1. INTRODUCTION Problems from collapses of near-surface workings have plagued mining operations for years, not just at remote mine sites but often also in urban settings, (Plate 1). Civil tunnel collapses to surface are also not uncommon in urban areas. Assessing the risk for whether or not any near surface mine opening or civil tunnel might collapse and break through to impact surface infrastructure is challenging; while defining an appropriate minimum rock crown cover thickness that should be left above an opening is a particularly difficult design task. An attempt to address such problems in a mining context, nearly 20 years ago lead to the development of the Scaled Span empirical design guidelines for crown pillar rock thickness dimensioning (Golder Associates, 1990, Carter, 1992, Carter and Miller, 1995). These initial guidelines, which were mainly developed looking at steep orebody geometries, were targeted at helping mining engineers define potential collapse risk levels for new or abandoned mined openings, and also for establishing critical crown thickness dimensions for new designs. Plate 1: Two urban crown collapse examples ARMA 08-282 Logistic Regression improvements to the Scaled Span Method for dimensioning Surface Crown Pillars over civil or mining openings Carter, T.G., Cottrell, B.E., Carvalho, J.L, and Steed, C.M.. Golder Associates Ltd., Mississauga, Ontario Copyright 2008, ARMA, American Rock Mechanics Association This paper was prepared for presentation at San Francisco 2008, the 42 nd US Rock Mechanics Symposium and 2 nd U.S.-Canada Rock Mechanics Symposium, held in San Francisco, June 29- July 2, 2008. This paper was selected for presentation by an ARMA Technical Program Committee following review of information contained in an abstract submitted earlier by the author(s). Contents of the paper, as presented, have not been reviewed by ARMA and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. ABSTRACT: Assessing collapse risk for any near surface mine opening or for a civil tunnel is challenging, while defining an appropriate minimum rock cover thickness that should be left above the crown is a particularly difficult design task. The difficulties of predicting behaviour and of correctly dimensioning crowns is borne out by the fact that in recent years there have been a number of highly publicized near-surface mine working cave-ins and urban tunnel collapses, with breakthroughs to surface that have in several cases seriously damaged surface infrastructure. The fact that problems from collapses of near-surface workings have plagued mining operations for years led to the development nearly 20 years ago of the Scaled Span empirical design guidelines for crown pillar rock thickness dimensioning. This paper presents an updated methodology and a new chart developed through use of logistic regression techniques aimed at improving the ease of application of the Scaled Span method of Surface Crown Pillar dimensioning for assessing collapse risk. Updated aids to application of the guidelines are presented for applying the mining Scaled Span concept for determining both mine opening and civil tunnel cover thicknesses of variable quality.
Transcript
1. INTRODUCTION
Problems from collapses of near-surface workings have plagued mining operations for years, not just at remote mine sites but often also in urban settings, (Plate 1). Civil tunnel collapses to surface are also not uncommon in urban areas. Assessing the risk for whether or not any near surface mine opening or civil tunnel might collapse and break through to impact surface infrastructure is challenging; while defining an appropriate minimum rock crown cover thickness that should be left above an opening is a particularly difficult design task. An attempt to address such problems in a mining context, nearly 20 years ago lead to the development of the Scaled Span empirical design guidelines for crown pillar rock thickness dimensioning (Golder Associates, 1990, Carter, 1992, Carter and Miller, 1995). These initial guidelines, which were mainly developed looking at steep orebody geometries, were targeted at helping mining engineers define potential collapse risk levels for new or abandoned mined openings, and also for establishing critical crown thickness dimensions for new designs.
Plate 1: Two urban crown collapse examples
ARMA 08-282 Logistic Regression improvements to the Scaled Span Method for dimensioning Surface Crown Pillars over civil or mining openings Carter, T.G., Cottrell, B.E., Carvalho, J.L, and Steed, C.M.. Golder Associates Ltd., Mississauga, Ontario
Copyright 2008, ARMA, American Rock Mechanics Association This paper was prepared for presentation at San Francisco 2008, the 42nd US Rock Mechanics Symposium and 2nd U.S.-Canada Rock Mechanics Symposium, held in San Francisco, June 29-July 2, 2008. This paper was selected for presentation by an ARMA Technical Program Committee following review of information contained in an abstract submitted earlier by the author(s). Contents of the paper, as presented, have not been reviewed by ARMA and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.
ABSTRACT: Assessing collapse risk for any near surface mine opening or for a civil tunnel is challenging, while defining an appropriate minimum rock cover thickness that should be left above the crown is a particularly difficult design task. The difficulties of predicting behaviour and of correctly dimensioning crowns is borne out by the fact that in recent years there have been a number of highly publicized near-surface mine working cave-ins and urban tunnel collapses, with breakthroughs to surface that have in several cases seriously damaged surface infrastructure. The fact that problems from collapses of near-surface workings have plagued mining operations for years led to the development nearly 20 years ago of the Scaled Span empirical design guidelines for crown pillar rock thickness dimensioning. This paper presents an updated methodology and a new chart developed through use of logistic regression techniques aimed at improving the ease of application of the Scaled Span method of Surface Crown Pillar dimensioning for assessing collapse risk. Updated aids to application of the guidelines are presented for applying the mining Scaled Span concept for determining both mine opening and civil tunnel cover thicknesses of variable quality.
Over the years these original mining assessment approaches have been extended to civil tunnelling situations and to shallow dipping mining situations (Carter et al., 2002). The problem increasingly though has been that economic, logistical or scheduling incentives oftentimes drive engineers to push the limits on opening dimensions and/or on crown thickness, or on skipping the necessary step of carrying out sufficient site investigation, such that excavations sometimes get planned much closer to the rock surface than is realized; thereby perhaps unknowingly unnecessarily increasing failure risk.
2. SCALED SPAN EVALUATION
Designing for stability of near surface crown pillars over excavated openings requires an understanding of many factors including the excavation geometry, the characteristics of the rock mass, data on stress conditions, overburden loads, and ultimately an understanding of the relative degree of risk (factor of safety) associated with the planned near surface excavation, (Hutchinson, 2000). Collected mining experience, as illustrated on Figure 1, has shown that there are two distinct time periods when a higher percentage of crown pillar failures seem to occur: – immediately on excavation and/or within a few years of service life and – much later, with this latter peak in the time dependency data being to some extent rock quality controlled, and related to degradation of the rockmass fabric over time.
Figure 1 – Time Dependency of Failure (from Carter & Miller, 1996, based on over 400 case records and nearly 50 failures in the Golder-Canmet Crown Pillar database)
Appropriate assessment of risk, to assess the likelihood of failure, must govern decisions for both the design of new excavations, as well as for remediation of existing mine workings. The Scaled Span concept and the initial design chart (Figure 2) were originally formulated to assist decision makers assess the degree of risk associated with their underground excavation designs. The most recent advances to the Scaled Span approach (as outlined later in this paper) are based on the application of logistical regression analyses to an expanded crown pillar database to help provide a more accurate and expedient means for assessing crown failure risk.
Figure 2 – The Original Scaled Span Chart Plotted as Scaled Crown Spans versus Rock Mass Quality
2.1. The Original Scaled Span Chart The original Scaled Span chart (Figure 2), was introduced in 1989 based on a review of 70 mine case records encompassing over 200 thin or problematic near-surface mine openings including 30 documented failures (Carter, 1989). In concept the chart was merely a development and refinement of the standard Q and RMR tunnelling charts and tabulations which basically related opening span to rock quality, (Barton et al, 1974, Bieniawski, 1976) but extended in a similar manner to the Mathews Method open stope design chart (Potvin et al, 1989) to consider the third dimension. In this case this was not specifically done through introduction of a hydraulic radius term, but rather by dimensionally scaling the excavation’s span. The intuitive principle on which this scaling was developed was that – as the size of an underground excavation increases, so does the degree of failure risk and the likelihood of collapse of that structure’s roof, or “crown pillar”. Conversely, as the rock mass quality into which the excavation is made increases, so the likelihood of failure of the crown decreases (because rock block size increases and intrinsic rock mass competence and strength improves, thereby helping to resist failure). It was recognized that:
⎟⎟⎠
⎞⎜⎜⎝
⎛=uSL
TfStabilityCrown h
γθσ
where increased stability for any rock mass quality would be reflected by an increase in:
T, the rock crown thickness hσ , the horizontal insitu stress
and/or…in θ , the dip of the foliation or of the underlying opening, and;
where decreased stability for any crown would result from increases in:
S, the crown span L, the overall strike length of the opening
γ , the mass (specific gravity) of the crown and/or…in u, the groundwater pressure.
Noting that this stability expression could be split into two, based on (a) mined opening geometry and (b) rockmass characteristics, led to the development
of the basic deterministic assessment approach of comparing the dimensions of the required opening geometry, as characterised by the Scaled Crown Span (CS), against a critical rockmass competence, Qcrit (at which failure might be expected) as defined from the boundary between the failure cases and the stable cases, as shown on Figure 2. On this chart the scaling expression, CS, used to plot the case records (which characterizes the overall geometry of the opening and crown, and also incorporates some account of excavation opening shape as defined by foliation dip), is as follows:
( )( )5.0
cos4.011 ⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=θ
γR
S STSC
where: S = crown pillar span (m); γ = rock mass unit weight (tonnes/m3); T = thickness of crown pillar (m); θ = orebody/foliation dip, and; SR = span ratio = S/L (crown pillar span/crown pillar strike length) The Scaled Spans for all of the original case records were plotted against rock quality, where the RMR/Q classification scales were positioned relative to each other on the basis of Bieniawski’s 1976 correlation relationship; RMR76 = 9 · ln(Q) + 44 where RMR76 values were categorized according to the 1976 codings, thereby maintaining an equivalence with GSI, (Hoek et al., 1995, Marinos & Hoek, 2000). In the original work, rather than merely defining the critical rockmass quality Qcrit for the given case and using this for comparing with the Scaled Span, a limiting span at which failure might be expected was defined based on a regression fit to the data. As it was noted that the best fit dividing line between the failed and stable case records was similar in shape and closely matched the original empirical “unsupported span” curve proposed by Barton et al, 1974 for tunnelling and natural cavern cases, a similar power curve regression fit was formulated to define the Critical Scaled Span (SC or SCrit); viz:
)(sinh3.3 0016.043.0 QQSC =
….where the hyperbolic sinh term was introduced to attempt to fit the trend of significant non-linearity to increased stability at high rockmass competence.
It should be noted that in the scaling expressions the influence of groundwater and clamping stresses has not been explicitly accounted for. Rather it is expected that these aspects will be considered by the practitioner in defining and if necessary derating the defined rockmass quality Q and/or RMR76 / GSI assessments appropriately. With increasing use of the method, however, the effect on stability of orebody dip and obliquity of opening geometry were increasingly found to be inadequately characterized. On detailed evaluation it was found that this arose because the original relationships had been mainly derived from steeply dipping case studies (ie. with excavation/foliation dips, θ >40˚). As a result, in 2002 supplemental improvements were made to the definition expressions to better account for shallow dipping geological structure influences. 2.2. Revised Approach for Shallow Dip Openings The 2002 extension of the Scaled Span concept to incorporate shallow dipping stopes was predicated on further study of failing and stable case records of shallow geometry (Carter et al., 2002). From this work it was found that with increasing obliquity, crown failure was initiated predominantly by hangingwall delamination and/or voussoir buckling, rather than by direct crown failure either through sloughing of the orebody core or by propagation along the ore/host rock contact margins.
In this revision to the basic relationships, the controlling “span” at crown level was redefined to take into account the equivalent effective span, SEFF of the hangingwall projection up to the elevation of the stope crown, as shown on Figure 3, and as determined from the cave line, ξL and with the failure geometry post-collapse of the crown and hangingwall estimated by drawing break lines up from the base of the stope.
An updated definition for the controlling “actual” crown span, S, was therefore introduced based on analogies to soft rock caving behaviour, guided by the concepts of critical, super-critical and sub-critical influence, caving and break-lines, and the concept of an effective span as illustrated in the diagrams on Figure 3.
The logic for this definition was, in concept, that the mode of failure associated with a shallow dipping stope in blocky rock, while intrinsically different from caving in coal, (ref. left hand sketches on Figure 3) would still tend on a gross scale to encompass a similar extent of ground deformation, as crudely might still be postulated based on inferred cave and break angles. Also, matching of stability assessments for actual cases supported use of this modification to defined “actual” spans as a means for better characterizing an effective controlling span appropriate for most shallow to intermediate dipping openings.
Figure 3 – The Zone of Influence and Cave Angles Associated with Shallow Dipping Stopes
To apply this correction, an expression for the effective crown span (SEFF) as defined as “the actual span at crown level plus the horizontal projection of the hangingwall span-length (within the cave line extent)”, was developed as follows:
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
LHEFF LSS
ξθθ
tansincos
where: S = actual crown span of the stope (m); LH = (hangingwall length of the stope (m); Lξ = cave angle, and; θ = stope/ore body dip Note that this expression has been modified from the original version published by Carter et al. 2002, as this only considered the hangingwall projection, which tended to underestimate true effective spans for stopes of wide extraction geometry, where the crown span itself was also of significant dimension.
While not critical to apply this correction for all moderately dipping cases steeper than about 45° this expression could be applied ubiquitously if adverse hangingwall behaviour were anticipated. Typically though, for angled stopes with relatively steep dips (ie., greater than approximately 45˚) the competence of the stope crown itself or the strength of the ore/host rock contacts has generally been seen from the database records to control behaviour.
Control in the intermediate dip range (40°-50°), however appears from the case records to differ appreciably dependent on the competence of the rock mass comprising the ore zone and hangingwall (back). If the ore zone is weaker than the host rock mass, then stability seems to always be controlled by crown failure through ravelling of the ore or by shear on the ore contact margins, and as such in defining the controlling span, use should be made of the original uncorrected true stope span, S.
If, however, hangingwall rock mass quality is adverse, hangingwall failure might be more likely, and in this case the shallow mode effective span criteria should be adopted.
When the dip of the stope is very shallow but not actually flat (ie., in approximately the 15˚ to 20˚ range), the hangingwall length will again be the main control on crown pillar stability, and the effective span calculated based on the back length needs to be used for the calculation of CS.
Definition of an appropriate span for nearly flat dipping stopes (ie., with dips of less than about 15˚), such as would be found in many seam mining situations (gold and platinum reefs, bedded gypsum, coal etc), will depend on the extraction length between abutments or between pillars. For this range of dips, it is suggested that a span appropriate to the controlling room or panel longitudinal or transverse dimensions be selected. Obviously in the effective span expression, the seam thickness, S, would then be ignored in this calculation.
2.3. Application to Assess Failure Risk In its most basic form the Scaled Span method can be applied deterministically to assess failure risk by simply comparing the Scaled Span (CS) to the Critical Span (SC), and calculating an approximate Factor of Safety, Fc ≈ SC/CS. If this is done and the Scaled Span, CS exceeds the Critical Span, SC, as defined by assumed rock quality, (as was the case for the stope crown illustrated in Plate 2), unless the stope had been sufficiently supported, or fill approaches had been used in mining; for any opening with Fc less than 1.0 the likelihood of failure would be predicted to be high.
Plate 2: Assessing Collapse Risk for Remediation
While straightforward, with increasing use of the Scaled Span approach being made for checking whether or not abandoned mine workings needed remediation, this deterministic Factor of Safety approach wasn’t found to provide sufficient discrimination for making expensive decisions, particularly at the high consequence end of the risk scale (where these needed to be made more on the basis of estimates of the likelihood of failure). As both regulators and mine operators wished to rank the severity of perceived problem excavations using risk matrix approaches, similar to Figure 4, so that the likelihood of failure and potential consequences could be better assessed as a means for prioritizing which excavations were of most concern, attempts were made in the mid-1990’s to develop a better probabilistic assessment methodology for use with the scaling expressions.
Figure 4 – Typical Risk Matrix for Crown Failure
Multiple probabilistic spreadsheet analyses using @Risk with Latin Hypercube sampling were therefore conducted on case records from the database where sufficient rock quality information was available to characterize rock mass quality variability. Plotting the SC/CS ratios as a cumulative frequency distribution from this assessment suggested that the variability was approximately normally distributed, which then allowed a crude error function fit to be formulated between the assessed probability of failure (Pf) and the SC/CS quotients, (Carter, 2000); as follows:
⎥⎦⎤
⎢⎣⎡ −
−=4
19.21 c
fF
erfP
where: Pf = Probability of failure; Fc ≈ an approximate Factor of Safety
= SC/CS, and; erf( ) is the standard error function Although simple to apply in order to estimate failure probability, use of this expression was found to consistently overestimate failure risk for very low
probability situations. While not a problem from the viewpoint of helping prioritize crown pillar cases for defining which needed more detailed follow-up assessment, (the original purpose for developing the expression), the fact that the expression consistently over-estimated Pf values by 5-10% compared with more rigorous probabilistic methods, has somewhat constrained widespread acceptance of this formula for more than ranking assessment purposes. This, in turn, also partially prompted development of the approach outlined below.
3. IMPROVED SCALED SPAN GRAPH
Since the introduction of the original Scaled Span chart in 1989, further updates, including the addition of several hundred new database case records and modifications to the span definition to account for shallow dipping workings, have not necessitated any revision to the basic concepts. Ongoing development of the Method has generally been concentrated more on refining the analysis approach and on adding new crown pillar case records (over 500 cases to date), to improve the applicability of the method. The basic information required for conducting a Scaled Span analysis has remained essentially consistent, viz:
• defining Thicknesses of Rock Crown cover • documenting Opening Spans and tunnel,
cavern, drift or stope dimensions • outlining data on Dip/Orientation of the
principal structural fabric of the rock mass • calculating and assigning Rockmass quality
classification values, either as Rock Mass Rating (RMR76, Bieniawski, 1976) or as GSI (Marinos and Hoek, 2000) or as Q (Barton et al. 1974, Barton, 1976), including assessment of water and stress terms, and
• tabulating and assigning available data on rockmass density and Hoek-Brown friction parameters, (Hoek & Brown, 1988)
A significant improvement is however advanced in this paper based on recent application of logistical regression as a statistical tool aimed towards computing iso-probability of failure contour intervals to be drawn as an overlay over the original Scaled Span assessment chart. This approach has taken advantage of the statistical distribution of the case record Scaled Span data thus allowing
probability intervals to be calculated and quantified. This provides a great advance now for application of the Scaled Span method as now that a set of iso-probability lines having been derived, very rapid assessment is now possible of risk for any known crown geometry and rockmass quality. Such application of logistic regression analysis to rock mechanics related data sets is not new, the Mathew’s Stability Method having been previously tackled in this manner by Mawdesley et al. 2004. Because the Scaled Span Method involves a similar type of database structure (ie. with stable vs. non-stable points), this data set also lends itself to application of logistic regression modelling. 3.1. Logistical Regression Applied to the Scaled
Span Method The basic relationship for the Critical Span Line defining SC, which was developed by regression fitting selected cases on either side of the stable versus failed line from the original database constitutes an example of the use of classical linear regression as a commonplace statistical analysis method where continuous dependent variable data is available. This however is a simplification of the actual situation for the crown pillar database points as they have variability with respect to more than one variable and ordinary linear regression cannot do full justice to assessing the probability of occurrence of an event when there are discrete, ordinal, or non-continuous outcomes for each event. The advance that has been made in applying logistic regression to the crown pillar database case records is that the analysis has been carried out by loglinear modelling the transformed multivariate categorical (crown pillar) data, (as per Demaris, 1992, Liao, 1994). To achieve this, a special form of the general log-linear model known as the Logit Model has been used for assessment of the crown pillar data set. Here the “logit” itself, is the natural logarithm of the odds, or the log odds that a particular crown case will fail or not (ie., it is an indicator of the relative probability that the outcome of a variable of interest will fall into one of two categories). In logit modelling of the crown pillar problem, it is convenient to express the log odds of failure (or stability) as a function of the set of explanatory variables (ie., in this case rockmass quality Q and the Scaled Span, CS for each and every case record).
To start with, if a straightforward random variable Y is examined using a linear model, (as would be the case for classical regression), its expectation could be specified as a linear combination of K unknown parameters (explanatory variables):
( ) ∑=
==K
kkk xYE
1
βμ
This is the ordinary linear model and is reminiscent of a linear regression model. In order to generalize this model, a variable η, which links the function:
∑=
K
kkk x
1
β to μ is now introduced.
Although this function, which relates η to μ does not necessarily have to be linear, it has to be specified and the choice of this “link” function distinguishes the different members of the generalised linear models. For the logit model chosen for analysis of the crown pillar data set, this link was defined as follows:
⎥⎦
⎤⎢⎣
⎡−
=μ
μη1
log
This arrangement specifies a logit model that takes a binary outcome variable (ie., failed or stable). In the interpretation of such a model the most appropriate way of testing the viability of the deduced relationship is to use a test to evaluate the goodness-of-fit of the model. In a classical regression model, an F test is used; in a logit model, the most commonly used test is the likelihood ratio statistic, G2, which generally follows the chi-squared distribution:
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛=
JI
ji ij
ijij m
nnG
,
,
2
ˆlog2
The log likelihood value computed by this methodology then would represent the likelihood that the data would be observed given the parameter estimates. One way to interpret the size of the log likelihood is to compare the model value to the initial or to a baseline value, i.e., assuming all the βk coefficients are equal to zero. This approach applies quite well for evaluation of the Scaled Span Method, wherein the event of concern is that either the failure of the crown pillar
has occurred (1), or has not (0). This can then be applied directly to the original data series of mine failure cases to assess the probability of failure based on the independent variables, Q, and CS. For those familiar with ordinary regression, the following parallels between linear and logistic regression terms help in understanding the method: Ordinary Regression Logistic Regression
Total sum of squares -2 × baseline of log likelihood
Error sum of squares -2 × model log likelihood
Regression sum of squares
-2 × difference between baseline and model log likelihoods
F test for model Chi-squared for log likelihood diffVariance Pseudo-variance
In order to utilize the approach for defining the controls applicable to the Scaled Span (CS) vs rock quality Q data, a logistical regression probability function of the following form was applied:
( )zezf −+
=1
1)(
where SCQz lnln 21 ββα ++=
… and where )(zf represents the logit probability function = an ‘s’ shaped curve between 0 and 1.
The predicted log odds value in this expression is z, α is the ‘intercept’ to the curve, and β1, and β2 are the constant terms appropriate to the regression data series analysed; in this case rockmass quality, Q and the Scaled Span, CS for each case record
The logistical regression process is fairly involved, therefore statistical software was utilized to perform the regression and calculate the logit probabilities for each data point as well as the intercept and constant terms. From this, cumulative frequencies of the logit values for the failed and stable cases were compared to assess the relative probability of occurrence (0.005 to 99.5 %) for each logit value.
The logit values were then applied to the above equations using the calculated intercepts and constant terms as well as an arbitrary value of Q in
order to back calculate the critical span line (Sc) for each probability of occurrence. These critical span lines are of significant value for practitioner use as they form iso-probability contours which follow the trend between the failed and the stable populations.
3.2. Improved Scaled Span Chart for Assessing Probability of Failure
The original Scaled Span database included 70 mine case records encompassing over 200 thin or problematic crown pillar geometries. The up to date crown database now includes 110 mine cases, with over 500 case records encompassing very shallow to steeply dipping excavation geometries under variable rock mass conditions. As outlined above and plotted on Figure 5, iso-probability contour lines, which delineate stability intervals within the Scaled Span chart, have now been generated based on the results of the logistic regression modelling. Each iso-probability contour line (denoted as a probability of failure contour interval) inherently represents a Critical Span line for varying acceptable limits of the probability of failure (0.5% to 99.5% Pf). This in turn has thus allowed definition also of the limiting 50% probability of failure (Fc = 1) line, analogous to the previous Critical Span expression, as follows:
44.058.3 QSc = (50% Probability of Failure) It is of note that in comparison to the original Critical Span equation, this updated Critical Span line for 50% probability of failure, (and which incorporates the updated database records) follows very closely the original 1990 best fit line (Golder Associates, 1990; Carter 1992, Barton et al., 1974). One striking occurrence though that is evident from the distribution of probability of failure intervals, as very evident from the updated Scaled Span chart on Figure 5, is that failure probability tolerance based on the Scaled Span (CS) variance becomes very low with poor to extremely poor rock mass quality. For these poor quality rockmasses a slight increase in excavation span, or decrease in rock quality significantly increases failure likelihood for the crown pillar. On the other hand, for good to very good quality rockmasses a much higher tolerance for variation in excavation geometry is evident.
0.001 0.01 0.1 1 10 100 1000
0.1
1
10
100
0.5%
0.5%
5%
10%
15%
20%50%
50%
80%
85%
90%
95%
99.5%
EXCEPTIONALLY POOR
EXCEP.GOOD
EXTREMELY GOOD
VERYGOODGOODFAIRPOORVERY POOREXTREMELY
POOR
ROCK MASS QUALITY INDEX, Q
VERY POOR POOR FAIR GOOD VERY GOOD
0 20 40 60 80 1009070503010
ROCK MASS RATING, RMR76 = GSI
ORIGINAL CRITICAL SPAN LINESc=3.3Q0.43SINh0.0016(Q)(GOLDER ASSOCIATES ,1990; BARTON, 1974)
50% PROBABILITY OF FAILURE (CRITICAL SPAN)CONTOUR LINESc=3.58Q0.44(GOLDER ASSOCIATES, 2008)
Cs=S[γ/({T(1+Sr)}{(1 - 0.4 cosθ)})]0.5
WHERE: S = CROWN PILLAR SPAN (m) T = THICKNESS OF CROWN (m) γ = ROCK MASS SPECIFIC GRAVITY Sr = SPAN RATIO = S/L L = LENGTH OF CROWN (m) θ = OREBODY/FOLIATION DIP, deg.
GENERALIZED LOGISTIC MODEL PROBABILITY OF FAILURE EXPRESSIONPf=100/[1+441exp(-1.7*Cs/Q0.44)]
Figure 5 – Updated Scaled Span Chart with Probability of Failure Contour Intervals
Table 2 – Acceptable Risk Exposure Guidelines - Comparative Significance of Crown Pillar Failure (modified from Carter & Miller, 1995)
Design Guidelines for Pillar Acceptability/Serviceable Life of Crown Pillar
Class Probability of Failure
%
Minimum Factor of
Safety
Maximum Scaled Span,
Cs (= Sc)
ESR (Barton
et al. 1974)
Expectancy Years Public Access
Regulatory position on
closure
Operating Surveillance
Required
A 50 – 100 <1 11.31Q0.44 >5 Effectively zero < 0.5 Forbidden Totally unacceptable Ineffective
B 20 – 50 1.0 3.58Q0.44 3
Very, very short-term (temporary mining purposes only ; unacceptable risk of failure for temporary civil tunnel portals
1.0 Forcibly Prevented
Not acceptable
Continuous sophisticated monitoring
C 10 – 20 1.2 2.74Q0.44 1.6
Very short-term (quasi-temporary stope crowns ; undesirable risk of failure for temporary civil works)
G <0.5 >>2 0.69 Q0.44 0.8 Very long-term (permanent crowns over civil tunnels) >100 Free Of no concern None
required
As can be seen from examination of Figure 5, this updated Scaled Span analysis chart allows almost direct assessment of the probability of failure for any given excavation opening geometry, crown pillar thickness and controlling rockmass quality.
Precision in definition of the spread and variability of controlling rockmass quality is however critical for correct application of this chart, and indeed of the whole Scaled Span method. To achieve this precision in defining the appropriate rockmass quality for use with the chart, proper understanding needs to be gained of (i) the presumed failure mechanism for the pillar under examination, and (ii) the propensity if any for weathering and/or degradation that could lead to variations in rockmass quality over time, (Carter and Miller, 1996). Although the updated Scaled Span chart and evaluation methodology can be used to rapidly assess failure risk, cognizance must always be given to the fact that longevity of a given excavation will differ dramatically dependent on the rock type and initial quality, and whether or not support has been designed for permanence, or not.
It is important to note that as additional case records are added to the database, that the data continues to support the 1996 observations that suggests that from the viewpoint of long-term stability, there appear to be two basic, quite different rock mass behavioural characteristics; i.e.,
− the essentially non-degradable, competent rock types (hard igneous and metamorphic types and well cemented sedimentary units) which exist, tend not to spall and hence seem to survive,
… and …
− the degradable, weathering susceptible, weak or highly fragmented rock types, that most commonly fail in due course of time, due to disaggregation and spalling.
These latter rockmasses are of most concern when applying the Scaled Span methodology, as they are notoriously difficult to properly characterize.
It is recommended, therefore, that when examining such rockmasses using the updated probability chart that some parallel evaluation also be conducted looking at cave mechanics approaches, specifically checking whether bulking will be a feasible restraint on cave raveling or whether chimney caving might be a problem (Bétournay, 2004).
While the logistic regression relationships shown on Figure 5 and listed in Table 2, as derived from analysis of the more than 500 points in the crown pillar database, help towards improving our means to better estimate the relative probability of failure (Pf) for any crown and opening geometry (as defined by the Scaled Span, (Cs) for any inferred rockmass quality (Q), this must not be done blindly.
Consideration must be given to the most likely failure mechanism and range of probable rockmass quality. Because of the fact that failure probability tolerance based on the Scaled Span (CS) variance becomes so very low for poor to extremely poor rock masses, it is recommended that computations for these types of rock masses always consider a range of probable rockmass qualities, rather than simply undertaking deterministic assessments of crown stability using single value Q estimates. Sensitivity deterministic evaluations can be carried out assuming a mean and a spread of say one standard deviation of probable rockmass quality, along the lines of the approach suggested in Carter and Miller, 1995. Alternatively, rockmass quality variability can be treated more rigorously using probabilistic analysis methods, ranging in complexity from two point estimation methods (such as used by Hoek, 1989 for examination of inter-dependency of rockmass quality variables on Factor of Safety calculations for crown pillar evaluations), through to use of Monte-Carlo and/or Latin Hypercube types of random variable simulation models.
Irrespective of the complexity or simplicity of the analysis approach chosen, only when a viable failure mechanism has been postulated and an appropriate range of rockmass quality defined can the controlling rock mass segment, critical to the stability of the crown pillar, the contact margins or the hangingwall or footwall be selected for Q characterization for use of the chart. Given these stipulations, direct assessment of an appropriate probability of failure can analytically be derived using the following expression:
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
⋅+
=44.0
7.1
4411
100(%)Q
Csf
e
P
… where the Cs and Q values for substitution into the expression can be deterministically specified (or read directly from the chart), or the expression can
itself be probabilistically solved by specifying distributions for the rock quality input variable, Q.
As is evident from Figure 6, which plots the critical span intercept values for a Q of 1, this probability of failure curve fit relationship is near normally distributed for most of its range, with excellent matching between the logistic regression intercept points and the exponential curve fit up to a Pf of around 75%, with increasing divergence above. This mirrors the behaviour of the previously proposed relationship based on the Latin Hypercube sampling evaluation (Carter, 2000), and reflects the increasingly non-gaussian distribution shape of the data spread as one moves to higher risk likelihoods.
As with the previous expression, it has been chosen to attempt to match the curve fit to the lower probability end of the distribution as this is the area which is, in general, of most concern and where this updated expression gives much improved results as compared with the earlier relationship. Comparison of the two expressions, in fact shows that the earlier relationship can overestimate the failure probability by up to 10% in certain parts of this tail segment of the probability distribution, particularly for Pf <5%.
Thus, while the conservatism that is inherent in the original relationship has generally benefited initial screening assessments, its inbuilt pessimism at the low end of the probability scale has, in some circumstances, been seen as a hindrance to proper decision-making, where choices were needed to be made between expensive remediation measures in order to develop “walk-away” closure solutions.
It is hoped therefore, that, in conjunction with the application of additional statistical treatment in the generation of rock mass quality distributions, this new chart will improve efficiency in assessing inherent risk associated with any new excavation under design, or any old excavation potentially requiring remediation measures.
3.3 Factor of Safety Considerations
While use of the probabilistic approaches for crown pillar assessment are quite widely accepted in the regulatory framework for assessing levels of acceptable risk for closure considerations, quite often decision makers are more comfortable working with a factor of safety approach. However, establishing a definitive Factor of Safety (FoS) for a given crown pillar is not a trivial matter as it is possible for a designed excavation to not only have a high mean factor of safety for a given mechanism (ie. plug failure, Hoek, 1989), but also at the same time have a high probability of failure if there exists a wide range of scatter in the statistical distribution of the calculated factor of safety values (as a result perhaps of wide variability in possible input parameters). Industry accepted inter-relationships between the two, such as laid out in Table 2 for crown pillar cases, now with the development of the logistic relationship to Cs and Q however have allowed much better assessment than possible hitherto using Fc ~ Sc/Cs as previously proposed as a crude safety factor.
Previous attempts at defining an approximate factor of safety term (Fc) simply by calculating the quotient between the critical span (Sc) for a given rock mass quality, and the scaled span (Cs), although providing a reasonable first estimate, significantly diverges from expected values at the top and bottom ends of the probability scale.
Making a more direct comparison between Pf and FoS utilizing the range specified in Table 2, and the newly developed logistic expression, suggests the following improved Factor of Safety estimate:
6.0cFFoS =
… where Fc=Sc/Cs, as previously defined.
This FoS expression is a significant improvement on the straight use of Fc as a crude safety factor estimate as it yields values that more closely follow industry accepted guidelines presented in Table 2.
Figure 6 – Probability of failure as a function of Scaled Span, CS for Q = 1
As with the basic probability of failure relationship, but expressed in terms of Fc, as follows:
( )Fcf
eP 6
4411
100(%) −⋅+
=
Use of this improved FoS (or indeed use of the basic Fc relationship for anything other than deciding which side of unity ~ corresponding with a Pf of 50% - ie. the critical span line) a case lies, should be treated with some caution. Considerable care should also be taken in applying this FoS term for anything other than defining risk exposure guidelines, in much the same way as can be done using the basic probability of failure expression. In fact neither should be used as the sole basis for design.
5. RISK EXPOSURE GUIDELINES
In designing new underground excavations, or designing remedial measures for old excavations near surface, such as shown on Plate 2, engineers and managers often are required to make decisions according to an acceptable level of risk associated with a particular situation. For new excavations, such as a subway or water intake tunnel in an urban area, tolerance to risk is limited and the acceptable degree of risk must be very low. However, for remediation of existing 50 year old mine workings in a desolate region, still on mining property, the acceptable degree of risk against crown pillar failure could be higher. A higher acceptable risk tolerance allows for more cost effective remedial measures, such as fencing and signage to be utilized rather than adopting a more costly, arguably safer, alternative such as backfilling or capping or plugging. Table 2 in the version previously published by Carter and Miller in 1995 gave guidelines for levels of risk that could be deemed acceptable for a particular closure or operating situation. To aid use of this table with the updated Scaled Span chart, equations for each iso-probability contour interval have been added into an extra column in the table. This has been done in the hope that dividing the Acceptable Risk Exposure Guidelines in terms of the iso-probability lines on the updated Scaled Span chart will help increase confidence in assessing crown pillar stability risk for decision makers, moving forward on their specific projects.
ACKNOWLEDGEMENTS
Part of the original work to develop the Scaled Span concept was funded by CANMET under Contract No.23440-8-9074/01-SQ, other parts were funded by the Ontario Ministry of Northern Development and Mines or conducted as part of programmes funded by various mining companies. Over the past almost two decades of use of the procedures, studies for many of these same mining companies have contributed to advances in current understanding. While specific thanks is due to these organizations that have supported the work by supplying unpublished, often confidential information to form a case study data base, acknowledgements must also go to many colleagues at Golder Associates and other consultancies and universities who have made major contributions to the formulation and testing of some of the methods outlined herein.
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