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Rock mass strength with non-persistent joints
B.H. Kim, M. Cai & P.K. Kaiser
Geomechanics Research Centre, MIRARCO Inc., Laurentian University, Ontario, Canada
ABSTRACT: Although tensile strength can often be neglected for weak or highly jointed rock masses, the strength of rock bridges in moderately fractured and blocky hard rock masses contributes significantly to their self-supporting capacity and should be considered in engineering design. This self-supporting capacity can often be attribut

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1
INTRODUCTION To provide a practical tool to estimate the rock mass strength for use with the Hoek-Brown failure crite-rion (Hoek & Brown 1980, 1988, Hoek et al. 2002), the
GSI
system was introduced (Hoek et al. 1995). The value of
GSI
ranges from 0 to 100. The
GSI
sys-tem consolidates various versions of the Hoek-Brown criterion into a single simplified and general-ized criterion that covers all of the rock types nor-mally encountered in rock engineering. A
GSI
value is determined from the interlocking structure and joint surface conditions. Although the importance of joint persistence on the overall rock mass strength has long been identified, the impact of joint persis-tence on rock strength is underrepresented in most current rock mass classification systems (Kim 2002). In addition, joints are often assumed to be fully persistent for stability analysis of tunnels or slopes. This oversimplification may lead to an over-estimation of the number of removable blocks near the excavated face, resulting in excessive expendi-ture on rock support. On the other hand, if the joint persistence characteristics are properly considered, the block size can be accurately estimated, and safe and economic design of the rock structure can be achieved. For an optimum design of underground structures in rock, it is therefore important to estimate the rock mass strength that constitutes the rock mass as close to reality as possible. For this purpose, using statisti-cal methods and numerical methods, such as UDEC (Itasca 2004), we have studied how rock mass strength is affected by various non-persistent joints. Combinations of joint spacing, dip direction, dip an-gle, and length have been statistically generated with the help of orthogonal arrays using the experimental design approach. Statistical analyses and numerical analysis were then performed to elucidate the influ-ence of joint persistence on the overall rock mass strength. 2
JOINT PERSISTENCE AND GSI 2.1
Joint persistence
Fractures usually occupy only a part of the surface extended by the joint plane to a given rock volume. Persistence is the term used to describe the area ex-tent or size of a discontinuity within a plane (Brady & Brown 1992). In jointed rock masses, rock bridges exist due to the non-persistent nature of joints. A rock bridge is defined as a small bridge of
Rock mass strength with non-persistent joints
B.H. Kim, M. Cai & P.K. Kaiser
Geomechanics Research Centre, MIRARCO Inc., Laurentian University, Ontario, Canada
ABSTRACT: Although tensile strength can often be neglected forweak orhighly jointed rockmasses, the strength of rock bridges in moderately fractured and blocky hard rock masses contributes significantly to their self-supporting capacity and should be considered in engineering design. This self-supporting capacity can of-ten be attributed to the distributed load bearing capacity of rock bridges separating joint segments and frac-tures. A rock bridge provides an effective cohesion to the joint and a rock block cannot fall or slide until all the rock bridges fail. In this study, we generate various combinations of geometric conditions of discontinui-ties using UDEC. Non-persistent joints generated in the UDEC models are assigned distributed joint proper-ties randomly. A degradation model is developed in UDEC to simulate the combined failure process of intact rock and joints. Equivalent rock mass strength considering joint persistence is obtained and compared to theo-retical results. The objective is to analyze how the joint persistence will increase the overall rock mass strength in jointed hard rock masses, with the focus on the verification of a previously proposed approach to consider the influence of joint persistence on the rock mass strength by the concept of equivalent block vol-ume in the GSI system.
intact rock separating coplanar or non-coplanar dis-continuities. Persistence estimation can be done by comparing either the sum of the trace length relative to a characteristic length of a colinear scan line or the sum of individual joint surface areas to the sur-face of a coplanar reference area (Dershowitz & Einstein 1988). In practice, since the true joint area is almost impossible to measure accurately, the per-sistence is often estimated based on the trace length. An illustration of persistent and non-persistent joints is presented in Figure 1.
Persistent joints Non-persistent joints
Figure 1. Illustration of joint persistence.
Consensus exists on the fact that rock bridges play an important role in stabilizing a rock mass by their resistance to removable blocks. A rock block cannot fall or slide from an excavation or slope until the appropriate rock bridges have failed. The rock bridge failure involves the failure of the intact rock, which can be an order of magnitude stronger than the rock mass (Kemeny 2005). The importance of rock bridges or non-persistent joints on the stability of rock slopes has been studied among others by Einstein et al. (1983), Nichol et al. (2002), Sjöberg (1996), and the effect of rock bridges on the strength or deformation properties of rock masses has been discussed by Kemeny & Cook (1986), Shen et al. (1995), Kemeny (2003) and many others. If the joints are not persistent, the rock mass strength is higher and the global stability is en-hanced. Diederichs & Kaiser (1999) demonstrated that the capacity of a 1 % rock bridge area equiva-lent to 100 cm
2
per 1 m
2
total joint area in a strong rock (UCS > 200 MPa) is equivalent to the capacity of at least one cablebolt. Consequently, the apparent block volume should be larger for rock masses with non-persistent joints. Cai et al. (2004) proposed that if the joint lengths are only about 20 % of the refer-ence length, the equivalent block size is about five times larger than that with persistent joints. Thus, the presence of discontinuous joints has a significant effect on the properties and behaviour of rock masses and should be carefully considered for the engineering characterization of rock masses. The issue of joint persistency is important but, at the same time, it is one of the most difficult issues to be addressed in rock mechanics. In the present study, we use the distinct element method, in com- bination with the experimental design technique, to systematically investigate the effect of joint persis-tence on the mechanical behaviour of jointed rock masses. 2.2
Block size and GSI value with non-persistent joints
The rock mass classification systems, such as
RMR
(Bieniawski 1973, 1976),
Q
(Barton et al. 1974), and
GSI
(Hoek et al. 1995) systems, are based on accu-mulated engineering experience and are excellent tools to characterize the complicated mechanical properties of in-situ rock masses during the design of rock structures. Since the mechanical and hydrau-lic behaviors of jointed rock masses are dominated by their discontinuities, most rock mass classifica-tion systems focus on determining specific values which represent discontinuity characteristics for de-sign purposes. Because there are so many elements or factors that influence the engineering properties of rock masses, and the inherent variability of the value is very large, it is almost impossible to exactly reflect all factors into a rock mass classification scheme. Consequently, the classification items and scores in most widely used rock mass classification systems are based on the experience and subjectivity of the srcinators. In some cases, certain assump-tions are essential, which may lead to inevitable er-rors in rock mass characterization. For example, the
Q
and
GSI
rock mass classification systems do not consider joint persistence explicitly. Joint persis-tence is only indirectly referred as “block interlock-ing” in the
GSI
system, using descriptive terms. In addition, joints are often assumed to be fully persis-tent for the stability analysis of tunnels or slopes. This oversimplification may lead to overestimation of the size of the joints and hence the numbers of removable blocks near the excavated faces, resulting in excessive expenditure on rock support. Con-versely, if the joint persistence characteristics are properly considered, the block size can be estimated accurately and a safe and economic design of the rock structure can be achieved. The GSI system is widely used in the design of underground structures and rock slopes. Since it can provide a complete set of design parameters, includ-ing the strength of the rock masses, it is ideally suited for design using numerical modeling tools. As an alternative approach to the srcinal system, which is qualitative in nature, Cai et al. (2004) proposed a quantitative approach, using block volume and joint surface condition factor, to utilize the
GSI
system.
Once the block volume (
V
b
) and the joint surface condition factor (
J
c
) are known, the
GSI
value can be determined from the quantified chart or from the following equation (Cai & Kaiser 2006): ,ln0253.0ln0151.01
ln9.0ln79.85.26
bcbc
V J V J GSI
−+++=
(1) where
J
c
is a dimensionless factor and
V
b
is in cm
3
. For jointed rock masses with non-persistent joints, the equivalent block volume is calculated from (Cai et al. 2004):
3321321
321
sinsinsin
p p psssV
b
γ γ γ
=
(2) where
i
s
,
i
γ
and
i
p
are the joint spacing, the angle between joint sets and persistence factor respec-tively. Correlation analysis was performed by the authors of this paper to relate the block size gener-ated from the numerical tools (UDEC and 3DEC) to the ones calculated using Eq. (2), and good corre-spondence between the two has been found (Kim et al. 2006). Using Eq. (2), the effects of joint persistence on the block size (area in 2D and volume in 3D) deline-ated by discontinuous joint sets can be considered quantitatively. In this fashion, the interlocking due to the presence of discontinuous joints is indirectly considered by increased block size in the
GSI
sys-tem. Using this scheme, utilization of the
GSI
sys-tem for the estimation of rock mass strength contain-ing non-persistent joints can be achieved. A few examples are presented in the next section. 3
ROCK MASS STRENGTH WITH NON-PERSISTENT JOINTS 3.1
Statistical analysis using experimental design
Because the inherent variability of the spatial char-acteristics of joints, (e.g., orientation, length, spac-ing, etc.) is significant and the number of relevant parameters in determining a joint network is large, it is impractical to run a parametric study by varying each parameter individually. Therefore, in order to save time but at the same time to honour the statisti-cal distributions of the parameters, it is necessary to develop an experimental design to collect sufficient sample data and develop a statistical analysis method to draw accurate conclusions from the col-lected sample data. The specific questions that the experiment is in-tended to answer must be clearly identified before carrying out the experiment. One must also attempt to identify known or expected sources of variability in the experimental units since one of the main aims of a designed experiment is to reduce the effect of these sources of variability on the result. The experimental design technique is used to as-sist data acquisition, to decrease experiment error, and to provide statistical analysis tools to describe the degree of errors. Two basic principles guide sta-tistical experimental design: replication and ran-domness. Replication means that the same results can be obtained by the same experimental condi-tions. Randomness is required to ensure objectivity by putting the experiment objects into different con-ditions or by randomly arranging their experimental order. Furthermore, it is necessary to have a homo-geneous experimental environment when applying probability theory. The principle of orthogonal array design is to eliminate the biasing effect due to other factors, when the influence of a specific factor is in-vestigated. When there are many parameters to be studied, the main effect of each parameter and some of the reciprocal actions are estimated, while other reciprocal actions are disregarded to reduce the number of tests. The benefit of orthogonal array de-sign is that it calculates the parameter changes from experimental or field-mapping data, facilitates the easy preparation of input data for analysis of vari-ance, and allows the consideration of many parame-ters in experiment or simulation without increasing the test scale (Devore 2000, Mendenhall & Sincich 1995). Let us assume that there is an orthogonal array presented as)3(
49
L
. This means that four factors or parameters could be used in the experiment and there are three stages in which the parameters can change their values. Eighty-one experiments are re-quired to obtain results that are statistically signifi-cant and representative. If the orthogonal array is used, however, only nine experiments are needed. In experiments, a treatment is something that re-searchers administer to experimental units. For ex-ample, in a rock mass with three joint sets each set may have a different orientation. Treatments are administered to experimental units by 'level', where level implies an amount or magnitude. For example, if the experimental units were given as 15
˚
, 45
˚
and 60
˚
dip angle, those amounts would be three levels of the treatment. 'Level' is also used for categorical variables, such as joint set A, B, and C, where the three are different kinds of joint set, not different levels of the same set. For the application at hand, orthogonal arrays are used to define the combination of joint spacing, dip direction, angle, and length of different joint sets that simulate discontinuous blocks with various shapes and sizes.
3.2
Estimation of GSI and rock mass strength with non-persistent joint using statistical simulation
Various block sizes are generated by statistical simulation using the Latin Hypercube method. First of all, the generation of the blocks can be achieved considering continuous and discontinuous joints based on Eq. (2), then different
GSI
values with and without considering the persistence were estimated using Eq. (1). When the block size is calculated, the considera-tion of the joint persistence obviously influences the estimation of the
GSI
value. Fully persistent joints allow the
GSI
value an opportunity to be underesti-mated while the classification is performed. Conse-quently, the
GSI
estimated by discontinuous block volume always presents higher values than the
GSI
by continuous block sizes. Figures 2 and 3 show respectively how the GSI value and the rock mass strength are related to joint persistence. The persistence factors are assumed to be 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 for es-timation of the
GSI
value. The rock mass strength in Figure 4 was calculated using an equation proposed by Hoek et al. (1995).
Figure 2. Ratio of GSI vs. persistence factor. Figure 3. Ratio of rock mass strength vs. persistence factor.
In these figures, the data variation range for the same persistence factor means that different joint spacings are used to calculate block volumes. In other words, upper bound data have wider joint spacings than those in the lower bound data for the same joint persistence factor. According to these figures, a maximum of 30% increase of the
GSI
value and approximately 50% increase of the rock mass strength can be expected if the joint persistences are drastically different. It is obvious that the rock blocks tend to be underesti-mated if joints are assumed to be persistent, as is the case in most designs. This can lead to more remov-able blocks than actually existed in-situ. In addition, a poor understanding of the rock bridge strength may lead to lower rock mass strengths, and conse-quently, to excessive expenditure on rock support. In the next section, numerical simulations using UDEC are performed to further verify the findings. 4
ESTIMATION OF ROCK MASS STRENGTH WITH NON-PERSISTENT JOINT USING NUMERICAL MODELLING 4.1
Introduction
The aim of this investigation is to understand how joint persistence influences the overall rock mass strength. UDEC has been chosen to model triaxial tests of fractured samples with different joint persis-tence. In the UDEC model, plane strain is assumed and the out-of-plane thickness is 1 unit. One has to be aware that there is no direct way to generate very short and isolated joints in UDEC. However, the short joints can be generated as different material properties are assigned in the model. It is possible that short joints can be represented as through-going discontinuities with only a certain section of the dis-continuity allowed to slip. The ends of the disconti-nuity are prevented from slipping by setting the fric-tional resistance to a high value over these regions. This approach has already been verified using some simple examples (Itasca 2004). Heterogeneity is an extremely important factor that influences the rock mass strength. UDEC pro-vides user-defined functions like the FISH and ran-dom number generator which can be used to gener-ate normal and other distributions. Using these functions, specific FISH functions have been devel-oped to generate heterogeneous material and joints of varying lengths. 4.2
Validation of tool for the modeling
First of all, both the mean and the standard deviation values in the random generator were adjusted to as-sign normal distribution material properties random-ly to element zones. Figure 4 presents one example

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