure 7). 
The dotted lines represents the potentially unstable zone, where with suitable support system, the 
openings may likely be made to stand stable. However, the points beyond the dotted lines are totally 
unstable. This assessment though seemed empirical, was found suiting as a simple to use design curve 
for the construction of underground openings in the limestone mines in Korea of a similar rock mass 
condition. 
 
 
Figure 6. Correlation between Rock Mass Rating 
(RMR) and shape factor(S) 
 
Figure 7. Modified critical span curve 
representing the datasets of the mines sites of the 
present investigation 
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5. CONCLUSIONS 
 
The critical span curves proposed by Lang(1994) as well as the modified curve suggested by Ouchiet 
al.(2004) did not give good assessment of the stability of the underground limestone mine opening of 
the mines sites considered in the present study. However, a satisfactory assessment is obtained using 
the modified critical span curve as well as the modified Mathews’ stability graph. The relationship 
between RMR and the unsupported span and stability factor gave reasonably good correlations for the 
limestone mines considered in the present investigation. The critical width as well as the limiting 
height at which the openings are stable, calculated using the linear equations obtained by the statistical 
regression analysis, gave a comparable results with those measured in the field. A modified Lang’ s 
critical span curve is found to be a easy to use chart for the 6 limestone mines. However the stability 
assessment is more dependent on the quality of data and since the evaluation of RMR and Q still 
adopts subjective methods, care must be taken not to promote over-confidence in the final results of 
stability assessment without realizing the nature of the data and the relative importance of the location. 
Empirical methods therefore can best be used as precursors of the stability assessment. 
 
 
REFERENCES 
 
Barton, N., R. Lien, and J. Lunde, 1974, Engineering classification of rock masses for design of tunnel 
support, Rock Mech., vol. 6, pp183-236. 
Bieniawski, Z.T., 1989, Engineering Rock mass classifications, published by John Wiley & Sons, 
251p. 
Lang, B., R. Pakalnis, S. Vongpaisal, 1991, Span Design in wide cut and fill stope at Detour Lake 
Mine, 93rd Annual General Meeting, Canadian Institute of Mining, Vancouver, Paper No. 142. 
Lang, B., 1994, Span Design for entry type excavations , MASC Thesis, University of British 
Columbia, Vancouver, BC. 
Mathews, K.E. et al., 1980, Prediction of stable excavation spans for mining at depths below 1000m in 
hard rock, Report to Canada Centre for Mining and Energy Technology (CAMMET), 
Department of Energy and Resources. 
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