In-situ Rock Spalling Strength near Excavation Boundaries ...

Rock Mech Rock Eng (2014) 47:659–675 DOI 10.1007/s00603-013-0437-0

ORIGINAL PAPER

In-situ Rock Spalling Strength near Excavation Boundaries M. Cai • P. K. Kaiser

Received: 19 April 2013 / Accepted: 15 May 2013 / Published online: 25 May 2013 Ó Springer-Verlag Wien 2013

Abstract It is widely accepted that the in-situ strength of massive rocks is approximately 0.4 ± 0.1 UCS, where UCS is the uniaxial compressive strength obtained from unconfined tests using diamond drilling core samples with a diameter around 50 mm. In addition, it has been suggested that the in-situ rock spalling strength, i.e., the strength of the wall of an excavation when spalling initiates, can be set to the crack initiation stress determined from laboratory tests or field microseismic monitoring. These findings were supported by back-analysis of case histories where failure had been carefully documented, using either Kirsch’s solution (with approximated circular tunnel geometry and hence rmax = 3r1-r3) or simplified numerical stress modeling (with a smooth tunnel wall boundary) to approximate the maximum tangential stress rmax at the excavation boundary. The ratio of rmax/UCS is related to the observed depth of failure and failure initiation occurs when rmax is roughly equal to 0.4 ± 0.1 UCS. In this article, it is suggested that these approaches ignore one of the most important factors, the irregularity of the excavation boundary, when interpreting the in-situ rock strength. It is demonstrated that the ‘‘actual’’ in-situ spalling strength of massive rocks is not equal to 0.4 ± 0.1 UCS, but can be as high as 0.8 ± 0.05 UCS when surface irregularities are considered. It is demonstrated using the Mine-by tunnel notch breakout example that when the realistic ‘‘as-built’’ excavation boundary condition is honored, the ‘‘actual’’ inM. Cai (&) Bharti School of Engineering, Laurentian University, Sudbury, ON, Canada e-mail: [email protected] P. K. Kaiser Centre for Excellence in Mining Innovation, Sudbury, ON, Canada

situ rock strength, given by 0.8 UCS, can be applied to simulate progressive brittle rock failure process satisfactorily. The interpreted, reduced in-situ rock strength of 0.4 ± 0.1 UCS without considering geometry irregularity is therefore only an ‘‘apparent’’ rock strength. Keywords Rock strength Apparent rock strength Actual rock strength Crack initiation Spalling failure Heterogeneity Irregular excavation boundary Brittle rock failure Mine-by tunnel

1 Introduction The determination of the mechanical properties of a massive or jointed rock mass remains one of the most difficult tasks in the field of rock mechanics. Due to practical difficulties, it is impossible to conduct excavation scale tests to determine the rock mass strength in-situ and the in-situ rock mass strength is often estimated using empirical approaches that are based on back-analyses, using case histories where failure was carefully documented. One of the most widely used failure criteria for estimating rock and rock mass strength is the generalized Hoek–Brown criterion in combination with GSI (Geological Strength Index) (Hoek and Brown 1997; Hoek et al. 2002). This empirical approach works well for rock masses with GSI\65 but is not applicable when brittle rock failure dominates and the fundamental assumption of the GSI is no longer valid (Kaiser and Kim 2012), or when both the GSI value and the uniaxial compressive strength of the rock are high (Cai et al. 2004b). For massive rocks, GSI = 100, the in-situ strength of massive rocks should theoretically be the same as that defined by the laboratory strength tested on intact rock samples.

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The intact rock strength is, by the ISRM Suggested Methods, obtained in laboratory using diamond drilling cores with a diameter around 50 mm and a length to diameter ratio of two, and the field or in-situ strength is then obtained by applying empirical relations. Empirical data show that the in-situ strength of massive rocks is in fact lower than the laboratory strength, i.e., the wall strength is less than the uniaxial compressive strength obtained from laboratory tests. One of the widely analyzed rock mechanics experiments is the notch failure observed at the 3.5 m diameter circular Mine-by tunnel at the Underground Research Laboratory (URL) in Canada. The tunnel was excavated in massive granite without joints and an extensive stress measurement program revealed that the maximum, intermediate, and minimum principal stresses at the Mine-by tunnel 420 level are r1 = 60, r2 = 45, and r3 = 11 MPa, respectively (Martin 1997). The tunnel axis is nearly parallel to the intermediate principal stress direction. The maximum principal stress is inclined with an inclination angle of 15° with respect to horizontal. The maximum wall tangential stress, according to the Kirsch’s solution, is rmax = 3r1 r3 = 169 MPa. Because the average uniaxial compressive strength (UCS) of the rock is about 220 MPa ([rmax) and the tunnel was carefully excavated to avoid blasting damage, no failure should be expected at the tunnel site. However, well-developed notches occurred in the roof and floor, which led to the conclusion that the wall rock strength was much lower than 169 MPa (Martin 1997; Read et al. 1998). In fact, Read et al. (1998) argued that the field rock strength at the Mine-by tunnel site was 120 MPa (or about 0.55 UCS). Martin et al. (1999) suggest that for tunnels excavated in massive rocks, the in-situ rock strength can be approximated by 0.41 ± 0.1 UCS; this means that for the Mine-by tunnel, the in-situ strength is about 90 ± 22 MPa. Exadaktylos and Stavropoulou (2008) proposed to estimate the in-situ rock strength for design purpose using 0.3 UCS. If this criterion were applied to the Mine-by tunnel, the in-situ strength would be around 66 MPa. A few questions can be asked at this point. Why is the in-situ rock strength of a massive rock significantly lower than its laboratory strength? Did we interpret the field data correctly? What is the dominant mechanism that dictates in-situ rock strength? If we properly answer these questions, we can gain confidence in determining the in-situ rock strength for the design of rock engineering structures. This paper presents a new perspective on in-situ rock strength of massive rocks. After reviewing some fundamentals of brittle rock failure, including crack initiation and propagation stresses and presenting some previous views or interpretations on in-situ rock strength, the influence of excavation boundary on rock strength is discussed.

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M. Cai, P. K. Kaiser

A new interpretation of the actual in-situ rock strength is presented and this approach is illustrated using the Mine-by tunnel case history.

2 Fundamentals of Brittle Rock Behavior 2.1 Damage Initiation and Propagation in Laboratory Tests A typical stress–strain relations obtained from a uniaxial compression test is presented in Fig. 1, where rcc is the crack closure stress level, rci is the crack initiation stress level, and rcd is the crack damage stress level that is close to the long-term rock strength. These three stress levels represent important stages in the development of the macroscopic damage and failure process of brittle rocks. In the light of microscopic observation in laboratory, we can summarize the brittle failure process of crystalline rocks into five distinct stages (Brace et al. 1966; Bieniawski 1967; Martin 1997; Cai et al. 2004a): Stage I: Closure of microcracks. Stage II: Linear elastic deformation. Stage III: Microcracks start to initiate at a crack initiation stress level of approximately 0.3 to 0.5 UCS; microcracks are tensile in nature, and they propagate in a stable fashion under compression loading; microcracks propagate mainly in the direction parallel to the maximum compressive stress; rock starts to dilate from the beginning of this stage. Stage IV: Onset of crack interaction or unstable crack propagation starts at a crack damage stress level of approximately 0.7 to 0.8 UCS; rock dilation rate increases rapidly. Stage V: Peak strength is reached; macro-crack or shearband formation normally follows after the peak strength is reached, leading to macroscopic failure. The crack initiation and propagation stress levels can be determined using AE monitoring and/or volumetric strain measurements. It is confirmed from long-term response testing that the crack damage stress (rcd) corresponds to the long-term rock strength (Lajtai and Schmidtke 1986). In the static fatigue test, the specimens will not fail under long-term loading if the applied stress is less than the crack propagation stress. When the applied stress is higher than rcd, the specimen will eventually fail if the loading time is sufficiently long. The closer the applied stress is to UCS, the shorter the time to failure will be. 2.2 Damage Initiation Stress In-situ It has been recognized through empirical observation that rock failure in massive to moderately jointed hard rock

In-situ Rock Spalling Strength near Excavation Boundaries

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Fig. 1 Stress-strain diagram of a brittle rock showing the stages of crack development (Martin 1997)

tunnels starts when the tangential stress at the excavation boundary exceeds 0.3 to 0.5 times of the rock’s unconfined uniaxial compressive strength (UCS) (Martin 1997). Using microseismic monitoring data from the Mine-by tunnel and combining with a 3D elastic stress analysis to determine the stress magnitudes ahead of the tunnel face, Martin (1997) found that the in-situ damage (crack) initiation stress can be approximated by a constant deviatoric stress given by r1 – r3 = 70 to 75 MPa, or about 0.33 UCS. This is very similar to the crack initiation stress determined from the laboratory study, and implies that for massive rocks such as the granite at the URL, the field behavior of damage initiation is similar to that of the laboratory behavior. AE monitoring was also conducted at the ZEDEX (Zone of Excavation Disturbance EXperiment) project undertaken ¨ spo¨ Hard Rock Laboratory (HRL). The stress at AE at the A event locations was calculated using Examine3D, and the average in-situ crack (AE) initiation stress was found to be r1 - r3 & 25 MPa or 0.13 UCS for an average uniaxial compressive strength of 195 MPa. This result puzzled the researchers because they had expected a crack initiation stress level comparable to that observed at the Mine-by tunnel at about r1 - r3 & 0.33 UCS or 64 MPa (Emsley et al. 1997). The low crack initiation stress level identified at the ZEDEX project was attributed to the fact that the rock mass contains joints. When the generalized crack initiation concept (Cai et al. 2004a) was used, the crack

initiation stress level was found at r1 - r3 & 0.55 UCSrm, for diorite rock mass with a GSI value of 73 and a rock mass strength of UCSrm = 45.3 MPa. Instead of using the UCS of intact rocks, the generalized crack initiation concept uses the uniaxial compressive strength of the rock mass (UCSrm) to define the crack initiation stress level. These two case histories clearly demonstrate that the insitu crack initiation stress for jointed rock masses can be very low when compared with the uniaxial compressive strength (UCS) of the intact rock. However, the in-situ crack initiation stress for massive rocks is very close to the crack initiation threshold observed in laboratory test. Under similar loading condition, rocks in the field will also go through the complete deformation process from crack initiation to crack propagation and finally reach the peak strength. Consequently, the in-situ strength of massive rocks should not be significantly different from that determined in the laboratory tests. 2.3 Scale Effect When discussing rock and rock mass strengths, the issue of scale effect (or size effect) cannot be ignored. It has been confirmed from laboratory and in-situ tests that there is a scale effect in rock strength (Mogi 1962; Pratt et al. 1972; Hoek and Brown 1980; Goodman 1989; Carter 1992; Lockner 1995). Test results show that if the specimen length to diameter ratio is kept constant, the peak strength

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is reduced for larger specimens. For intact rocks, the scale effect is predominantly related to rock heterogeneity in the form of microcracks and grains of different mineral composition. Unstable crack propagation is affected by stored strain energy (which is a function of specimen size). Hence, the size effect can be caused by the strain energy effect (Einstein et al. 1970). Larger specimens are more likely to contain more flaws that lead to a lower strength. For jointed rock masses, the scale effect is mainly attributed to the fact that larger volumes contain more joints that can form more blocks, which provide greater freedom to develop failure. Hoek and Brown (1980) proposed the following equation to assess the scale effect on intact rock strength, 0:18 50 rcD ¼ rc50 for D 200 mm ð1Þ D where rc50 or UCS is the uniaxial compressive strength of a 50 mm diameter specimen and rcD is the uniaxial compressive strength of a specimen with a diameter of D (mm). Eq. (1) shows that for a maximum specimen diameter of 200 mm, the rock strength rc200 is about 0.78 times of rc50. In addition, test data show that the rock strength ratios of rcD/rc50 are almost constant at around 0.8 for D between 150 and 200 mm. Test data for norite with specimen sizes from 20 to 200 mm (Bieniawski 1972) fit the trend defined by Eq. (1) very well. Eq. (1) should not be used for D [ 200 mm because there are no test data in the dataset used for its development. As far as we know, the largest laboratory test specimen ever tested for the Lac du Bonnet granite has a diameter of 294 mm (Jackson and Lau 1990). Large cylinder specimens of other rock types with diameter up to 900 mm had been tested in laboratory (Singh and Huck 1973). For the Lac du Bonnet granite, although there was variability in the test result, the rock strengths reach an asymptotic level for D greater than 150 mm. Large-scale field tests conducted by Pratt et al. (1972) on quartz diorite, to a maximum specimen length of 2.7 m, showed that the rock strengths reached an asymptotic level for specimen length greater than 300 mm. Bieniawski and Van Herdeen (1975) tested coal cubes with a maximum specimen length of 2.0 m and found that most of the strength decrease with increasing specimen size occurred when the side reached 0.5 m. It should be noted that the boundary conditions of the field tests by Pratt et al. (1972) and Bieniawski and Van Herdeen (1975) are different than those tested in laboratory. Others studied the effect of size and stress gradients on the failure strength around boreholes (Hoek 1965; Ewy and Cook 1990; Carter 1992; Martin et al. 1994). For example, Martin et al. (1994) conducted laboratory compression tests on blocks containing a circular hole. The maximum tangential stress (rhf) at the periphery of the hole when failure

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occurred was assumed equal to the unconfined compressive strength. For small boreholes, the tangential stress at the borehole boundary required to cause breakout is at least twice the uniaxial compressive strength of the material. Although scale-dependent behavior was observed for smaller holes, rhf was almost identical to the uniaxial compressive strength when the hole diameter was greater than 75 mm. This type of model test is limited in sample size (the maximum hole size tested was 103 mm) due to load capacity constraint of the test frame. However, the phenomenon of failure stress at the borehole boundary approaching the uniaxial compressive strength as the borehole size increases indicates that there is a consistency between the laboratory rock strength tested at a smaller scale and the field rock strength at a larger scale for massive rocks. Note that the excavation (hole) boundary in these tests was smooth. These test results demonstrate that the maximum tangential stress (rhf) at which failure occurs at the smooth tunnel boundary should be equal to the in-situ uniaxial compressive strength of the rock. In addition, the in-situ strength of crystalline rocks can be related to laboratory rock strength using an empirical relation shown in Eq. (1).

3 Previous Views on In-situ Rock Strength 3.1 In-situ Rock Strength Defined By Damage Initiation Threshold Some researchers (Stacey 1981; Myrvang 1991; Pelli et al. 1991; Martin 1995) have reported that the field strength of massive rocks is approximately 0.5 UCS. Using data from square tunnels in South Africa, Hoek and Brown (1980) noticed that when the far-field maximum stress magnitude exceeded 0.15 UCS, spalling failure occurred around the excavation. Using data from South African gold mine tunnels, Wiseman (1979) found that for unsupported tunnels, the sidewalls deteriorated rapidly when the sidewall stress concentration factor (rmax/UCS = (3r1 - r3)/UCS) reached a value of about 0.8. Here, r1 and r3 are the in-situ maximum and minimum principal stresses and rmax (=3r1 - r3) is the maximum tangential stress at the circular tunnel boundary. It should be noted that none of the tunnels surveyed by Wiseman (1979) was even approximately circular in cross-section. Martin (1997) and Diederichs (2007) stated that in-situ rock strength in the near field around excavations can be defined by the damage initiation threshold. For most brittle rocks, the damage initiation threshold is between 0.3 to 0.5 UCS (Brace et al. 1966; Kaiser et al. 2000; Bieniawski 1967; Martin 1997; Cai et al. 2004a). Although only shown in a qualitative fashion, the in-situ unconfined rock strength


is depicted in Fig. 2 as about half of the rock strength defined from laboratory test. Many explanations for the low in-situ rock strength can be found in the literature. As one moves from the very specific loading and geometrical conditions of a typical laboratory test to a comparable sized sample adjacent to an excavation, a number of factors may compound to reduce the in-situ upper bound strength from the laboratory yield envelope to the lower bound defined by crack initiation (Martin 1997). These factors include pre-existing damage, the result of enhanced crack propagation influenced by low confinement and surface interaction effects, stress rotations, and loss of effective confinement into the tunnel wall due to progressive slabbing (Diederichs 2007). Diederichs (2007) further argues that in-situ strength reduction (from laboratory UCS) is due to the geometry difference between laboratory tests and in-situ conditions. The cylindrical geometry of the laboratory specimen gives rise to hoop tension (circumferential) which, in turn, induces radial constraint on dilating extension cracks, suppressing unstable propagation. In-situ, and in particular near an excavation wall, cracks are free to propagate. Hence, the in-situ rock strength is lower than UCS. While this is a plausible explanation of the strength difference between in-situ and laboratory, it may not be the main reason why such a difference exists. 3.2 Influence of Loading Path/Stress Rotation Martin (1997) and Read et al. (1998) argued that a reduction of the in-situ rock strength relative to that obtained from laboratory tests could be attributable to the

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complex loading path occurring in the anterior region (i.e., region ahead of the tunnel face). In the laboratory, the strength is obtained via a monotonically increasing loading path where as the in-situ strength is mobilized essentially by unloading the rock through a complex loading path involving stress rotation. After examining 3D stress results from an elastic stress model, they argued that the complex stress path could result in damage around a tunnel and reduced the rock strength near an advancing tunnel face. These researchers stated that depending on the location and severity of the stress changes in the anterior region of a tunnel, the rock will experience damage or ‘‘preconditioning’’, potentially leading to a reduction of cohesive strength. Using the URL Mine-by tunnel data, they showed that for the excavations at the 420 level, failure occurred only in those tunnels where the peak tangential stress exceeded about 120 MPa. Because the in-situ crack initiation stress was defined by the line of r1 - r3 = 70 to 75 MPa (shown in Fig. 3 as a dashed line indicated by rci), any stress path leading to a stress level higher than the crack initiation threshold would result in rock damage and related cohesion loss. They suggested that the rock mass strength at the point of peak tangential stress concentration on the tunnel periphery had been reduced from the longterm strength of 150 MPa (about 0.7 UCS with UCS = 220 MPa) to a lower value due to accumulated damage. For the Mine-by test tunnel, they concluded that the preconditioning accounts for the local reduction of rock strength to 120 MPa (about 0.55 UCS). It should be noted that the peak tangential stress was calculated from 3D linear elastic numerical analysis with perfectly smooth tunnel boundary.

Fig. 2 Schematic composite strength envelope illustrated in principal stress space which highlights low in-situ rock strength as compared to the uniaxial compressive strength obtained from lab specimens (Kaiser et al. 2000)

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Fig. 3 Comparison of critical stress paths in the roof and floor of the Mine-by test tunnel. The deviatoric stresses increases above r1 - r3 = 70 MPa (the crack initiation threshold). The rock mass ahead of the face in the roof region experiences tensile loading. These conditions combine to locally weaken, or ‘‘precondition’’, the rock mass near the tunnel periphery (Read et al. 1998)

3.3 Influence of Loading Rate Strain rates reported to be of relevance in rock mechanics range from 10-14 to 108 1/s (Olsson 1991). Rock strength is loading rate dependent (Lankford 1981; Lajtai et al. 1991; Cai et al. 2007b). Test results show that the crack propagation stress corresponds to the long-term rock strength (Lajtai and Schmidtke 1986). Laigle (2006) provided an explanation for the reduced field rock strength at the Mine-by tunnel site. The researcher stated that because the in-situ loading rate due to tunnel excavation (about 10-9 to 10-10/s) was much lower than the loading rate (about 10-5/s) used in laboratory test, hence, the in-situ strength should be about 50 % of the laboratory strength. Based on the loading rate influence, Laigle (2006) concluded the field strength at the Mine-by tunnel was about 100 MPa (for a laboratory UCS of 200 MPa). While the loading rate does influence the rock strength, it is unlikely that it is the main reason that leads to a low in-situ rock strength. 3.4 Depth of Failure—Linking Stress to Rock Strength Predicting the demand on rock support is challenging due to the need to integrate stress, geometry, geology, seismicity, and variability in geotechnical and seismological models. For this purpose, knowledge of the depth of failure around excavations is needed. One approach to estimate the depth of failure is based on empirical equations that relate depth of failure to the virgin field stress via the concept of rmax introduced by Wiseman (1979). rmax is an indicator of maximum stress level at a given tunnel location considering the far-field and mining-induced stress magnitudes, orientations and, most importantly, the stress ratios.

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Kaiser et al. (1996), based on field observation of brittle rock failure, proposed to estimate the depth of brittle failure (df, see Fig. 4) in massive rocks through the following empirical equation, df rmax ¼ 1:34 0:57ð0:05Þ a UCS

ð2Þ

where rmax = 3r1 - r3, a—radius of the opening, UCS— uniaxial compressive strength of rocks. r1 and r3 are the in-situ or mining-induced maximum and minimum principal at the location where the depth of failure is to be determined. Martin et al. (1999) added additional data to the database and proposed a similar equation to estimate the depth of brittle failure and the data are presented in Fig. 4. The tunnels surveyed have either a circular crosssection or a D-shaped section. Where the tunnels are D-shaped, an effective tunnel radius is used. It is seen from Fig. 4 that failure initiates when rmax/UCS & 0.41 ± 0.1. In other words, this figure suggests that the in-situ wall rock strength is 0.41 ± 0.1 UCS. In the following, it will be shown that the implied field rock strength of 0.3 to 0.5 UCS is mainly a result of oversimplified tunnel wall geometry.

4 Apparent and Actual In-situ Rock Strengths near Excavation Boundaries 4.1 Hypothesis It becomes evident from the above discussion that the insitu crack initiation stress for massive rocks is very close to the crack initiation threshold observed in laboratory test, and when loading continues, rocks will go through the


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Fig. 4 Depth of failure plotted as a function of rmax/UCS (modified from Martin et al. (1999)

complete deformation process from crack initiation to crack propagation and finally reach the peak strength, whether in the laboratory or in-situ. Therefore, the in-situ rock strength of massive rocks should not be significantly different from that determined in the laboratory. However, interpretations described above suggest that the in-situ rock strength is 0.3 to 0.5 times of the uniaxial compressive strength obtained from laboratory test. Based on the following hypothesis, it will be demonstrated that the ‘‘actual’’ rock strength for massive rocks in the field may be much higher. The hypothesis proposed here is that the actual in-situ rock strength of massive rocks is close to the scaled peak strength determined from laboratory UCS test. For crystalline rocks, scale effect and the influence of timedependent (or loading rate dependent) processes may reduce the long-term strength of larger volumes of intact rock to between 0.7 and 0.8 UCS or even slightly lower depending on the mineral composition of the rock, but certainly not as low as 0.3 to 0.5 UCS. It is suggested here that the previous conclusion on in-situ rock strength near the crack initiation stress was reached based on the interpretation of field data without honoring the true excavation boundary condition of the excavation. In the following, the strength back-calculated in the manner described above, with simplified excavation shapes and an elastic strength index (rmax/UCS), is called the ‘‘apparent’’ rock strength, and the rock strength interpreted by honoring the true geometry of the excavation, and therefore the actual stresses, is called ‘‘actual’’ rock strength. 4.2 Excavation Boundary Profiles Over the last two decades, the importance of rock material heterogeneity on rock strength has been recognized and widely investigated. By considering microscopic strength and deformation variability, it is possible to simulate the

rock failure process realistically (Cundall 1994; Cai 2011; Tang 1997; Potyondy and Cundall 2004; Cai et al. 2007a). Because of the material heterogeneity (stress raisers), local tensile stress concentrations are generated from which failure propagates by extensional straining. Today, progressive rock failure processes can be simulated realistically by various means, such as BEM (Napier et al. 1997; Shen et al. 2002), FEM (Tang 1997; Ma et al. 2012), DEM (Potyondy and Cundall 1999, 2004; Cundall et al. 2008), FEM/DEM combined method (Roberts et al. 1999; Cai and Kaiser 2004; Cai 2008; Elmo et al. 2011), and MNN-DDA method (Shi 1991; Ma et al. 2009). Figure 5a presents the design and the actual ‘‘as-built’’ drift (horizontal tunnel in a mine) profiles at a D-shaped drift location in a Canadian mine. The drift was excavated using the drill and blast technique. Despite the intention to build a smooth walled drift, the actual drift boundary profile is highly irregular. When this drift is highly stressed, failure will initiate from locations with localized stress raisers causing stress concentrations that exceed those calculated by the stress index rmax/UCS. The Mine-by tunnel was excavated using perimeter line drilling and a mechanical breaking excavation method (Martin 1997) with the intent of avoiding blasting damage to the excavation walls. Unlike a tunnel excavated by TBM, the line drilling holes created a very rough tunnel boundary as shown in Fig. 5b. This tunnel profile causes localized stress concentration that differ significantly from the simplified stress condition calculated for a smooth walled tunnel. As is illustrated below, these localized stress concentrations act as initiators for progressive rock failure processes as the tunnel face advances. By modeling a smooth excavation boundary as an approximation of reality, the actual stresses near the opening are underestimated; for example, by modeling the Mineby tunnel with a smooth, circular tunnel boundary, the maximum tangential wall stress is 169 MPa, significantly

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666 Fig. 5 a Designed and ‘‘asbuilt’’ drift profiles in a deep mine in Canada; b tunnel excavated by perimeter line drilling and mechanical breaking at Mine-by tunnel (Martin 1997)


“As-built” profile

Design Drift Profile Design profile

Blast Holes

σ1=60MPa σ3=11MPa

4.02 m

(a) less than that calculated with the actual tunnel shape (Fig. 7). Cai et al. (2004a) noticed that the irregularity of the surface of the Mine-by tunnel contributed to the failure process of notch formation, and their preliminary elastic stress analysis showed that the boreholes used for breaking out the tunnel profile locally increase the maximum tangential wall stress to 195 MPa. This suggests that the in-situ rock strength could be higher than 169 MPa. Therefore, when assessing in-situ rock strength, material heterogeneity, geometric variations in the form of irregularities causing stress raisers must be considered. This is particularly important when modeling brittle rock failure near excavation boundary where stress raisers have a major influence on the fracture propagation process. As the tunnel face advances, the maximum tangential stress at the wall increases gradually (in TBM tunnels) or cyclically in drill and blast tunnels. In both cases, though, surface geometric variability increase the local stress, which in turn affects extensional or spalling failure processes. In the following discussion, we study rock failure processes near irregular tunnel excavations. The objective is to seek the actual in-situ rock strength, distinct from the apparent rock strength. 4.3 Apparent and Actual In-situ Rock Strengths at the Mine-by Tunnel As mentioned above, the notch failure observed at the Mine-by tunnel has been extensively studied and contributed to the development of ‘‘brittle rock mechanics,’’ leading to a better understanding of spalling failure. As shown in Fig. 5b, the wall boundary of the Mine-by tunnel is not smooth, i.e., it is not a perfect circular tunnel but rather a circular tunnel with highly irregular wall

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(b) geometry and a realistic simulation of the rock failure process should consider this wall geometry in detail. Hence, an ‘‘as-built’’ model with half-barrels of the perimeter boreholes was constructed (Fig. 6). A very fine Phase2 FEM mesh with eight node elements was used to increase the stress analysis accuracy. The far-field in-situ stresses are: r1 = 60 MPa, r3 = 11 MPa, r2 = 45 MPa (out of plane stress). r1 is inclined with an inclination angle of 15° with respect to horizontal. The tunnel excavation in the modeling is completed in 16 stages using the load split technique in Phase2. The load split option allows users to split the field stress-induced load between any stages of the model, rather than applying the entire load in the first stage. This allows us to apply the field stress load gradually as excavation progresses. The external boundary was five tunnel diameters away from the tunnel and fixed; however, in Fig. 6a only a small part of the model around the tunnel is shown in order to present the detailed meshes of the model. For comparison, the maximum principal stress distributions from an elastic stress analysis for the simplified (virtual) cases and ‘‘as-built’’ (actual) tunnel wall boundary are presented in Fig. 7. Although limited to a small area near the tunnel boundary, it is evident that the maximum principal stress in the ‘‘as-built’’ model can be as high as 240 MPa, i.e., higher than the average rock strength of UCS = 220 MPa obtained from the laboratory test. On the other hand, the maximum principal stress in the simplified or ‘‘virtual’’ smooth walled model is around 169 MPa. A slight stress calculation error is seen in Fig. 7c because the smooth tunnel wall is not perfectly smooth due to segmented tunnel boundary approximation in the FEM model. Next, the Mohr–Coulomb model with brittle material parameters is used to simulate the progressive rock failure. The elastic modulus is 60 GPa, the tensile strength of the


2

(a)

-2

-1

0

1

Fig. 6 a FEM mesh for the Mine-by tunnel simulation with ‘‘as-built’’ wall geometry near the tunnel boundary; b zoom-in plot showing detailed meshing on the floor. Higher order 8 node quadrilateral elements are used in the stress analysis

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-2

-1

0

1

2

(b)

rock is set to 30 MPa, and the Poisson’s ratio is 0.3. The cohesions and frictional angles are 59 MPa and 22° for the peak and 0.1 MPa and 45° for the residual strength, respectively. The corresponding peak and residual uniaxial compressive strengths are 175 MPa and 0.5 MPa, respectively (Fig. 8). The reason for choosing a low residual cohesion and high residual friction angle is that when a brittle rock fractures, its cohesive strength is lost almost completely and the remaining strength component is the frictional strength. For the peak strength, the dominant strength component is the cohesive strength (Martin 1997).

Based on laboratory test results on rocks, one can plot the peak and residual strength envelopes and determine the strength parameters. A common feature of the residual strength envelopes for brittle rocks is that the residual cohesion is very small and the residual friction angle is higher than the peak friction angle. For example, the peak and residual friction angles for the Tailuko marble are 26° and 41°, respectively (Hsiao et al. 2011). Other test data analyzed by Golchinfar and Cai (2012) show that the residual friction angles of hard rocks vary primarily between 40° and 55°.

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(a)

(b)

Sigma 1 (MPa) 240.0 220.0 200.0 180.0 160.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 -20.0

Sigma 1 (MPa) 240.0 220.0 200.0 180.0 160.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 -20.0

1m

(c)

Sigma 1 (MPa) 240.0 220.0 200.0 180.0 160.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 -20.0

0.2 m

1m

(d)

Sigma 1 (MPa) 240.0 220.0 200.0 180.0 160.0 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 -20.0

0.2 m

Fig. 7 Distributions of the maximum principal stress from elastic stress analysis: a smooth, virtual geometry; b actual geometry; c and d are zoom-in figures showing enlarged areas in roof a and b, respectively

The final notch breakout shape, as represented by the yielded elements, is presented in Fig. 9. Both the notch extent and depth on the roof are very close to that recorded from the field observation shown in Fig. 11b (Martin 1997). The degree of failure on the floor in the field is less than what is simulated in the model because of the influence of excavation debris and gravity on progressive rock failure. Because of using load splitting to resemble the effect of tunnel face advance, the failure process is gradual as shown in Fig. 10 (and Fig. 9 for the final stage). The first element yield occurs at Stage 10 near the excavation borehole boundary at a location where the highest tangential wall stress is expected (Fig. 10a). This stage corresponds to

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70 % of stress loading that accounts the effect of tunnel face advance. Failure continues to appear on the excavation borehole boundary at other locations in subsequent stages (as the tunnel face is advanced) and tensile failure appears on the sidewall at Stage12. Subsequent loading (or tunnel face advance) leads to a progressive deepening of the notch as can be seen in Figs. 9 and 10c–f. The progressive development of the breakout in the roof, traced from the failure maps in Figs. 9 and 10, is presented in Fig. 11a and compared with the recorded notch development from field observation in Fig. 11b. It can be seen that the simulated progressive notches development in the roof resemble the actual, observed shapes well. Profiles 2–5


2

Fig. 8 Peak and residual strength envelopes of the rock

0

1

0.523

-2

-1

stage number: 16

-2

-1

0

1

2

Fig. 9 Final simulated notch shape at the end of excavation (Stage 16), comparable to the observed shape presented in Fig. 11b

in Fig. 11b was recorded based on direct monitoring over a 5-month period. It was observed that in the field that notch breakout initiated at the location about 0.5 to 1 m from the tunnel face, immediately after each excavation round was taken (Martin 1997). The failure progressed to deeper ground as the tunnel face was advanced. The progressive development of the notch in the floor could not be observed because the floor always contained

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tunnel muck until the whole tunnel advance was completed. The modeled lateral extent fits well but the depth of notch is overestimated. The good correspondence between the model simulation and field observation of the notch shape and its progressive development in the roof demonstrates that the model geometry and material properties used to reproduce the observed breakout are representative for the massive rock mass at the Mine-by tunnel. This example also demonstrates the need for an accurate representation of the actual, ‘‘as-built’’ tunnel wall geometry. Figure 12 for Stages 11 and 16 presents the distributions of yielded elements around the notch in the roof, along with the maximum principal stress (r1). At Stage11 (before the tunnel face is further advanced), the maximum principal stresses near the excavation boreholes are higher than 169 MPa. For an element located at the tunnel boundary where r3 is zero, when r1 is greater than the uniaxial compressive wall strength of 175 MPa, localized yielding occurs. As the tunnel further advances and additional load is applied, spalling failure gradually propagates and eventually forms the notch. The maximum principal stresses near the tunnel boundary of the notch zone, indicated by yielded elements, are very low (between 0 and 25 MPa). This is so since a brittle failure model is used and the residual strength is primarily governed by the frictional strength (Fig. 8). In the failure zone where the confinement is small, the corresponding residual rock strength is also small. The rocks sustain relatively high residual stresses only in the notch tip area where the confinement is high. It is seen from this simulation that the peak rock strength assigned to the elements is in the order of 175 MPa, or about 0.8 UCS, for UCS = 220 MPa. When Martin (1997); Read et al. (1998); Diederichs (2007) and others interpreted the in-situ rock strength at the Mine-by tunnel, the actual, ‘‘as-built’’ tunnel wall geometry was simplified as a circular tunnel with smooth wall boundary. When such a tunnel is subjected to the in-situ stress field, the maximum tangential wall elastic stress after the tunnel excavation is 169 MPa (Fig. 7a, c). When a peak wall strength of 175 MPa is used, no failure will occur. Thus, peak strength values significantly lower than 169 MPa had to be used to simulate rock failure and the notch breakout in previous studies. For example, the peak uniaxial compressive strengths used by Hajiabdolmajid et al. (2000) and Diederichs (2007) were 100 MPa and 103.8 MPa, respectively; the peak in-situ rock strength interpreted by Martin (1997) and Read et al. (1998) was between 100 and 120 MPa. For comparison, the Mohr–Coulomb model with brittle material parameters is used to simulate the breakout with smooth wall boundary, and the final notch breakout is presented in Fig. 13. Compared with the model parameters

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(a)

(b)

1m

(c)

1m

(d)

1m

(e)

1m

(f)

1m

1m

Fig. 10 Progressive development of the notch geometry in the roof and floor of the Mine-by tunnel (Stages 10 to 15)

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(a)

5

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(b) 4

3 2

1

1 2 3 4 5

Stage-10 Stage-12 Stage-13 Stage-14 Stage-16

Fig. 11 a Simulated progressive development of the notch geometry in the roof and floor; b observation of progressive development of the notch geometry in the roof and floor of the Mine-by tunnel (Martin 1997)

(a)

(b)

Sigma 1 (MPa) 205.0

Sigma 1 (MPa) 205.0

185.0

185.0

165.0

165.0

145.0

145.0

125.0

125.0

105.0

105.0

85.0

85.0

65.0

65.0

45.0

45.0

25.0

25.0

5.0

5.0

-15.0 Shear Tension

-15.0 Shear Tension

0.2 m

0.2 m

Fig. 12 Distribution of yielded elements and the maximum principal stress in the zoom-in notch area in the roof: a Stage11; b Stage16

used above with irregular wall boundary, only the peak cohesion is altered to try to match the simulated notch shape to the observed one; the residual strength parameters (residual cohesion = 0.1 MPa, residual frictional angle = 45°) are the same as those used in the simulation with irregular wall boundary. To simulate the notch shape well, a peak cohesion of 48 MPa instead of 59 MPa used previously has to be used (with the peak frictional angle being kept as 22°). This corresponds to a peak uniaxial compressive strength of 142 MPa, or 0.65 UCS (for UCS = 220 MPa).

Obviously, such an interpreted in-situ rock strength, either from previous studies by others or from this study, is not the actual rock strength. We call it ‘‘apparent’’ rock strength because an approximation of the actual, ‘‘as-built’’ tunnel wall boundary is made in the numerical stress analysis. In other words, an omission was introduced in the interpretation models by ignoring the boundary irregularity. When the model geometry does not honor the reality truthfully, a set of adjusted material parameters had to be used to ensure that the simulation results resemble the field observation.

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There is nothing wrong with such an approach of using simplified excavation geometry, as long as the interpreted rock strength and the numerical model are used consistently, i.e., similar simplified models are used in design. However, when we are talking about the actual in-situ rock strength, the approach that employs a simplified excavation geometry could lead to significantly underestimated field rock strength. For the Mine-by tunnel case, instead of an in-situ strength of about 175 MPa, an underestimated field strength in the order of 100 to 120 MPa (previous studies) and 140 to 145 MPa (this article) would result. Note that a 100 MPa field strength is close to 0.45 UCS (for UCS = 220 MPa), a stress level that corresponds to the crack initiation stress level in the laboratory test results. This observation had led to the recommendation of using the crack initiation stress in laboratory specimens and insitu to approximate the field rock mass spalling strength (Diederichs 2007). It is evident from this study that such a spalling strength is a ‘‘apparent’’ rock strength and not the ‘‘actual’’ rock strength. A disadvantage of using the apparent rock strength for design is that this strength depends on the excavation profile, or on how the results are interpreted. An excavation with a smooth wall will have a higher apparent rock strength than an excavation with a

Fig. 13 Simulated notch breakouts with smooth wall boundary (unit: m)

Fig. 14 Apparent and actual insitu rock strength envelopes of brittle rocks (color figure online)

Realistic “as-built” tunnel boundary profile

σ 1/UCS

“Actual” in-situ rock strength

Shear failure

Spalling limits Simplified smooth tunnel boundary profile

Laboratory rock strength

“Apparent” in-situ rock strength Spalling limits Damage initiation threshold

1.0 0.8 -

Spalling failure

0.6 No damage

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σ 3/UCS Tensile failure

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0.1

0.2

0.3

0.4

0.5


very rough wall. For example, the spalling strength interpreted by Stacey and Yathavan (2003), for some mine drifts in South African, was 0.2 UCS, and in one case was as low as 0.04 UCS. Spalling failure is a phenomenon of brittle rock failure near the excavation boundary. Visible extension fractures are developed due to high stress loading. It is well known that material heterogeneity is one of the main factors that cause local tensile stress that initiate and propagate cracks during extensional straining. The composite strength envelope illustrated in Fig. 2 was obtained based on interpretation of observed rock failure using simplified stress analysis (e.g., Kirsch’s solution or numerical simulation with smooth wall boundary), leading to ‘‘apparent’’ strength parameters and apparent in-situ rock strength envelope as shown in Fig. 14 (red lines). As discussed above, the interpreted ‘‘apparent’’ rock strength varies significantly depending on the degree of simplification of the tunnel wall geometry. Hence, two envelopes, with interceptions with the normalized major principal stress axis at 0.3 and 0.6, are shown in Fig. 14. These two envelopes define the ‘‘lower’’ and ‘‘upper’’ bounds of the apparent in-situ rock strength. Most data support a data range of 0.4 ± 0.1 and the apparent strength envelopes can be used in design when real excavation boundaries are approximated by smooth walls in the model. On the other hand, the ‘‘actual’’ in-situ rock strength of massive rocks is much higher, and a representative strength envelope (thick black line) is shown in Fig. 14. It has an interception with the normalized major principal stress axis at about 0.8 or possibly lower in the long-term. The actual strength envelopes can be used in design when real ‘‘as-built’’ excavation boundaries are considered in the model. The spalling limits (r1/r3) are also illustrated in Fig. 14 for both the actual and apparent strength envelopes. Because spalling occurs near the excavation boundary, the confinement that defines the spalling limit is small, in the order of less than UCS/10 (Kaiser et al. 2010). For a massive rock, its strength envelope is not drastically different from that of the intact rock defined from laboratory test (also shown in Fig. 14 as blue dashed line).

5 Conclusions Field rock strength is not measured, but rather than interpreted. This paper demonstrates that rock strength obtained from back-analyses of failure in brittle massive rock masses may lead to an underestimation of the actual rock mass strength when simplified boundary geometries are used in the back-analysis. When a smooth excavation boundary is used in the numerical model of a back-analysis, a reduced strength value has to be introduced to match the observed depths of failure. Thus, the strength value obtained in this fashion, called

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‘‘apparent’’ rock strength, underestimates the ‘‘actual’’ rock strength. Therefore, existing approximations of in-situ strength of massive rocks as 0.3 to 0.5 UCS can only be used to approximate field rock strength when simplified model geometries are used. The interpreted correspondence of the crack initiation stress with the (apparent) in-situ rock mass strength is therefore accidental and not generally applicable. The investigation presented in this paper shows that the actual in-situ rock strength is much higher, which means that using the crack initiation threshold can significantly underestimate the in-situ rock mass strength. The main reason for this underestimation is found to result from the neglect of excavation boundary irregularity. Brittle rock failure is a progressive process; it starts from crack initiation and propagation to coalescence of the fractures and eventually reaches the peak strength. Furthermore, for excavation problems, the tangential stress near the excavation boundary is the maximum principal stress. This is the driving force to cause crack initiation, propagation, and coalescence. Any irregularity near the excavation boundary will act as a stress raiser, so that the rocks can actually fail at a lower far-field stress level, which means that the actual rock strength should be higher than what is inferred from stress analysis without considering the effect of stress raisers (boundary irregularities). Numerical simulation using the Mine-by tunnel shows that when the peak in-situ strength defined by 0.8 UCS is used, the observed notch in the roof can be captured when the actual tunnel boundary created by perimeter excavation boreholes is accurately reflected in the model. Although the ‘‘apparent’’ rock strength significantly underestimates the field rock strength, it is still useful in design, but only if equivalent simplified excavation boundaries are used. However, these strengths are not actual rock mass strength and thus should not be used to infer rock properties in other geometric conditions or when designing pillars. The ‘‘actual’’ rock strength is interpreted using the true excavation boundary geometry and excavation process. It is therefore important to simulate the progressive rock failure realistically and to consider excavation shape in back-analyses for strength determination. When such an approach is taken, actual rock strength parameters are obtained from back-analyses and these parameters can then be used in design with anticipated excavation geometries. Acknowledgments Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada is greatly appreciated.

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