An investigation on the respone of hard-rock pillars in Mining

Pillar Design in Hard Brittle Rocks-An overview of

using appropriate analysis tools in the assessment of

pillar failure modes in deep mining.

Course: Applied Rock Mechanics

-MINE818

Student Name: Ioannis Vazaios

Student ID: 10123567

Course Instructor: James F. Archibald

April, 2014

1

Abstract One of the most common techniques applied in deep mining and underground spaces is the Room and

Pillar method in which columns of rock (or ore) support the opening made during excavation or extraction

processes. Therefore, pillars are a vital and fundamental component of underground operations in which

this technique is applied, creating the need of accurate assessment of the pillar response.

The assessment of pillars includes two basic steps: 1. Estimation of the stresses induced on the pillar due

to excavation (Demand), and 2. Estimation of the strength of the pillar needed to maintain its own

stability and the stability of the opening (Capacity). Additionally, it is equally important to define the

expected failure mode as this can create great implications in the analysis and consequently to design

procedure.

Observations of pillar failures in Canadian hard-rock mines indicate that the dominant mode of failure is

progressive slabbing and spalling which is a brittle type of behaviour. Empirical formulas developed for

the stability of hard-rock pillars suggest that the pillar strength is directly related to the pillar width-to

height ratio. These include linear shape effect formulas, power shape effect formulas and size effect

formulas which have been derived using specific data sets and thus they are subjected to limitations as it

makes their general application for other sites rather difficult. In this particular paper an overview on the

empirical methods for pillar strength assessment is provided and specific details will be provided for

specific empirical formulas.

However, most of these empirical formulas, although once considered a rather practical tool, have been

substituted nowadays by the use of sophisticated numerical tools which make possible the analyses of

projects of different complexity depending on the in-situ stresses, geometrical features of the project and

the mechanical properties of the rockmass in-situ. The application of this kind of analysis though has to

be made with caution, as these methods are rather sensitive to the input parameters used and the

constitutive models assumed. More specifically, two of the most widely used rockmass failure criteria are

the Mohr-Coulomb failure criterion and the empirical Hoek-Brown criterion but their application in brittle

rockmass behaviour is not appropriate and thus failure modes like spalling cannot be captured by them.

By making specific modifications though, spalling behaviour is possible to be simulated at a specific

extent by using the Hoek-Brown failure criterion. Numerical analyses using software Phase2 distributed

by Rocscience have been conducted in order to illustrate this and their results will be discussed.

2

Contents

Abstract....................................................................................................................................................... 1

List of Figures ............................................................................................................................................. 3

List of Tables .............................................................................................................................................. 5

1. Introduction......................................................................................................................................... 6

1.1 Pillar strength assessment ................................................................................................................. 6

1.2 Role of pillars in Mining ................................................................................................................... 6

2. Pillar Design Methodology ................................................................................................................. 8

2.1 Pillar stress and strength determination ............................................................................................. 9

2.2 Empirical design methods ............................................................................................................... 12

2.2.1 Linear Shape Effect Formulas .................................................................................................. 12

2.2.2 Power Shape Effect Formulas .................................................................................................. 13

2.2.4 Effective Pillar Width .............................................................................................................. 14

2.2.5 The Size Effect Formulas ......................................................................................................... 14

2.3 Numerical modelling techniques ..................................................................................................... 17

2.3.1 Continuum Methods ................................................................................................................. 17

2.3.2 Discontinuum Methods ............................................................................................................ 18

2.3.3 Hybrid Methods ....................................................................................................................... 18

3. Pillar Failure Mechanisms................................................................................................................. 19

4. Rockmass Failure Criteria ................................................................................................................. 22

4.1 The Mohr-Coulomb criterion .......................................................................................................... 22

4.2 The Hoek-Brown empirical criterion .............................................................................................. 23

5. Damage and spalling prediction criteria ............................................................................................ 25

5.1 Hard Rock Lower and Upper Bound Strength ................................................................................ 26

5.2 Hard Rock Strength using Hoek-Brown Empirical Criterion .......................................................... 27

6. Numerical simulation of hard rock pillars ......................................................................................... 30

6.1 Results of empirical method application ......................................................................................... 30

6.2 Results of numerical method analysis application ........................................................................... 31

6.2.1 The Geological Strength Index (GSI) rockmass classification system ..................................... 31

6.2.1 The numerical model................................................................................................................ 33

3

6.2.3 Pillar response for poor quality rockmasses (GSI=45) ............................................................. 34

6.2.3 Pillar response for medium quality rockmasses (GSI=65) ....................................................... 37

6.2.4 Pillar response for medium to good quality rockmasses (GSI=75)........................................... 40

6.2.5 Pillar response for good quality rockmasses (GSI=85) ............................................................ 43

6.3 Comparison between empirical and numerical methods, and constitutive approaches .................... 46

7. Closing remarks ................................................................................................................................ 47

References ................................................................................................................................................ 48

List of Figures Figure 1 Sketch of stream lines in a smoothly flowing stream obstructed by three bridge piers (Hoek and

Brown, 1980) ............................................................................................................................................ 10

Figure 2 Plan view of geometry for tributary area analysis of pillars in uniaxial loading (Brady and

Brown, 1992, after Maybee, 2000) ........................................................................................................... 10

Figure 3 Typical room and pillar layout showing load carried by a single pillar assuming total rock load to

be uniformly distributed over all pillars (Hoek and Brown, 1980) ............................................................ 11

Figure 4 Pillar stability graph by Hudyma (1988) ..................................................................................... 13

Figure 5 Hedley and Grant's (1972) method for determining pillar stresses (Hedley and Grant, 1972) .... 16

Figure 6 Hedley and Grant's (1972) estimation of pillar stresses and strengths (Hedley and Grant, 1972) 16

Figure 7 The size effect in the uniaxial complete stress-strain curve (Hudson and Harrison, 1997). ........ 19

Figure 8 The shape effect in uniaxial compression (Hudson and Harrison, 1997). ................................... 20

Figure 9 Principal modes of deformation behaviour of mine pillars (Brady and Brown, 1985) ................ 21

Figure 10 Schematic illustrating the difference between pillar skin bursts and pillar bursts (Maybee, 2000)

.................................................................................................................................................................. 21

Figure 11 The Mohr-Coulomb failure criterion (Hudson and Harrison, 1997) ......................................... 23

Figure 12 The Hoek-Brown empirical failure criterion (Hudson and Harrison, 1997) .............................. 24

Figure 13 Potential for spalling failure processes in intact rock based on compressive strength and tensile

strength (Diederichs, 2007) ....................................................................................................................... 26

Figure 14 a. Shear failure around a tunnel; b. spalling damage in hard rocks at high GSI; c. example of

brittle spalling and strain bursting in a deep mine opening; d. mechanisms of crack initiation in hard rock

(Carter et al., 2008) ................................................................................................................................... 27

Figure 15 The composite strength envelope illustrated in principal stress space (2D) to highlight the zones

of behaviour as bounded by the damage initiation threshold, the upper bound shear threshold (damage

interaction) and the transitional spalling limit (Diederichs, 2003) ............................................................ 27

Figure 16 Determination of damage initiation thresholds for rock under compressive loading using strain

and acoustic emissions (Diederichs et al., 2004) ....................................................................................... 28

Figure 17 Example of "peak" and small strain "residual" strength parameters for damage initiation and

spalling limits (Diederichs, 2007) ............................................................................................................. 29

Figure 18 Estimated Pillar Safety Factor based on empirical methods. .................................................... 31

Figure 19 Numerical model of a pillar created in Phase2. ......................................................................... 33

Figure 20 Numerical model illustrating the pillar under examination. At the upper right corner the

orientation of the geostatic stresses is presented. ...................................................................................... 33

4

Figure 21 Poor quality rockmass: Mohr-Coulomb criterion without dilation. Left: Yielded elements and

maximum shear strain contours when the first stope is excavated. Right: Trajectories of the post-mining

principal stresses and maximum shear strain contours. ............................................................................. 35

Figure 22 Poor quality rockmass: Mohr-Coulomb criterion with dilation. Left: Yielded elements and

maximum shear strain contours when the first stope is excavated. Right: Trajectories of the post-mining

principal stresses and maximum shear strain contours. ............................................................................. 35

Figure 23 Poor quality rockmass: Typical Hoek-Brown criterion. Left: Yielded elements and maximum

shear strain contours when the first stope is excavated. Right: Trajectories of the post-mining principal

stresses and maximum shear strain contours. ............................................................................................ 36

Figure 24 Poor quality rockmass: Modified Hoek-Brown criterion. Left: Yielded elements and maximum

shear strain contours. Right: Trajectories of the post-mining principal stresses and maximum shear strain

contours. ................................................................................................................................................... 36

Figure 25 Medium quality rockmass: Mohr-Coulomb criterion without dilation: Left: Yielded elements

and maximum shear strain contours. Right: Trajectories of the post-mining principal stresses and

maximum shear strain contours. ............................................................................................................... 38

Figure 26 Medium quality rockmass: Mohr-Coulomb criterion with dilation: Left: Yielded elements and

maximum shear strain contours. Right: Trajectories of the post-mining principal stresses and maximum

shear strain contours. ................................................................................................................................ 38

Figure 27 Medium quality rockmass: Typical Hoek-Brown criterion: Left: Yielded elements and



Figure 28 Medium quality rockmass: Modified Hoek-Brown criterion: Left: Yielded elements and



Figure 29 Medium to good quality rockmass: Mohr-Coulomb criterion without dilation: Left: Yielded

elements and maximum shear strain contours. Right: Trajectories of the post-mining principal stresses and


Figure 30 Medium to good quality rockmass: Mohr-Coulomb criterion with dilation: Left: Yielded

elements and maximum shear strain contours. Right: Trajectories of the post-mining principal stresses and


Figure 31 Medium to good quality rockmass: Typical Hoek-Brown criterion: Left: Yielded elements and



Figure 32 Medium to good quality rockmass: Modified Hoek-Brown criterion: Left: Yielded elements

and maximum shear strain contours. Right: Trajectories of the post-mining principal stresses and


Figure 33 Good quality rockmass: Mohr-Coulomb criterion without dilation: Left: Yielded elements and



Figure 34 Good quality rockmass: Mohr-Coulomb criterion with dilation: Left: Yielded elements and



5

Figure 35 Good quality rockmass: Typical Hoek-Brown criterion: Left: Yielded elements and maximum


contours. ................................................................................................................................................... 45

Figure 36 Good quality rockmass: Modified Hoek-Brown criterion: Left: Yielded elements and maximum


contours. ................................................................................................................................................... 45

List of Tables Table 1 Linear Shape Effect empirical constants from various researchers (after Lunder, 1994).. ........... 12

Table 2 Size effect formulae empirical a and b constant values from various researchers (after Lunder,

1994).. ....................................................................................................................................................... 14

Table 3 Salamon and Munro (1967) database summary for compiled cases. ............................................ 15

Table 4 Characterisation of blocky rockmasses on the basis of interlocking and joint conditions (Hoek and

Marinos, 2000). ......................................................................................................................................... 32

Table 5 Material properties used as input parameters for the numerical analyses. .................................... 34

6

1. Introduction Pillars can be defined as the in-situ rock between two or more underground openings. Hence, all

underground mining methods utilize pillars, either temporary or permanent, to safely extract the ore.

Pillars can also be used to other underground structures serving for example as storage facilities. In mines

rectangular pillars are often designed in regular arrays, such that should a single pillar inadvertently fail

the load could be transferred to adjacent pillars causing these to be overloaded. This successive

overloading process can lead to an unstable progressive domino effect whereby large areas of the mine can collapse. Back analysis was one of the key elements in pillar design since 1960 in order to make pillars with adequate capacity to maintain the stability of the underground openings. This is an approach

that has been used extensively in geotechnical engineering and was also adopted in mining operations.

This kind of approach has led to the development of empirical formulas in order to estimate the strength

of a pillar. However, the modern trend nowadays is the use of numerical tools which can successfully

implement in the design the in-situ conditions (stress field, rockmass mechanical properties etc.) and thus

provide more realistic results by taking into account the specific properties of a particular site.

1.1 Pillar strength assessment

The design of mine pillars is a part of the mining operations sequence of great importance and the methods that they may be used vary significantly. Empirical pillar strength determination methods for

hard rock mine pillars have been proposed (Hedley and Grant, 1972; Hudyma, 1988; Lunder and

Pakalnis. 1997), however, these pillar design methods have relied on observed and measured behaviour of

full scale pillars, both stable and failed. Since the empirical formulas are based on site specific case

studies, the use of these empirical methods to design pillars a priori is limited and they cannot be used

with a high degree of confidence in hard rock mining operations.

The limited range of use of empirical approaches created the need to develop methodologies of

designing hard rock mine pillars in underground mining operations which would result in:

Increased ore recovery

Improved safety through better pillar design

Improved knowledge of pillar loading and failure mechanism such that modifications to mining plans can be quantified

Hence, it is rather significant that pillars can be designed with confidence for a varying range of rock

types, pillar shapes and sizes, and varying in-situ stress regimes. One of the modern trends in estimating

the strength of a mine pillar is the use of rock strength criteria, e.g. Hoek-Brown, Mohr-Coulomb etc., by

implanting them in various numerical analysis methods, e.g. Finite Element Method (FEM), Finite

Difference Method (FDM) etc., which are not limited by specific site characteristics.

1.2 Role of pillars in Mining

Mine pillars are found in all underground mines and play a wide and varied role depending on the

situation in which they are used. Pillar types can be:

Protection pillars surrounding mine shafts

Temporary pillars that allow quick exploitation of mineable reserves

Barrier pillars which must remain stable for the duration of the life of a mine Pillars may be designed such that failure will occur, while other pillars may require that they remain

stable for the duration of their life. In general, the role of a mine pillar is to support the adjacent rockmass

for a given period of time during the mining operations. In order for a pillar to perform its designed role,

the strength of it and the load acting upon it must be assessed. If these two factors are not adequately

determined, the performance of the pillar may not be the one desired in a specific project.

Three major categories of pillars can be classified according to Salamon (1983, after Lunder, 1994):

7

1. Support Pillars: Include all pillars that are used in situations where undermined hangingwall rock support is provided by a series of pillars. They are usually laid out in a systematic manner. Examples

in hard rock mining operations include room-and-pillar stope pillars, post-pillars and rib pillars.

2. Protection Pillars: Are employed to safeguard installations for which failure is intolerable. Examples of installations to be protected are surface buildings, mine shafts and boundary pillars between two

adjacent mining operations. These pillars can also be referred to as shaft pillars, roadway pillars or

boundary pillars. A significantly high factor of safety is used in these situations to compensate for

potential errors associated with the assumptions made in pillar strength estimation.

3. Control Pillars: Are employed in situations where rockburst activity is anticipated or experienced. These pillars are designed so that failure will not occur and are designed to reduce the magnitude of

stress changes in a mine environment and so alleviate the risk of rock bursting.

Pillars must be of sufficient size and appropriate shape so that they can support the induced loads

throughout their design life. The impact of poorly designed pillars can result in the mine being deemed

uneconomic because of an overly conservative design. Conversely, overly optimistic strength estimates

can result in local or regional failure in the mine horizon, making a portion or all of the mineable resource

unrecoverable.

8

2. Pillar Design Methodology In this section a brief literature review has been undertaken in order to assess previous practices in

pillar design. In the following sections the procedures employed in pillar design and pillar strength

estimation will be defined. Most of these practices have been developed for horizontally bedded coal

deposits, and as a result, these techniques are primarily applicable to similar deposits.

The function of pillars in mining is to maintain the stability of the adjacent strata for the design life on

the pillars. Simplistically, the safety factor of a pillar can be given by Equation 1. From this Equation it

can be inferred that unless the strength of the pillar is exceeded, the pillar will not fail under specific

loading conditions. This Equation forms the basis of all strength formulae. The safety factor can

subsequently be used to compensate for errors when estimating the input parameters used for the strength

formulae. This, however, requires that strength and stress estimates be determined with the associated

variability in each.

PillartheonAppliedStress

StrengthPillarSF .. (1)

The assessment of pillar stress in non-tabular or irregular dipping deposits is a complex task. The

intact strength of a sample of rock can be determined reasonably accurately by testing laboratory samples.

However, applying the intact strength of a rock sample to make an assessment of the strength of full size

pillar is a rather complex procedure, as the rockmass in-situ is not either intact, homogenous or isotropic,

while the intact rock samples in the lab are considered to have this kind of properties.

A common approach for pillar design was to use experience obtained under similar mining conditions

as the undertaken project. However, this trial and error method was but occasionally successful, as it is

not based on fundamental engineering principles. In this chapter a number of empirical and deterministic

methods of estimating pillar stress and strength will be presented.

Pillar design follows the premise that in most cases it is desirable to design pillars that will maintain

their loading capacity throughout their design life, thus the pillar strength must be sufficient to support the

stresses that the pillar will be subjected to. Therefore, it is of great significance that the pillar strength and

the pillars stresses are accurately estimated.

A rockmass is a generally a non-homogeneous and anisotropic medium and as such the determination

of pillar strength is highly dependent on the factors like, but not necessarily limited to, the following:

The intact strength of pillar material

The pillar geometry (width, height, width/height ratio)

The structural features within the pillar

The material properties of the pillar, such as deformational characteristics

The effects of blasting on the pillar Design methods are largely based on the limit state of equilibrium, meaning that they are based on

equating stress to strength so that a stable equilibrium exists. This requires that an estimate of stress has to

be made with levels of accuracy commensurate to strength estimates. The actual pillar stress depends on,

but not limited to:

The in-situ stress conditions

The mining induced stress changes

The effects of geological features, such as faults and jointing

The shape and orientation of pillars

The spatial relationship between pillars and mine openings

The effects of groundwater

9

Potvin (1985, after Lunder, 1994) presents pillar design divided into four broad groups: heuristic,

empirical, theoretical and numerical methods. The aforementioned categories include the methods which

are usually applied to design pillars and to assess the pillar strength even nowadays. Potvin (1985, after

Lunder, 1994) stated that heuristic methods are the most widely and least sophisticated methods used for

designing mine pillars as this kind of design is generally based on the principle that what worked before could work again and it neglects the strength or loading conditions. Empirical methods are based on previous experience or experimental data. Its main difference from the heuristic methods is that the case

histories are studied and then applied to the future design of mine pillars, which resulted in development

of strength formulas for pillars. A significant disadvantage of these methods is that the majority of the

conducted studies were performed in coal mines and it has been extrapolated to hard rock conditions but

not in a comprehensive manner. On the contrary theoretical methods of pillar design attempt to utilize

mathematical concepts and input parameters upon a rigorous formulation. Rockmass conditions, however,

can be highly variable and the determination of the critical variables is a rather challenging task. The

complexity of theoretical approaches makes them difficult to use and time consuming. Some of the

theoretical approaches include the work by Wilson (1972, after Lunder, 1994), Coates (1965, after Lunder

1994), Grobbelaar (1970, after Lunder 1994) and Panek (1979, after Lunder 1994). Due to the

development of sophisticated numerical tools nowadays, numerical analysis is becoming more and more

popular in determining the strength and stresses of a pillar. With relatively small computational cost,

complex problems included in the layout of a mine can easily be simulated in order to examine the

response of a pillar. However, it has to be taken into account the fact that they highly depend on the input

parameters used, thus results have to be treated with caution. Numerical analysis of pillars and their

response will be discussed in detail in the following sections.

2.1 Pillar stress and strength determination

Determining the actual stress applied on a mine pillar is rather difficult. In the previous section some

of the factors affecting the applied stresses on a mine pillar were discussed. In Figure 1 a simplified

example is illustrated in order to show the theory of stress redistribution after the excavation as being

analogous to a stream flowing around bridge piers. Two typical methods of calculating pillar stress found

in the literature include the tributary area theory and the application of numerical methods. Tributary area

theory utilizes a simplified approach in order to determine the stresses by implying that the load on each

pillar is a function of the vertical column of rock immediately above each pillar as well as the above area

between an individual pillar and its adjacent pillars as illustrated in Figure 2. Provided that the pillars

have a regular geometry it is possible to express the average pillar stress as a function of the extraction

ratio. The average stress in a pillar found by the tributary area theory is expressed as a function of the

extraction ratio, e, by the following expression:

)1( e

zp

(2)

where z represents the in-situ stress acting normal to the pillar axis. According to Brady and Brown (1992) (after Maybee, 2000) the extraction ratio, e, can be expressed in terms of the dimensions given in

Figure 2 as:

cbcaabcbcae / (3)

10

The average stress (p) on a square most pillar, using tributary area theory is given by the following expression:

2

1

p

o

pW

Wz (4)

Where, , is the unit weight of the overlying rock, z, is the depth below the ground surface, Wo, is the excavation width and, Wp, is the pillar width, as illustrated in Figure 3.

Figure 1 Sketch of stream lines in a smoothly flowing stream obstructed by three bridge piers (Hoek and Brown, 1980)

Figure 2 Plan view of geometry for tributary area analysis of pillars in uniaxial loading (Brady and Brown, 1992, after

Maybee, 2000)

11

Figure 3 Typical room and pillar layout showing load carried by a single pillar assuming total rock load to be uniformly

distributed over all pillars (Hoek and Brown, 1980)

While the tributary area theory gives a good approximation of the pillar stresses for simple uniform

geometries, in reality neither the pillars have a square cross-sectional shape nor the total load applied on

them comes from the total height of the rockmass column (arching effects etc.); hence due to the

complexity of the problem the tributary area theory is but a rough approximation able to provide only an

order of magnitude of the stresses developing in the pillar. Since the mid 1980s and more intensively nowadays due to the advance of computer science numerical modelling has been used extensively to

establish the stress distribution in pillars. For the purposes of this paper, a number of numerical models

were examined in order to determine the stress distribution within the pillars using the Finite Element

Method (FEM) code Phase2 distributed by Rocscience. Other proposed methodologies include Pariseaus Inclined Stress Formulae (1982), Szwilskis Chain Pillar Formulae (1982), Hedley & Grants Formula for Inclined Pillars (1972) (after, Lunder, 1994).

Regarding the estimation of the strength of a pillar some of the most common approaches include

empirical, theoretical and heuristic methods, as it has already been mentioned. Empirical methods rely on

experience combined with geotechnical terms related to the stability of the pillar in order to derive the

strength formula. On the contrary, theoretical approaches are derived mathematically to describe the

expected performance of a pillar when it is subjected to loading for a given set of input variables. While

both of the aforementioned methods have a rather strong background support, heuristic methods could be

better described as a rule of thumb techniques for designing pillars that may, however, disregard valid input parameters that affect the pillar strength. From the aforementioned approaches only the empirical

methods will be discussed in the following sections and they will be compared to the numerical analyses

conducted for this papers investigation purposes.

12

2.2 Empirical design methods

A number of empirical methods for pillar for pillar strength determination have been proposed by

various researchers include the following:

The Linear Shape Effect Formulas

The Power Shape Effect formulas

The Size Effect Formulas The aforementioned techniques relate the geometrical features of the pillar including the pillar width and

height, the intact rock strength and the safety factor to estimate pillar strength. The width of the pillar is

measured normal to the major principal stress induced in the pillar while its height is measured parallel to

the major principal stress induced in the pillar. All these formulae can be written in the following general

form illustrated in the following Equation.

b

a

Sh

wBAKP * (5)

in which Ps is the pillar strength, K is a term related to the material strength making the pillar, w is the

pillar width, h is the pillar height and A, B, a, b are empirically derived constants. A and B constants have

been determined by various researchers and are illustrated in Table 1. In the following subsections the

empirical methods proposed by Obert & Duvall (1967), Bieniawski (1975) and Hudyma (1988) are

discussed.

Table 1 Linear Shape Effect empirical constants from various researchers (after Lunder, 1994).

Source A B w/h

Bunting (1911) 0.700 0.300 0.5-1.0

Obert and Duvall (1967) 0.778 0.222 0.5-2.0

Bieniawski (1968) 0.556 0.444 1.0-3.1

van Heerden ((1974) 0.704 0.296 1.1-3.4

Bieniawski (1975) 0.640 0.360 1.0-3.1

Sorenson & Pariseau

(1978) 0.693 0.307 0.5-2.0

2.2.1 Linear Shape Effect Formulas

For this approach it is assumed that pillars of equal width/height ratios will have equal strength,

independent of the volume of the pillar and the relationship connecting pillar strength and pillar

width/height ratio is assumed to be linear. Continuing in the following sub-sections the work of Obert &

Duvall (1967), Bieniawski (1975) and Hudyma (1988) will be discussed.

2.2.1.1 Obert and Duvall (1967)

Obert and Duvall (1967) reported data obtained from a series of unconfined compressive strength tests

performed by Obert et al. (1946, after Lunder, 1994) on specimen coal pillars of varying width/height

ratios. The Equation proposed could be used to estimate coal pillar strength. Obert and Duvall (1967)

suggest that the strength parameter K that should be used in the formula is the strength of a specimen of

pillar material with a width/height ratio of one. However, this formula does not include a term to account

for the size effect on strength. Additionally, Obert and Duvall (1967) have made no recommendations

about the size of the specimen to be used for the determination of parameter K. They do however suggest

that a safety factor should be within a range between 2 and 4 in order to account for the size effect on

strength.

13

h

wKPS 22207780 ..* (6)

in which Ps is the pillar strength (MPa), K is the unconfined compressive strength of a cubical pillar

specimen (MPa), w is the pillar width (m) and h is the pillar height (m).

2.2.1.2 Bieniawski (1975)

Based on a tests of large scale coal specimens Bieniawski (1975) proposed Equation 7. The formula

was a result of performing in-situ tests on large scale coal specimens over a period of eight years

including a total of 66 tests in which the specimens varied in side length from 0.6m to 2.0m and had a

width/height ratio between 0.5 to 3.4. Bieniawski (1968) originally proposed the values for the empirical

constants of 0.556 and 0.444 used in this Equation could be used to describe the strength of the pillar.

h

wKPS 34006400 ..* (7)

2.2.1.3 Hudyma (1988)

Hudyma (1988) presented a method entitled the Pillar Stability Graph Method for determining the strength of open stope rib pillars based upon data derived from Canadian hard rock underground mining

operations. The method was derived by processing data collected on 47 case histories of pillars that had

been classified as being stable, sloughing or failed. The geometric data along with predicted pillar loads

were related to derive the Pillar Stability Graph illustrated in Figure 4. Three distinct regions were defined based on pillar observations. The valid range of pillar width/height ratios for this method is

between 0.5 to 1.4.

Figure 4 Pillar stability graph by Hudyma (1988)

2.2.2 Power Shape Effect Formulas

For these approaches it is assumed that the strength of the pillar is no more a linear function of the

width/height ratio but it a function of the square root of this ratio. The formula is defined by Equation 8.

This expression has been proposed by Zern (1926, after Farmer, 1982), Holland (1956, after Farmer,

1982) and Hazen & Artler (1976, after Farmer, 1982).

14

h

wKPS * (8)

2.2.4 Effective Pillar Width

The aforementioned methodologies assume that the shape of the cross-sectional area of the pillar is

square. Several researchers suggested that pillars of rectangular cross-sectional shape will have higher

strength than their square counterparts due to the confinement provided in the long dimension. Therefore,

several suggestions have been made on modifying them in order to take into account the aforementioned

increase in confinement as it is discussed later in this section. In all cases the pillar width in the strength

formulae is replaced by an effective pillar width term.

Sheorey & Singh (1974, after Lunder, 1994) suggested that width term should be substituted by the

effective pillar width term and they proposed that the effective width would be the average of the length

of the two pillar sides. Their work was based on small-scale sample testing of various rectangular

dimensions. In the work of Wagner (1980, after Lunder 1994) and Stacey & Page (1986, after Lunder,

1994) the width term has been proposed to be replaced by an effective pillar width term defined by

Equation 9.

R

AW

p

e *4 (9)

in which We is the effective pillar width (m), Ap is the cross-sectional area of the pillar (m2) and R is the

circumference of the pillar (m).

Although the aforementioned methods extrapolate the strength of a square pillar to a rectangular pillar,

it becomes evident that an upper limit of the pillar strength attributed to the increase in the pillar side

length has to be defined.

2.2.5 The Size Effect Formulas

In this section the Size effect Formulas will be discussed. A general form of the formula is illustrated

in Equation 10. The effect of this type of formula is that by increasing the size of the pillar, the strength of

a pillar of equal shape is going to decrease.

b

a

Sh

wKP * (10)

in which a and b are constants derived empirically. In Table 2 proposed values for these two constants

respectively are presented including the work of various authors.

Table 2 Size effect formulae empirical a and b constant values from various researchers (after Lunder, 1994).

Source a b

Steart (1954) 0.5 1.0

Holland-Gaddy (1962) 0.5 1.0

Greenwald et al. (1939) 0.5 0.833

Hedley and Grant (1972) 0.5 0.75

Salamon and Munro (1967) 0.46 0.66

Bieniawski (1968) 0.16 0.55

Sheorey et al. (1987) for slender

pillars 0.5 0.86

Continuing, the work of Salamon and Munro (1967), Hedley and Grant (1972), and Sheorey et al.

(1987) will be discussed in the following sections. It has to be noted that as a result of the dimensionally

unbalanced nature of the formulae, quantities of length must be in feet and strength has to be in

15

pounds/square inch. Going from Imperial to metric units, term K has to be reduced to compensate for the

dimensionally unbalanced conversion.

2.2.5.1 Salamon and Munro (1967)

Salamon and Munro (1967) conducted their research on square pillars in South African coal mines.

Their database consisted of 125 case histories, the 98 of which were classified as stable and 27 as

collapsed, as it is illustrated in Table 3. The applied loads on the pillars were estimated by using the

tributary area theory and from the statistical assessment of this data the empirical strength constants in

Table 2 were derived. Salamon and Munro (1967) used the coal strength constant K for the entire pillar

cases in the dataset. The parameter value was determined statistically from all the case histories without

reference to the actual intact coal strength at each mining operation.

Table 3 Salamon and Munro (1967) database summary for compiled cases.

Group Stable Collapsed

Number cases 98 27

Depth (ft) 65-270 70-630

Pillar height (ft) 4-16 5-18

Pillar width (ft) 9-70 11-52

Extraction ratio 37-89 45-91

w/h ratio 1.2-8.8 0.9-3.6

2.2.5.2 Hedley and Grant (1972)

Hedley and Grant (1972) proposed their pillar design method by processing data from uranium mines

in the Elliot Lake district of Ontario, Canada. Their dataset consisted of 28 pillar case histories, from

which 23 were stable, 2 were partially failed and 3 were completely failed. Hedley and Grant (1972)

formulae is described by Equation 11.

750

50

.

.

*h

wKPS (11)

in which Ps is the strength of the pillar (psi), K is the strength of 30cm cubic sample (0.7*UCS for 50mm

diameter samples, 179 MPa or 26,000 psi for Elliot Lake rocks, w is the width of the pillar (ft) and h is

the height of pillar (ft).

The applied pillar stress was determined using the tributary area theory with modifications in order to

take into account the horizontal in-situ stresses (Figure 5). That was because the Eliot Lake uranium

mines occur in dipping orebodies. This work represents one of the few instances where hard rock pillar

data were used to develop a pillar strength formula. Hedley and Grants (1972) work was based on the Salamon and Munros (1967) method in order to derive their relationship. The development of a Size Effect Formula for pillar strength, however, requires a database in which pillars of various sizes have to

be included and the Hedley and Grants (1972) formulae was derived from pillars of similar size; hence the use of a Size Effect Formulae cannot be fully justified. Additionally, the use of only three pillars in

failure to develop a strength relationship leaves the potential for a wide margin of error.

16

Figure 5 Hedley and Grant's (1972) method for determining pillar stresses (Hedley and Grant, 1972)

Figure 6 Hedley and Grant's (1972) estimation of pillar stresses and strengths (Hedley and Grant, 1972)

2.2.5.3 Sheorey et al. (1987)

As in most of the aforementioned cases, Sheorey et al. (1987) investigated stable and failed pillars for

coal mines in India and proposed the empirical strength formula described by Equation 12. The database

used to conduct this method is comprised of 23 failed and 20 stable pillar observations. Sheorey et al.

(1987) have also proposed a second formula for slender pillars (w/h

17

860

50

270.

.

*.h

wP cS (13)

in which Ps is the strength of the pillar (MPa), c is the unconfined compressive strength of a 25mm cube of pillar material, pillar width (m), pillar height (m) and H the depth below surface (m).

Generally, an advantage of empirical strength formulae is that they are based on observations thus

incorporating data from full size mine pillars. However, they do not make an attempt to explain the

mechanism of how a pillar is loaded or fails.

2.3 Numerical modelling techniques

Numerical modelling techniques have been widely used during the few last decades in order to help

deal with geomechanical, geotechnical and mining problems in which the level of complexity is rather

significant and empirical or other methods seem to be inadequate. Numerical models are able to

determine stress redistributions around the excavation boundaries within a triaxial stress field employing

either two-dimensional or three-dimensional analyses more accurately than the tributary area theory for

complex geometrical or geological environments, as the simplistic approach of this methodology is not

able to describe the complexity of the given conditions.

Numerical modelling methods including the Boundary Element Method (BEM), the Finite Element

Method (FEM), the Finite Difference Method (FDM), the Discrete Element Method (DEM) etc. use

mathematical formulas to solve stress related problems. Common applications of numerical modelling

include almost every single field of engineering and today they are widely accepted. Today numerical

methods include sophisticated tools and software which can deal with the complexity of various

problems. However, even these sophisticated tools are subjected to various limitations. Numerical models

are created using numerical codes and algorithms which must have specific inputs in order to produce the

requested outputs for a problem. However, poor input data will result in poor output datasets. In mining

and geotechnical problems input data consists of in-situ stress conditions (whether they are measured or

estimated), the geometrical features of the opening, the rock material properties etc.. Due to the high

complexity of the problems, and thus of the models, approximations are adopted in order to simplify

them, such as the assumption that rock material is homogeneous, isotropic, elastic etc., which is not the

case in either geotechnical or mining engineering in which geomaterials are characterized by non-

homogeneity, anisotropy and elasto-plastic response. Although possible approximations may be adopted

in order to perform the analysis, the results obtained by using numerical methods are still going to be of

higher quality than using the tributary theory and empirical methods. Among the numerical techniques

employed nowadays are continuum and discontinuum techniques, and during the last decade hybrid

techniques combining the previously mentioned ones have started being developed.

2.3.1 Continuum Methods

In this category the Finite Element Method (FEM) and the Finite Difference Method (FDM) are

included. Continuum techniques assume a continuum medium in which the mathematical and physical

approximations are required to be made throughout the region of interest, which is the problem domain.

Both of the aforementioned techniques utilize approximate numerical solutions for the partial differential

Equations governing the problem within the problem domain.

A finite difference approximate the value of some derivative of a scalar function u(x) at a point xo in its domain, e.g. u(xo) or u(xo), relies on a suitable combination of sampled function values at nearby points. This gives an approximate solution, based on the selected step, to an exact problem when the FD

Method is employed. On the contrary, in the FE Method the problem domain is discretized into a series of

different elements connected with each other through shared points (nodes). This provides the physical

approximation of continua in order to calculate stresses and strains within the medium. The partial

differential Equations governing the problem are solved exactly at nodes at which adjacent elements

connect, as long as the force and moment equilibrium are satisfied. The result of this is an exact solution

for a differential approximation to the problem. More particularly for the FE Method, the response of the

18

model, and thus the solution of the problem, highly depends on the boundary conditions applied for each

single case; hence boundary effects have to be limited in order to secure the accuracy of the solution.

However, increasing the dimensions of the model in order to minimize boundary effects has a significant

impact on the computational cost in terms of time and output space, although this can be mitigated by

manipulating the density of the mesh. The FE Method will be discussed more thoroughly in the following

sections where the numerical model used will be described in detail.

2.3.2 Discontinuum Methods

Considering rockmasses as a continuum is a rough approximation in order to describe fairly complex

problems. This approximation can only be valid though in some cases, where the scale of the project does

not let it be affected by the anisotropy of the material. However, when discrete discontinuities are likely

to affect a mining project and the anisotropy of the material cannot be neglected the continuum techniques

are inadequate and cease to be valid for application. That led to the development of techniques capable of

dealing with discontinua problems. The premise is to iteratively use Newtons second law on multiple elements of a discretized domain. The elements however do not share common nodes. On the contrary

each element is an independent entity which interacts with its adjacent elements through an assigned

contact interaction property resulting in the generation of contact forces between them.

2.3.3 Hybrid Methods

Based on the aforementioned techniques, in the early 1990s the combined Finite Element-Discrete

Element Method (FEM-DEM) started being developed. By using this method it is possible to consider

systems comprising millions or even billions of particles, and its application is considered rather

significant for simulation of failure, fracture, fragmentation and collapse of solid materials. The most

important advantage of this particular methodology is the faster solution that it provides as the system is

considered continua before fracturing. However, this technique is subjected to significant limitations in

terms of computational cost if sophisticated algorithms for contact detection and interaction are not

employed.

19

3. Pillar Failure Mechanisms Knowledge of the complete stress-strain curve for pillars is rather important in order to fully

understand the modes of failure that might be experienced at the site. An early indication of the response

of a pillar in the field can be inferred by testing rock specimens in the lab. In Figure 7 the complete stress-

strain curve is illustrated for varying specimen sizes but with the ratio of length to diameter remaining

constant. The main effects are that both the compressive strength and the brittleness of the specimen are

reduced as the size of the specimen increases. The specimen contains micro-cracks (which are a statistical

sample from the rock micro-crack population); therefore the larger the specimen, the greater number of

micro-cracks and hence the greater the likelihood of a more sever flaw. The supplementary effect to the

size effect is the shape effect, when the size (e.g. the volume) of the specimen is preserved but its shape

changes. In Figure 8 the complete stress-strain curve is illustrated in uniaxial compression. It can be

observed that the strength and ductility increase as the aspect ratio, defined as the ratio of diameter to

length, increases. According to Hudson et al. (1972, after Maybee, 2000), as failure propagates, the actual

cross-sectional area of the sample decreases, however, the stress is still calculated using the original

cross-sectional area of the sample. For squat samples (small L/D ratio), the actual cross-sectional area of

the sample during testing does not vary significantly from the original.

Figure 7 The size effect in the uniaxial complete stress-strain curve (Hudson and Harrison, 1997).

20

Figure 8 The shape effect in uniaxial compression (Hudson and Harrison, 1997).

In the field, however, there are three modes of pillar failure which are commonly observed.

Structurally controlled failure.

Stress-induced progressive failure.

Pillar bursts. In most cases, rockmasses contain pre-existing failure planes (bedding, joints or other discontinuities);

hence structurally controlled failure may occur when the pillars are orientated unfavorably with respect to

the discontinuities existing within the rockmass. When a pillar fails along these planes, it is usually in the

form of shear movement along that plane of weakness (Figure 9.b and d). This specific type of failure is

often observed as corners of pillars coming off along well-defined planes. Where structure is oriented

along the vertical axis of the pillar, a buckling failure mode can be assumed (Figure 9.e).

Another type of failure, as it has already been mentioned, is progressive failure due to the stresses

induced in the pillar. This is observed as slabs spalling off the walls of the pillars. This progressive

spalling mode of failure, also known as hour-glassing (Figure 9.a) due to the distinct shape of the pillar as

slabs of material are falling off, can be observed in squat pillars. Due to the little confinement and the

existence of high tangential stresses, cracking occurs and slabs parallel to the direction of the major

principal stress in the pillars are formed. At the early stages of the failure the core of the pillar remains

intact due to the high confinement pressure it still has; hence the pillar retains most of its load capacity.

However, as material starts falling off the pillar and spalling occurs, lead to a redistribution of the stresses

applied within the pillar towards its intact core. The loss of slabs relaxes the confinement on the adjacent

intact rock material and further damage occurs to the newly exposed pillar walls. If this type of failure is

not mitigated and it is allowed to propagate further into the intact material, a critical cross-sectional area

is formed which suffers from an extreme concentration of stresses leading to the failure of the pillar.

Other progressive types of failure include internal splitting as the result of pillar movement along soft

partings (clay) at the top and bottom of the pillar (Figure 9.c).

21

Figure 9 Principal modes of deformation behaviour of mine pillars (Brady and Brown, 1985)

Another failure mode that can be encountered is pillar burst. In order to have this kind of failure two

conditions have to be met: a. the stress in the pillar must exceed the strength; and 2. the local mine

stiffness must be less than that of the pillar. Based on the work by Martin (1997) and Kaiser et al. (1996),

when the pillar stress exceeds 1/3 of the uniaxial stress compressive strength of the rock the first

condition is generally met. Pillars undergoing burst failure may either fail entirely or partially while the

core remains intact (Figure 10).

Figure 10 Schematic illustrating the difference between pillar skin bursts and pillar bursts (Maybee, 2000)

22

4. Rockmass Failure Criteria How a rock fails, either in terms of the precise details of each micro-crack initiation and propagation

or in terms of the total structural breakdown as many micro-cracks propagate and coalesce, is not exactly

known. In both cases, the process is extremely complex and it cannot be subjected to convenient

characterization through simplified models. Traditionally, stresses have been regarded as the cause and strains as the effect in materials testing and as consequence, early testing and standards utilized a constant stress rate application; hence it was natural to express the strength of a rock material in terms of

the stress achieved in the test specimen at failure. Uniaxial and triaxial compression tests of rock

materials are among the most common laboratory testing techniques in rock mechanics and rock

engineering. Therefore, the most obvious means of expressing a failure criterion is to have strength as a

function of the principal stresses. However, with the advent of stiff and servo-controlled testing machines

and the associated preference for strain rate control, perhaps the strength could be expressed in the form

of principal strains. Another possibility includes more eclectic forms of control such as constant rate of

energy input, leading to more sophisticated possibilities for strength criteria expressed in the form of

principal stresses and strains.

However, the number and variation of the failure criteria which have been developed, and which are

more commonly used are limited. Among them, the Mohr-Coulomb criterion expresses the relation

between the shear stress and the normal stress at failure. The plane Griffith criterion expresses the

uniaxial tensile strength in terms of the strain energy required to propagate micro-cracks, and expresses

the uniaxial compressive strength in terms of the tensile strength. The Hoek-Brown criterion is an

empirical criterion derived from a best fit to strength data plotted in 1-3 space. From the aforementioned criteria the Mohr-Coulomb and Hoek-Brown criterion will be discussed further in the

following sections.

4.1 The Mohr-Coulomb criterion

The plane along which failure occurs and the failure envelope are illustrated in Figure 11 for the two

dimensional case, together with some of the most basic expressions associated with the criterion. From

the initial principal stresses, the normal stress and shear stress on a plane at any angle can be found using

the transformation Equations as represented by Mohrs circle. Utilizing the concept of cohesion, which is the shear strength of the rock when no normal stress is applied, and the angle of internal friction, which is

the equivalent to the angle of inclination of a surface sufficient to cause sliding of a superincumbent block

of similar material down the surface, the linear Mohr envelope is generated. The failure envelope (or

locus) defines the limiting size for the Mohrs circles. In other words, - coordinates below the envelope represent stable conditions, while the - coordinates o the failure locus represent limiting equilibrium. Coordinates - above the failure envelope represent unstable (and thus not real stress combinations) under static loading conditions. The criterion has been developed for compressive stresses and the tensile

strength of the rock cannot be determined adequately, as usually is seriously overestimated. In order to

mitigate this problem a tensile cut-off is usually utilized to provide a realistic value for the uniaxial tensile

strength.

However, the Mohr-Coulomb criterion is not usually adequate to describe the response of a rockmass

except from the cases in which high confinement pressure is applied. In these cases the Mohr-Coulomb is

adequate to describe the response of the rockmass as the material does fail through the development of

shear planes. At lower confining pressures, as in the unconfined case, the failure gradually occurs due to

the increase in the density of the micro-cracks. These micro-cracks are formed approximately sub-parallel

to the major principal stress; hence this type of frictional criterion does not directly apply. When the

confinement pressure is high, however, this linear relationship between the stresses can be observed and

as illustrated in Figure 11 the failure plane will be oriented at =450+/2. In the case of the existence of significant water pressure is porous materials; its influence can clearly

be observed as the Mohr circle is moved to the left (as the effective stresses decrease) by an amount equal

to the water pressure. Therefore, the criterion takes into account the possibility of going from a stable

23

condition to an unstable (failure) one. Although there are difficulties associated with application of the

criterion, it remains one of the most widely used due to the small number of the input parameters required

which make it a rapidly calculable method for engineering practice. Additionally, it is rather significant in

the case of discontinuities.

Figure 11 The Mohr-Coulomb failure criterion (Hudson and Harrison, 1997)

4.2 The Hoek-Brown empirical criterion

The Hoek-Brown criterion has been created to describe the response of rockmasses at failure based on

an empirical method as it is derived from a best-fit curve to experimental data. The criterion is plotted in

the 1-3 space as shown in Figure 12. The criterion in its original form (1980) was expressed as:

5.02331 cici sm (14)

where 1,3 are the major and minor principal stresses respectively, ci is the unconfined compressive strength of the intact rock, m and s are material constants for a specific rock type. The criterion in its

latest form (2002) has slight differences when compared to its initial format and is expressed as: a

cibci sm

331 (15)

where mb is the reduced material constant given by the following expression:

D

GSImm ib

1428

100exp (16)

GSI: Geological Strength Index after Marinos and Hoek, 2000 (Section 6.2.1).

D: Disturbance Factor taking values between 0-1 (undisturbed to very disturbed rockmass)

mi: constant of the material for the intact rock.

s is the material constant given by the following expression:

D

GSIs

39

100exp (17)

a is the constant of the criterion affecting the curvature of the envelope given by the following expression:

24

3

20

15

6

1

2

1eea

GSI

(18)

Although constants m and s arise from the curve fitting procedure, there is an element of physical

interpretation associated with them. Regarding the s parameter, it relates the degree of fracturing of the

rockmass and thus it is analogous to cohesion. If the rock is completely intact then s becomes equal to 1

(it can be derived from Equation 17 by making the GSI parameter equal to 100 or by substituting 3=0 into the criterion). It has to be noted that for an intact rock specimen with a response obeying to the Hoek-

Brown criterion, if 3=0, then 1=ci which is the intersection of the failure locus with the 1 axis (Figure 12). For a highly fractured rockmass s tends towards zero with the strength of the rockmass reducing

significantly. Regarding parameter m, it is related to the degree of particle interlocking present in the rockmass and thus with friction. If the rock is intact then parameter m is equal to mi and if it is fractured

this parameter decreases as it can be observed from Equation 16. Inter-criteria relations can be found in

particular linking the Hoek-Brown criterion with the Mohr-Coulomb criterion, i.e. linking m and s

parameters with c and .

Figure 12 The Hoek-Brown empirical failure criterion (Hudson and Harrison, 1997)

25

5. Damage and spalling prediction criteria The problem of spalling damage and associated strain bursting in deep mining excavations, deep

tunneling and shaft boring creates significant issues. Spalling can occur in drill and blast excavations

when blast damaged is limited, but it is particularly problematic in bored tunnels and shafts. Spalling

damage in shafts and orepasses can propagate in an uncontrolled fashion during operation, reducing

drastically the service life span of the structure. This kind of damage and failure mode is the same as that

which is prevalent in the larger scale mining environments. Continuing, in order to describe this kind of

failure mode the following definitions can be observed:

1. Spalling occurs when visible extension fractures develop under compressive loading. Spalling is associated with hard rock excavations. Hard rocks are brittle in nature but the failure mode is not

violent. This particular process governs rock damage and failure processes in crystalline rocks near

excavation boundaries under high stress. Spalling can be either violent or nonviolent, while in some

cases it may be time dependent. When the opening remains unsupported and under anisotropic in-

situ stresses, notch geometries can be formed often confused with wedge fallout.

2. Strain bursting occurs when violent rupture of rock walls under a high stress condition. In spalling rocks the extension fractures can happen before the actual rockburst or strain burst. It is the

instability created by the formation of parallel and thin spall slabs that provides for the sudden

energy release. In most cases of moderately extensive brittle spalling failure, the rock located beyond

the zone created by spalling is reasonably competent and the excavation is typically self-stable after

the failure occurs provided, though, the local stress condition remains the same.

For non-isotropic stress fields (Ko1) the failure geometry is often notch shaped and the surrounding ground is self-stable after failure. Normally spalling involves the creation of fractures parallel to the

maximum compressive stress and to the excavation boundary. When the propagation of these fractures is

unstable beyond the grain scale of the rock, this results in the typical spalling process observed in mining

excavations, rock pillars and deep tunnels. When strain bursting develops, the failure process must be

brittle and dilational and the rockmass around the failing rock must be comparatively soft, subjected to

large instantaneous deformations into the zone previously occupied by the failing rock.

The mechanics of spalling are inherently brittle with strength loss occurring across the fracture.

However, the macroprocess of failure may not be. The development of parallel slabs may be significant

with depth while retaining load capacity parallel to the slabs. Likewise the process of spalling does not

need to incur large dilation until secondary yield process begins. Furthermore, kinematic instability

(buckling) shear-through, slab failure, bulging due to dilatant shear behind the zone of spalling, or release

along intersecting structure can subsequently and instantaneously lead to energy release, which was stored

in the slabs and the surrounding system, leading to strain burst and significant apparent buckling.

Following this, a significant decrease in the tangential stress occurs which leads to release of large

volumes of pre-spalled rock in an aseismic groundfall.

The extensile crack initiation as the result of various internal heterogeneities and strain discontinuities

within polycrystalline or clastic material is the primary form of damage, even under compression for hard

rock materials. The process of failure is controlled by the internal tensile strength of the rock. Under low

confinement conditions the extension of cracks leads to the familiar spalling processes which can be

observed in compressive samples under low confinement and around hard rock openings at depth. From

this, it is possible to develop a criterion for rock material susceptible to brittle spalling (as opposed to

plastic shear) based on the ratio between compressive and tensile strength (Lee et al., 2004). A lower ratio

indicates the potential dominance of extension cracking in the damage process defining the spall

potential. Stronger rocks generally result from a resistance to shear failure as it can be reflected in the

unconfined compressive strength. Additionally the stronger the material, the larger the buildup of strain

energy in the wall rock and in the surrounding rockmass defining the strain burst potential. This is

illustrated in Figure 13.

26

Figure 13 Potential for spalling failure processes in intact rock based on compressive strength and tensile strength

(Diederichs, 2007)

Volume change associated with bulking or strain bursting occurs due to the buckling and

discontinuous interleaving of fractured slabs; hence this not the mechanism assumed in traditional

plasticity theories when dilation is considered. The process of spall damage is incompatible with the

mechanics usually applied in the most commonly used constitutive models for continuum geomaterials.

Conventional models for Rock Mechanics such as the Mohr-Coulomb criterion and Hoek-Brown

criterion, mentioned earlier, are based on yield via continuum shearing. However, spalling is the result of

parallel slab formation driven by extension cracking, therefore after the formation of the slabs the rock

cannot be considered a continuum anymore and the discontinuous nature of it dominates the failure

mechanism. Of great importance is the lack of frictional behaviour in its mechanical sense. As the slabs

fail due to the result of extension cracking and remain parted, friction cannot play a role in the failure

mechanism. The lower bound for damage initiation is closely related to the internal mechanics of the

constituent crystals. The role of friction in crack initiation is rather limited as a result of the relatively low

coefficients related to internal cleavage planes and grain boundary structures. Although friction is not

important in subsequent crack propagation, confinement dependency does play a role in the intersection

of accumulating and propagating cracks to form failure surfaces (Diederichs, 2003). The

phenomenological result with respect to the upper bound strength of rock is similar to the conventional

frictional relationship. This allows the use of conventional models and constitutive relationships for

spalling applications, although the physical processes differ significantly. For example, in the Mohr-

Coulomb criterion, cohesion is a function of internal tensile strength while the friction angle represents

critical accumulation and interaction of extension cracks and the associated critical extension strain.

5.1 Hard Rock Lower and Upper Bound Strength

Lower bound in-situ compressive strength for excavations in hard rocks corresponds to an extension

crack damage initiation threshold. This threshold is a function of the nature and density of internal flaws

and heterogeneity of the rock material. Numerous researchers (Pelli et al., 1991; Martin et al., 1999;

Brace et al., 1966; Wagner, 1987; Castro et al., 1995) have shown that in massive hard rock excavations

failure begins when the tangential stresses at the excavation boundary exceed 33% to 50% of the

unconfined compressive strength of the rock. The aforementioned threshold when expressed relative to

laboratory unconfined compressive strength can be as low as 1/3 for igneous rocks, after Lajtai and Dzik

(1996), and as high as 1/2 for dense clastic materials, after Pestman and Van Munster (1996). The

27

mechanisms illustrated in Figure 14 can occur even at moderate confinement confining stresses, but they

can only propagate under conditions of locally low confinement. When confinement pressure is high

crack accumulation must occur earlier than crack coalescence leads to propagation. In this particular case

the upper bound yield strength in-situ is going to correspond to the long term strength of laboratory

samples (Figure 15). However, at low confining stresses near the excavation boundary spontaneous

propagation of initiating extension cracks may be possible. If that is that is the case, the yield strength of

the rockmass in-situ collapses to the damage initiation threshold as illustrated in figure 15.

Figure 14 a. Shear failure around a tunnel; b. spalling damage in hard rocks at high GSI; c. example of brittle spalling

and strain bursting in a deep mine opening; d. mechanisms of crack initiation in hard rock (Carter et al., 2008)

Figure 15 The composite strength envelope illustrated in principal stress space (2D) to highlight the zones of behaviour as

bounded by the damage initiation threshold, the upper bound shear threshold (damage interaction) and the transitional

spalling limit (Diederichs, 2003)

5.2 Hard Rock Strength using Hoek-Brown Empirical Criterion

In this section the application of the Hoek-Brown criterion in its generalised non-linear form for

rockmasses with GSI65 will be discussed. In such cases a different approach needs to be adopted in order to determine the correct input parameters rather than the normal GSI and the Unconfined

Compressive Strength of the intact rock (UCSi) and consideration needs to be given to spalling processes

and tensile cracking behaviour. Outlined by Diederichs et al. (2007) definition of spalling behaviour for

28

modelling in an inelastic Hoek-Brown formulation can be achieved carrying through the steps described

below and by taking into account Figure 16:

i. Determine UCS*, the onset of systematic cracking (B in Figure 16), from acoustic emission or radial strain data.

ii. Obtain a reliable estimate of tensile strength T (e.g. carefully executed Brazilian testing) iii. Set aSP=0.25 (ie., assuming peak conditions-spalling initiation). iv. If basic laboratory UCSi for the rock material is known, appropriate values of sSP and mSP have to

be estimated according to the following expressions:

SPaiSP UCSUCSs1

/* (19)

TUCSsm iSPSP / (20)

Figure 16 Determination of damage initiation thresholds for rock under compressive loading using strain and acoustic

emissions (Diederichs et al., 2004)

It has to be noted at this point that the aforementioned parameter determination applies only for the onset

of systematic cracking and thus define the peak or spalling initiation threshold curve (lower slope gradient curve on Figure 17).

However, initiating spall cracks can only propagate under low confinement and therefore a second

threshold is needed in order to define the condition constraining spalling behaviour. Based on the

experimental work of Hoek (1968, after Carter et al. 2008) and on numerical simulations described by

Diederichs (2007) a limit can be postulated creating a transition between spalling behaviour for low

confinement stress fields, governed by the initiation threshold, and a shear strength closer to the long term

strength limit for the intact rock (as extensile crack propagation is suppressed) at higher confinement

stress fields. This residual spalling limit can be approximated using aSP=0.75, sSP=0 and a residual mres value of approximately mi/3 (typically 5-10). By substituting the aforementioned spalling parameters into

the generalised non-linear Hoek-Brown formulation the two curves illustrated in Figure 17 are derived,

along with the equivalent Mohr-Coulomb thresholds, with the peak parameter Mohr-Coulomb envelope line defining the damage initiation and the residual parameter envelope line defining the spalling confinement limit.

29

Figure 17 Example of "peak" and small strain "residual" strength parameters for damage initiation and spalling limits

(Diederichs, 2007)

The shape of the residual limit curve is very sensitive to confinement, as provided by natural interleaving of spalled material, by rock reinforcement or other constraint methods. The determination of

the absolute magnitude of the imposed constraint in any given situation may not be always

straightforward. However, some measure of natural or applied constraint degree may be achieved by

using parameter D of the Hoek-Brown criterion, which is the dilation parameter of it, described in the

previous sections. Dilation should be set to zero in cases where unrestrained fallouts or loose retention

occurs. If light support is installed, like wire mesh, spot bolts etc., dilation of the material can be set by

adopting an appropriate non-associative flow rule substituting mdil for mb by using an appropriate value of

mdilmres/(8-10), where mres is the value specified in the yield function for residual strength. When using the aforementioned approach for evaluating the extent of likely inelastic radial

displacement, it is rather important to examine the plastic shear strain gradients within the yielded rockmass and determine if the rock is actually failing or is merely damaged. By observing the contour

plot of a numerical method software, the region enclosing large plastic strains should be consider as

spalling. On the contrary apparently yielded regions with very low relative strain values would more

likely correspond to stress paths above the damage threshold but below the spalling limit.

30

6. Numerical simulation of hard rock pillars In the previous sections some of the most common empirical methods for estimating the strength of a

pillar, the failure mechanisms of pillars and additionally the most common failure criteria used in Rock

Mechanics including the Mohr-Coulomb failure criterion, the Hoek-Brown failure criterion and the

modified Hoek-Brown criterion which describes the brittle behaviour of hard rocks. In this section, the

aforementioned will be summarized in the form of numerical examples and the response of a pillar will be

examined through application of some of the empirical and numerical methods described earlier. For this

particular case the following assumptions are made:

The unconfined strength of the rock is UCS=70MPa and its tensile strength is assumed to be T=2.8MPa.

Geostatic stresses are assumed to be principal stresses initially and the lateral pressure coefficient is ko=1.2. The vertical stress is assumed to be v=13MPa and the horizontal stress h=15.6MPa at the depth of interest.

The pillar is assume to have a width to height ratio wp/hp=1.

The width of the stope is assumed to be equal to the width of the pillar (ws=wp). Additional parameters for the application of the Mohr-Coulomb and Hoek-Brown criterions in numerical

simulation will be given in the following sections.

6.1 Results of empirical method application

In this subsection the results derived by using the empirical methods of Obert and Duvall (1967),

Hedley and Grant (1972), and Sheorey et al. (1987) for estimating the strength of the pillar will be

presented. For the estimation of the stresses induced on the pillar the tributary area theory will be used.

For each single of the aforementioned methodologies the following pillar strength values are

estimated:

Obert and Duvall (1967): Ps=49 MPa (According to Equation 6)

Hedley and Grant (1972): Ps=27.66 MPa (According to Equation 11)

Sheorey et al. (1987): Ps=18.90 MPa (According to Equation 13) In the aforementioned methodologies it was assumed that for Obert and Duvalls (1967), and Hedley and Grants (1972) formulas coefficient is K=0.7*UCS for this example, while for the Sheorey et al. (1987) formula parameter c is equal to the UCS value of the rock. Continuing, by using the tributary area theory the stress induced on the pillar is assumed to be equal to p=52MPa (According to Equation 4). In figure 18 the safety factor of each one of the applied methodologies is illustrated. For all cases the pillar is

expected to be unstable as the stress applied on it is greater than the estimated pillar strength. However,

the assumptions made for the input parameters are judged as rather conservative due to the inability of

defining accurately the coefficient parameters relative to the material. It has to be noted though, that

although all of them predict failure of the pillar they do not describe the failure mechanism and they do

not take into account the major principal stress orientation, as it is the horizontal geostatic stress in this

case.

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Figure 18 Estimated Pillar Safety Factor based on empirical methods.

6.2 Results of numerical method analysis application

In sections 4 and 5 different approaches of rockmass failure criteria were discussed. By using these

approaches the failure mechanisms of a pillar example will be discussed. In order to achieve this, the

numerical finite element method was applied by using software Phase2 distributed by Rocscience. This

subsection will be divided into four groups depending on the quality of the rockmass assumed based on

the GSI classification system in order to discuss the results.

6.2.1 The Geological Strength Index (GSI) rockmass classification system

The strength of a jointed rockmass depends on the properties of the intact rock pieces forming the rock

material and upon the freedom of these pieces to slide and rotate under the application of different stress

fields (magnitude and orientation). This behaviour is controlled by the geometrical shape of the intact

rock pieces and additionally to the condition of the discontinuity surfaces separating the intact pieces e.g.

pieces of rock which are angular with clean, rough surfaces result in a stronger rockmass than one

containing rounded pieces which are surrounded by weathered and altered material.

The Geological Strength Index (GSI) introduced by Hoek (1994) and Hoek, Kaiser and Bawden

(1995) provides a number which when combined with the intact properties of the material estimates the

reduction in rockmass strength for different geological conditions. The suggested classification system is

presented in Table 4 for blocky rockmasses. For other type of rockmasses the reader is advised to go back

to the work of Hoek et al. (2006) and Marinos et al. (2005). Before the introduction of the GSI system in

1994, the application of the Hoek-Brown criteri

An investigation on the respone of hard-rock pillars in Mining

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