Quantifying the error associated with the use of triaxial rock strength criteria in rock stability assessment around underground openings

Transcription

1 Quantifying the error associated with the use of triaxial rock strength criteria in rock stability assessment around underground openings by Roozbeh Roostaei A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Civil Engineering University of Toronto c Copyright 2014 by Roozbeh Roostaei

2 Abstract Quantifying the error associated with the use of triaxial rock strength criteria in rock stability assessment around underground openings Roozbeh Roostaei Master of Applied Science Graduate Department of Civil Engineering University of Toronto 2014 In this research, the importance of using polyaxial rock strength criteria rather than triaxial criteria is investigated when predicting stability of an underground opening. A 3D boundary element method program (Examine3D) is employed to compute the induced stress state around the planar end of an opening, and then the analysis is extended using MATLAB to determine the error associated with the use of triaxial criteria. A bivariate colour scheme is used to effectively visualize two variables on one plot, which is found to be helpful when assessing one variable is not conclusive and the reader needs to go back and forth on two plots. The effects of in-situ stress state and tunnel geometry in stability assessment and the associated error are discussed. ii

3 Acknowledgements First and foremost, I would like to express my gratitude to my advisor Professor John Harrison for his motivation, thoughtful guidance, critical comments, and immense knowledge. His guidance helped me in all the time of research. I could not have imagined having a better advisor for my graduate studies. He is certainly more than a supervisor to all his students. Of his unique advice, I will never forget the principle of least surprise and the story of the woolly pom-pom. I am also grateful to my colleagues Nezam Bozorgzadeh, Ke Gao and Greg Gambino. It would not be an enjoyable twenty-month of research without having you around. I extend my utmost appreciation for my friends. Above all, Negar for her patience and support, Amin, Mohammad and Patrick for their encouragements and Atena for her honest friendship. I would also like to thank my cousins Pouya and Pedram here in Canada. Without them, leaving home would be difficult. Most importantly, none of this would have been possible without the love and patience of my family. There are no proper words to convey my heart-felt gratitude for my mother. She has been a constant source of love, concern, support and strength all these years. I would also like to express my appreciation to my supportive and encouraging sister, Romina. The last but not the least, I extend my thanks to my cousins Mehrdad and Saeid for their support from the other side of the world. This dissertation is dedicated to the memory of my father. iii

4 Contents 1 Introduction 1 2 Peak strength criteria Triaxial strength criteria Effect of intermediate principal stress Polyaxial strength criteria Ottosen peak strength criterion Summary Stability analysis methodology Degree of polyaxiality Strength factor Estimation of strength factor using triaxial rock strength criteria Estimation of strength factor using polyaxial rock strength criteria Error in strength factor when using of triaxial criteria Summary Effective use of colour for visualization Typology of colour schemes Bivariate colour scheme for rock stability analysis Summary Numerical analysis of stress state around an advancing tunnel face Tunnel geometry and boundary element mesh used for 3D numerical analysis 35 iv

5 5.1.1 Effect of initial stress field on stability analysis Effect of geometry on stability and the error in prediction: elliptical tunnels Conclusions Conclusions Recommendations Bibliography 50 Appendix: MATLAB script 54 v

6 Chapter 1 Introduction Deep in the earth when a tunnel advances, the initial field stress is disturbed. Changing the magnitude and the orientation of stress components may cause a failure if exceeds the ultimate peak strength of the rock. Assessment of stability around an underground excavation, known as a common practice in engineering design, usually involves strength factor which determines the degree of overstress in the rock (Corkum, 1997). While it has been demonstrated that the strength of the rock is a function of three principal stresses, the stability around an excavation is conventionally assessed in terms of major and minor principal stresses, even in 3-dimensional modern software. Neglecting the influence of σ 2 on rock strength and assessing the state of stress in two dimensions is a substantial shortcoming of computer programs. A great number of polyaxial criteria has been proposed to overcome the deficiency of neglecting σ 2. The application of such criteria in 3-dimensions enables to estimate the strength factor accurately. Regarding the error associated with the 2-dimensional approach of numerical modelling software in assessment of strength factor, it is not known whether an over- or underestimation of actual state of stability is occurred. This study aims to investigate the stability around an opening, and discuss the consequences of an inappropriate approach in assessment of stability. 1

7 Chapter 2 Peak strength criteria Studies on rock mass behaviour begins with determination of the material properties, of which peak strength or ultimate strength is one of the most immediate ones to assess, and determines under which stress conditions rock fails to bear more and a failure occurs. A great number of criteria has been proposed to describe rock behaviour under different stress conditions, and predict rock peak strength. This chapter sets the target of reviewing triaxial failure criteria commonly being used, i.e. Mohr-Coulomb and Hoek-Brown, and discuss how they fail to predict accurately in all stress conditions, and thus has led rock engineers to propose various polyaxial criteria. At the end of this chapter, the advantages and disadvantages of using different types of failure criteria in rock strength prediction will be discussed. 2.1 Triaxial strength criteria In 1980, Hoek and Brown introduced an empirical failure criterion in terms of major and minor principal stresses, which has been widely accepted in rock strength prediction since then. Looking at the data from laboratory triaxial test results on isotropic rocks, it is evident that by increasing confining pressure, peak strength of the rock increases with a nonlinear parabolic trend (Eberhardt, 2012). In contrast to the famous linear 2

8 Chapter 2. Peak strength criteria 3 failure criterion, i.e. Mohr-Coulomb, the Hoek-Brown succeeded to predict nonlinear increasing effect of confining pressure in rock strength (Figure 2.1) (Labuz and Zang, 2012). Shear stress, τ n (MPa) Hoek-Brown (nonlinear) Mohr-Coulomb (linear) Normal stress, σ n (MPa) Figure 2.1: Comparison of Hoek-Brown and Mohr-Coulomb failure criteria plotted in σ n τ n space against triaxial test data for intact rock (from Eberhardt, 2012). Hoek and Brown (1980a) proceeded through trial and error to derive an equation which is in a good agreement with triaxial test data. They also aimed to derive a criterion with a mathematically simple equation, as well as possibility of extending to deal with anisotropic rocks (Hoek, 1983). The two latest goals were met by providing dimensionless parameters, which could be determined by empirical methods. The original form of Hoek-Brown failure criterion was introduced as: where: σ 1 = σ 3 + (m.σ c.σ 3 + s.σ 2 c ) (2.1) σ 1 and σ 3 are major and minor principal effective stresses, respectively, σ c is the uniaxial compressive strength, m and s are dimensionless constants empirically determined.

9 Chapter 2. Peak strength criteria 4 In terms of material properties, m is correspondent to frictional strength of the rock (Eberhardt, 2012) and always has a positive value ranges between corresponding to highly disturbed rock masses to hard intact rocks (Hoek, 1983). Figure 2.2 shows that failure envelope is inclined more steeply with larger values of m. Shear stress, τ n (MPa) m = 30 m = 15 m = Normal stress, σ n (MPa) Figure 2.2: Inclination of Hoek-Brown failure envelope as a function of m value plotted in σ n τ n space (from Eberhardt, 2012). The other constant s, analogous to the rock mass cohesion, is a measure of how fractured the rock is (Eberhardt, 2012), and varies from 0, when the tensile strength is almost zero for highly jointed rock mass, to 1 for intact rock material (Hoek, 1983). In this study, as well as many others, it is assumed that the rock is intact, so s = 1. Despite all advantages of using nonlinear Hoek-Brown failure criterion rather than linear Mohr-Coulomb, there are some shortcomings and limitations in practice, which will be discussed in following section Effect of intermediate principal stress As discussed in section 2.1, Hoek-Brown failure criterion was introduced in terms of σ 1 and σ 3. In other words, it is assumed that intermediate principal stress, i.e. σ 2, has no effect on rock strength, or is equal to σ 3, and that is why this criterion fits on triaxial compression test data reasonably perfect. However, many experiments have shown that

10 Chapter 2. Peak strength criteria 5 when σ 2 increases to a larger value than σ 3, i.e. polyaxial stress states σ 1 > σ 2 > σ 3, strength of rock changes. Studies on effect of intermediate principal stress were started by conducting triaxial tests in compression (σ 2 = σ 3 ) and tensile (σ 2 = σ 1 ) stress state by Kärman (1911) and Böker (1915) on Carrara marble (Figure 2.3), and followed by Murrell (1963), Handin et al. (1967), and Mogi (1967) with different rock types. As shown if Figure 2.3, peak strength of the rock exposes a greater value in tensile tests, which suggests that, σ 2 has an increasing effect on rock peak strength when it increases from σ 3 to σ σ 2 =σ 1 Böker (1915) 500 σ 1 (MPa) σ 2 =σ 3 von Kármán(1911) Carrara marble σ 3 (MPa) Figure 2.3: Triaxial tests conducted by (Kärman, 1911) and (Böker, 1915) in compression (filled circles) and tensile (open circles) (summarized by Murrell, 1963, digitized by Jimenez and Ma, 2013) Later on, true-triaxial or the so-called polyaxial tests were conducted by Mogi (1971). Results from polyaxial tests suggest that, while stress state at a point changes from triaxial compression (σ 1 > σ 2 = σ 3 ) to triaxial extension (σ 1 = σ 2 > σ 3 ), the peak strength of material increases to a maximum value, before it decreases to a value higher than that

11 Chapter 2. Peak strength criteria σ 2 =σ σ 1 (MPa) σ 3 = 0 MPa Dunham dolomite σ 2 (MPa) Figure 2.4: Polyaxial tests conducted by Mogi (1971) in seven different groups with σ 3 varying in the range of 0 σ MPa (digitized by Haimson, 2006) of in triaxial compression condition (Figure 2.4). It is worth noting that, evaluation of an empirical criterion accuracy, in terms of taking σ 2 effect into account when fitting to the polyaxial data, is generally done in σ 1 σ 2 space, i.e. biaxial plane-strain condition (Eberhardt, 2012). It is evident that when triaxial failure criteria, such as Hoek-Brown, suffer from neglecting the influence of intermediate principal stress, they are shown as a line in σ 1 σ 2 space. Figure 2.6 shows how triaxial failure criteria behave in σ 1 σ 2 space. In order to have a better understanding of peak strength criteria in practice, they must be assessed in 3-dimensional stress invariants space. Figure 2.5 shows how strength criteria typically appear as a surface in σ 1 σ 2 σ 3 space. Any point in this space, representing a body subject to a particular stress state, that lies inside the area bounded by strength envelope, indicates that the body has not reached the critical value, and

12 Chapter 2. Peak strength criteria 7 Figure 2.5: Example of a failure envelope in 3-dimensional stress space (from Benz et al., 2008) any point lies on the surface defines a body which has reached the limiting value, and thus, failure may occur in material. Stress state cannot lie outside the limiting boundary in practice. However, assuming an elasticity analysis carried out, the stress state can surpass the strength envelope, which indicates that stress has exceeded the ultimate strength of material and a failure would occur if the analysis were carried out in plasticity (Rocscience Inc., 2009). Here discussion about stress state location in space is deferred until chapter (MPa) σ σ 2 =σ 3 σ 3 =60 MPa σ 1 =σ 2 σ 3 =90 MPa (MPa) σ σ 2 =σ 3 σ 3 =30 MPa σ 1 =σ 2 σ 3 =90 MPa σ 3 =60 MPa 100 σ 3 =0 σ 3 =30 MPa σ c = 60 MPa ϕ = σ 3 =0 σ c = 60 MPa m = 16 s = (MPa) σ 2 (a) Mohr-Coulomb criterion (MPa) σ 2 (b) Hoek-Brown criterion Figure 2.6: Sensitivity of the triaxial strength (σ 1 ) to the intermediate stress (σ 2 ) in triaxial criteria (after Colmenares and Zoback, 2002). Figure 2.7 shows how a point in principal stress space, in the form of P (σ 1, σ 2, σ 3 ), can be characterized by distance from the origin of the plane passing through P that

13 Chapter 2. Peak strength criteria 8 σ 1 Deviatoric plane (σ 1 + σ 2 + σ 3 = constant) σ 1 * P (σ 1, σ 2, σ 3 ) P θ o ξ ρ o Hydrostatic axis (σ 1 = σ 2 = σ 3 ) ρ 60 σ 3 o σ 2 * σ 3 * σ 2 (a) Principal stress space (b) Deviatoric plane Figure 2.7: Representation of a stress state in principal stress space and the deviatoric plane (from Lee et al., 2012). is perpendicular to the hydrostatic axis (σ 1 = σ 2 = σ 3 ), and the location of P within this plane. The plane containing P, which has a distance ξ from the origin, is generally referred to as the deviatoric plane or π-plane. The location of P within this plane may be characterized using the distance ρ and angle θ (see Figure 2.7.b); in this work, θ, the so-called Lode angle is defined as the departure of the stress state from σ1-axis, which is the projection of σ 1 -axis on the π-plane, and varies within the range of 0 < θ < π, which 3 represents the condition σ 1 > σ 2 > σ 3. A triaxial compression state (σ 1 > σ 2 = σ 3 ) is represented by θ = 0, while θ = π corresponds to a triaxial extension state (σ 3 1 = σ 2 > σ 3 ). The parameters ξ, ρ and θ may be written in terms of stress invariants as ξ = I 1 3, ρ = 2J 2, θ = 1 3 cos 1 ( J 3 J 3/2 2 ) (2.2) where I 1 is the first invariant of stress tensor, while J 2, and J 3 are second and third invariants of the stress deviator. Performance of different empirical strength criteria is usually assessed in π-plane, as well as meridian cross sections (ρ ξ planes). Figures 2.9 demonstrates that Hoek-

14 Chapter 2. Peak strength criteria 9 σ 1 * ρ θ ρ c ρ θ=0 θ=π/3 ρ t ρ c o ρ t σ 2 * σ 3 * - 3ccotφ ξ (a) Deviatoric plane (b) Meridian plane Figure 2.8: Mohr-Coulomb failure criterion in the deviatoric and the meridian plane (from Lee et al., 2012). Brown failure criterion is non-linear in form (in the meridian plane), which is the main advantage of it over Mohr-Coulomb (Figure 2.8) that performs linearly in the meridian plane. However, neglecting the effect of σ 2 makes both criteria perform linearly in π- plane in the range of 0 θ π 3 (see Figure 2.9.a and 2.8.a), and consequently prevents smoothness and continuity in triaxial compression and tensile. This results in irregular hexagons in deviatoric plane, and thus gradient functions of triaxial criteria become singular and make difficulties in their numerical implementation (Lee et al., 2012). σ 1 * ρ t ρ θ ρ c ρ ρ c θ=0 θ=π/3 o ρ t σ 2 * σ 3 * - 3σ c s/m ξ (a) Deviatoric plane (b) Meridian plane Figure 2.9: Hoek-Brown failure criterion in the deviatoric and the meridian plane (from Lee et al., 2012).

15 Chapter 2. Peak strength criteria Polyaxial strength criteria As discussed earlier, an appropriate failure criterion that can predict rock behaviour in all stress conditions needs to incorporate the effect of intermediate principal stress σ 2 on rock strength. For this purpose, several polyaxial failure criteria have been proposed, among which Pan-Hudson (Pan and Hudson, 1988), Zhang-Zhu (Zhang and Zhu, 2007), Jiang-Xie (Jiang and Xie, 2012), and the so-called HB-WW (Lee et al., 2012) are initially being referred here, since they are commonly used in different studies, and also take Hoek-Brown strength parameters (m, σ c ), which can be easily derived from simple laboratory tests, as inputs. Afterwards, a different criterion used in the present analysis, i.e. Ottosen failure criterion, is compared to those mentioned before. As seen in Figure 2.7, a failure surface can be geometrically defined as: F (ξ, ρ, θ) = 0 or F (I 1, J 2, θ) = 0 (2.3) which declares that any criterion, such as Pan-Hudson, which lacks the effect of term θ, and predicts an identical ρ value for triaxial compression and extension regimes, does not evaluate rock behaviour appropriately (see Figure 2.11) (Lee et al., 2012). Moreover, smoothness and convexity in both meridians and deviatoric plane are assets, as discussed, whereas it is shown that Zhang-Zhu does not satisfy smoothness requirement in triaxial extension regime (see Figure 2.11) (Jiang and Xie, 2012). Additionally, tests on rock samples in triaxial compression and extension (Figure 2.3) suggest that rock peak strength in triaxial extension is higher than that of in triaxial compression. Thus, keeping in mind that the Hoek-Brown strength parameters are derived from triaxial compression tests results, any criterion reduces to Hoek-Brown in triaxial extension lacks the strengthening effect of σ 2. Figure 2.11 summarizes the performance of all mentioned criteria in the deviatoric plane and in σ 1 σ 2 space. From these plots, it can be inferred that the only criterion that

16 Chapter 2. Peak strength criteria 11 takes the Lode angle into account, and meets the convexity and smoothness conditions, and also predicts a higher strength in tensile than that of in compression, is Ottosen failure criterion. The following section introduces this polyaxial strength criterion Ottosen peak strength criterion In 1977, Ottosen proposed a failure criterion for concrete with four parameters derived from triaxial compression and triaxial extension data (Ottosen, 1977). Taking tensile strength into account, Ottosen failure criterion overcomes the drawbacks of other polyaxial criteria that predict same values in compression and tensile, however has not been commonly used in rock failure prediction due to difficulty of obtaining parameters and complicated formulation (Ottosen and Ristinmaa, 2005). Figure 2.11 compares all mentioned failure criteria with conventional Hoek-Brown criterion, and clearly shows unlike the other criteria, Ottosen does not necessarily reduce to Hoek-Brown in triaxial extension. as: Ottosen peak strength criterion in general form of failure surfaces (Eq. 2.3) is given A J 2 σ 2 c where λ is a function of Lode angle θ and is defined as: J2 + λ + B I 1 1 = 0 (2.4) σ c σ c λ = K 1 cos(ψ) (2.5) and 1 Ψ = 3 cos 1 (K 2 cos 3θ) cos 3θ 0 π 3 1 (2.6) 3 cos 1 ( K 2 cos 3θ) cos 3θ < 0 Now it is clear that Ottosen criterion obtains four dimensionless parameters: A, B, K 1, K 2,

17 Chapter 2. Peak strength criteria 12 which are determined from experiment. Four failure stress conditions are used to obtain Ottosen parameters as illustrated in Figure 2.10 (Ottosen and Ristinmaa, 2005): 1. σ c : Uniaxial compressive strength, where σ 1 σ 2 = σ 3 = 0; 2. σ bc : Biaxial compressive strength, where σ 1 = σ 2 σ 3 = 0; 3. σ t : Uniaxial tensile strength, where σ 1 = σ 2 = 0 σ 3 ; 4. An arbitrary peak strength along the compressive meridian, representing by the point (x, y) in (I 1, J 2 ) space. Uniaxial Tensile Strength J Uniaxial Compressive Strength (x, y) Compressive Meridian Biaxial Compressive Strength Tensile Meridian I Figure 2.10: Failure states on compressive and tensile meridians used to obtain the four parameters of the Ottosen criterion (from El Matarawi and Harrison, 2014). To calibrate the Ottosen parameters, an analytical approach is suggested by Ottosen and Ristinmaa (2005), and reformulated by El Matarawi and Harrison (2014), to be compatible with sign convention in geomechanics, where stress is positive as it produces compression. This analytical approach results as following: γ = (y 3 x)(σ t σ bc ) (2.7) A = 3σ 2 c [γ 3(σ t σ bc )( y 3 σ c 1)] σ t σ bc [γ 3y(3y σ c 3)] (2.8) B = [γ 3y σ c 3 y 3 x ][ 1 3 σ t σ bc σ c y] σ t σ bc [γ 3y(3y σ c 3)] (2.9)

18 Chapter 2. Peak strength criteria 13 λ t = 3(B + σ c σ t A σ t 3σ c ) (2.10) λ c = 3( A 3 + B 1) (2.11) K 1 = 2 3 λ 2 t + λ 2 c λ t λ c (2.12) K 2 = 4( λ c K 1 ) 3 3 λ c K 1 (2.13) It appears from this formulations that A 0, B 0, K 1 0, 1 K 2 0 (El Matarawi and Harrison, 2014). Keeping in mind that using Ottosen failure criterion requires a large set of experimental data to obtain the parameters, with complex and long equations that may result in mistakes, it is used in this study to compare with the triaxial Hoek-Brown failure criterion. However, in many cases it is preferred to ignore the small discrepancy of other criteria and use a less complicated and time consuming method. 2.3 Summary This chapter reviewed a few of many rock peak strength criteria commonly being used in geomechanics. Conventionally, triaxial criteria in terms of major and minor principal stresses is used to predict failure in rocks, whereas it has been recognized that the intermediate principal stress has an increasing effect on rock peak strength. Many polyaxial criteria has been proposed, mostly derived based on empirical methods, with no one performing well in all conditions. One particular deficiency of many polyaxial criteria is predicting an equal peak strength for rocks in compression and tensile. Ottosen failure criterion, however, overcome this drawback by taking the experimental measurement of tensile strength.

19 Chapter 2. Peak strength criteria 14 Ottosen peak strength criterion has, of course, more complications in use rather than simple extensions of triaxial criteria in 3-dimension, but is elected in the present analysis since has not been commonly used in rock engineering and perhaps a need of investigation on application of that in geomechanics is necessary.

20 15 Chapter 2. Peak strength criteria σ1 * σ1 (MPa) Pan-Hudson o σ3 * 700 σ3=90 σ3= σ3= σ2 * σ3=0 MPa σ2 (MPa) σ1 * σ1 (MPa) Zhang-Zhu o 700 σ3=90 σ3= σ3=30 σ3* 300 σ2 * σ3=0 MPa σ2 (MPa) σ1* σ1 (MPa) Jiang-Xie 700 σ3=90 σ3= σ3=30 o σ3* 300 σ2 * σ3=0 MPa σ2 (MPa) σ1* σ1 (MPa) HB-WW 700 σ3=90 σ3= σ3=30 o σ3 * 300 σ2 * σ3=0 MPa σ2 (MPa) σ1 * σ1 (MPa) Ottosen o σ3* σ2 * 700 σ3=90 σ3= σ3= σ3=0 MPa σ2 (MPa) Figure 2.11: Comparison of different polyaxial criteria with Hoek-Brown in the deviatoric plane (left column) and in σ1 σ2 space (right column). Solid lines represent polyaxial criteria, and dashed lines represent Hoek-Brown failure criterion.

21 Chapter 3 Stability analysis methodology Triaxial strength criteria and their shortcomings in prediction of rock peak strength in polyaxial stress regime (σ 1 > σ 2 > σ 3 ) were discussed in chapter 2, followed by an outline of some polyaxial peak strength criteria along with a comparison of triaxial and polyaxial criteria envelopes in principal stress space. In this chapter, a method is discussed to investigate and compare the application of each form of failure criteria in assessment of rock stability. This study is primarily focused on prediction of failure around underground openings through investigation on induced stresses caused by excavation. Bedi and Harrison (2012) showed that stress state is always polyaxial around an underground excavation, regardless of in-situ stress condition being triaxial or polyaxial. While Lode angle is usually used to present the position of a stress point in π-plane, Bedi and Harrison (2012) introduced an alternative way to measure deviation of stress state from triaxial regime, called degree of polyaxiality, which varies between 0 for triaxial stress conditions and 1, representing maximum polyaxiality. This will be discussed in detail in section 3.1. Figure 3.1 illustrates degree of polyaxiality around a circular opening in an elastic ground for (a) triaxial and (b) polyaxial in-situ stress state. This analysis will be validated in the present study utilizing the boundary element program Examine3D. Afterwards, the method of stability analysis around excavation using both 2d and 3d criteria will be 16

22 Chapter 3. Stability analysis methodology 17 discussed with a particular focus on discrepancy between these two approaches. (a) triaxial in-site stress state (b) polyaxial in-site stress state Figure 3.1: Polyaxiality around a circular tunnel with two different in-situ stress conditions (from Bedi and Harrison, 2012). 3.1 Degree of polyaxiality As discussed in chapter 2, state of stress is represented by three invariants in 3d stress space: ξ, ρ, and the so-called Lode angle θ, representing the location of a stress point in π-plane from 0 degrees (triaxial compression) to 60 degrees (triaxial extension). A more convenient representation of stress location in π-plane to determine the deviation from being triaxial compression or extension regime, which is the main concern in use of triaxial strength criteria inappropriately, is defined as the ratio of the smallest intermediate Mohr circle to that of the largest one (Figure 3.2), and is given as (Bedi and Harrison, 2012): α = min(δ 1, δ 3 ) r (3.1) where : r = (σ 1 σ 3 ) 2, δ 1 = (σ 1 σ 2 ), δ 3 = (σ 2 σ 3 ).

23 Chapter 3. Stability analysis methodology 18 τ δ 3 δ 1 O σ σ 3 σ 2 σ 1 Figure 3.2: Mohr circle in 3d stress state (after Davis and Selvadurai, 2002). As shown in Figure 3.2, when δ 1 is equal to zero means σ 2 = σ 1, and thus the stress regime is triaxial extension (σ 1 = σ 2 > σ 3 ), degree of polyaxiality is minimum (α = 0). Likewise, when δ 3 is equal to zero and stress condition is triaxial compression (σ 1 > σ 2 = σ 3 ), degree of polyaxiality is minimum as zero. On the other hand, maximum degree of polyaxiality (α = 1) occurs when δ 1 = δ 3, i.e. σ 2 = (σ 1 σ 3 ), and stress state 2 is extremely polyaxial. As a part of this study, induced stress obtained from boundary element analysis is investigated in terms of degree of polyaxiality around a circular opening in an elastic ground. Here, the result of analysis is shown, and details about boundary element model is deferred until chapter 5. A circular tunnel in an elastic ground is assumed, and induced stresses around the planar end of the tunnel is calculated using the boundary element program Examine3D. In this study, the excavation is assumed advancing along the main axis, and thus the analysis is extended from 0.5 tunnel radius behind the working face to 1.5 radii ahead of the excavation. MATLAB is used to extend the analysis and visualize the results.

24 Chapter 3. Stability analysis methodology 19 Four different in-situ stress states are assumed as shown in Table 3.1. In all of these cases, vertical stress is taken as the minor principal stress (σ v = σ 3 ), while magnitude of horizontal and axial stresses change to produce other assumed conditions. Figure 3.5 displays degree of polyaxiality around and ahead of the tunnel in three cutting planes. As shown, one plane is taken perpendicular to the tunnel axis 0.5 radius behind the working face (plane A), and two longitudinal sections are taken horizontal (plane H) and vertical (plane V). Results from polyaxiality analysis confirm the earlier studies that show regardless of in-situ stress state being polyaxial or not, it will be disturbed after excavation and need to be investigated. As expected, disturbance of initial stresses can extend several tunnel radii ahead and around of the excavation where stress state eventually shows the tendency to the initial field stress. This suggests that using a simple triaxial criterion, such as Hoek-Brown, in failure prediction may not be appropriate in a three dimensional analysis. Table 3.1: in-situ stress conditions assumed in analysis in-situ stress states stress direction with respect to the tunnel axis hydrostatic triaxial extension triaxial compression polyaxial σ v σ 3 σ 3 σ 3 σ 3 σ h σ 3 σ 1 σ 1 σ 1 σ a σ 3 σ 1 σ 3 σ Strength factor The primary interest of this study is investigating stability around underground openings, particularly focusing on error associated with the use of an inappropriate strength criterion in calculations. In order to evaluate ground stability after excavation, induced stresses must be calculated and compared to the ultimate allowable stress, i.e peak

25 Chapter 3. Stability analysis methodology 20 strength of the rock. Here, strength factor (S.F.) also called strength reserve is being used to determine state of stress with respect to the ultimate allowable stress, and is given as (Corkum, 1997): S.F. = ultimate allowable stress induced stress (3.2) It appears from this relation that when the strength factor is equal to 1, induced stress reaches the peak strength, and S.F. < 1 indicates that the induced stress has exceeded the ultimate strength and a failure may occur. Obviously, greater values of S.F. indicate that the rock is more stable Estimation of strength factor using triaxial rock strength criteria A very simple method can be used to estimate strength factor in two dimensions. With the use of triaxial failure envelope in σ 1 σ 3 space, S.F. can be defined as the ratio of maximum principal stress at which a failure occurs for the current minimum principal stress (Corkum, 1997). In other words, σ 1 on failure envelope, corresponding to a certain σ 3, indicates rock peak strength. So, Eq.3.2 for conventional Hoek-Brown failure criterion can be written as: S.F. 2 = σ 1,HB σ 1,i (3.3) where σ 1,HB is the peak strength predicted by Hoek-Brown criterion, and σ 1,i is the induced major principal stress obtained from numerical analysis (Figure 3.3). A similar 2-dimensional technique can also be carried out in σ n τ n space to estimate the strength factor (Corkum, 1997).

26 Chapter 3. Stability analysis methodology 21 σ 1 peak strength induced stress σ 3 Figure 3.3: Estimation of S.F. using Hoek-Brown failure criterion Estimation of strength factor using polyaxial rock strength criteria When the stability of the rock is being assessed in a 3-dimensional model, estimation of strength factor in 2-dimensional space might be inaccurate. This problem arises mainly from neglecting the influence of intermediate principal stress. In order to investigate the strength factor more accurately, the peak strength of the rock needs to be estimated using a polyaxial criterion. As discussed in Chapter 2 the induced stress at a point is represented in σ 1 σ 2 σ 3 space. Strength factor in this space is evaluated by the location of stress point in 3- dimensional space with respect to the failure envelope. Thus, the ratio of ρ value, i.e. the distance from the origin of π-plane, on failure envelope to that of for the induced stress is regarded as the strength factor in 3-dimensional space, and thus, for the Ottosen peak strength criterion is given as: S.F. 3 = ρ OT ρ i (3.4) where ρ OT is the maximum ρ value in Ottosen π-plane with the same Lode angle as of induced stress, which has the distance of ρ i from the origin of the π-plane. Figure 3.4 illustrates how strength factor is estimated in the π-plane.

27 Chapter 3. Stability analysis methodology 22 peak strength σ 1 * induced stress ρ σ 2 * σ 1 * Figure 3.4: Estimation of S.F. using Ottosen failure criterion Error in strength factor when using of triaxial criteria Following discussions in sections and 3.2.2, another parameter, on which this study focuses most, is the error resulting from the use of a triaxial criterion to estimate strength factor, where a polyaxial criterion is recommended to use. Here it is assumed that the strength factor given by a polyaxial criterion (S.F. 3 ) is an accurate estimation, and thus error in strength factor that may exist when using a triaxial criterion is calculated as: % ɛ = 100 S.F. 2 S.F. 3 S.F. 3 (3.5) where S.F. 2 and S.F. 3 are strength factors predicted by the 2d and 3d criteria, respectively. We are therefore able to show in which regions around an opening we may underor overestimate the strength factor, and consequently end up with either an uneconomic or unsafe design. It appears from Eq. 3.5 that positive percentage error indicates that the triaxial criterion estimates the S.F. higher than what it actually is. In other words, a positive value means that the rock is actually more highly stressed than the 2d Hoek Brown criterion shows it to be, which may result in an unsafe design, and the rock reaches the ultimate

28 Chapter 3. Stability analysis methodology 23 strength before it is predicted. Whereas, a negative percentage error means that rock is not as much stressed as the triaxial criterion shows, and may lead to an uneconomic or over-conservative design. A series of numerical analysis are carried out to investigate the stability around the end of a circular tunnel. The boundary element program Examine3D is used to calculate induced stresses, and further analysis and visualization are done using MATLAB. Detailed explanation of the numerical modelling is deferred until Chapter 5, and here a preliminary analysis is presented in Figure 3.6 to provide examples of what has discussed so far. The initial field stress in this analysis is assumed polyaxial, i.e. σ 1 > σ 2 > σ 3, and the direction of principal stresses with respect to the tunnel axis is shown along the results in Figure 3.6. The plots that are shown in Figure 3.6 indicate the strength factor estimated using (a) triaxial and (b) polyaxial criteria. It is found to be very difficult to interpret the results of two strength factor plots. Therefore, another plot that shows the error in prediction of strength factor using the triaxial criterion is necessary (Figure 3.6.c) It can be inferred from these results that, in the certain stress state, a significant overestimation, i.e. more than %30, is occurred in the proximity of the wall. This means that, using Hoek-Brown to estimate the strength factor results in an unsafe design in the wall. The question could arise here that if an unsafe design results in an unpredicted failure. To find out the possibility of failure, an accurate estimation of strength factor is also needs to be looked at, i.e. S.F. 3. Thus, both strength factor estimation (Figure 3.6.b) and the error associated with an inappropriate evaluation (Figure 3.6.c) need to be investigated at the same time.

29 Chapter 3. Stability analysis methodology Summary In this chapter, firstly, state of stress around an underground excavation is assessed using the so-called degree of polyaxiality. Assuming different field conditions, it is confirmed that the use of triaxial criteria is not recommended regardless of in-situ stress state being polyaxial, triaxial or hydrostatic. Then, two different approaches, i.e. use of triaxial and polyaxial criteria, to estimate strength factor around an excavation are discussed. Assuming that a polyaxial criterion gives an accurate estimation of strength factor, the error associated to the use of a triaxial criterion can be calculated. Results of a preliminary analysis confirms the need for investigation of the error that the use of a 2d criterion in numerical modelling may produce. It is demonstrated that a sound conclusion requires to know an accurate estimation of strength factor along with the error of poor estimation.

30 Chapter 3. Stability analysis methodology 25 Vertical longitudinal section, V Cross section A, 0.5R behind tunnel face Horizontal longitudinal section, H (a) hydrostatic in-situ stress state (b) triaxial compression in-situ stress state (c) triaxial extension in-situ stress state (d) polyaxial in-situ stress state σ 1 σ 3 σ 2 Degree of polyaxiality (α) Figure 3.5: Degree of polyaxiality around a planar end of a circular opening (α = 1 indicates maximum polyaxiality)

31 Chapter 3. Stability analysis methodology 26 Vertical longitudinal section, V Cross section A, 0.5R behind tunnel face > Strength factor Horizontal longitudinal section, H (a) strength factor predicted using Hoek-Brown criterion. (b) strength factor predicted using Ottosen criterion. 1.0 <1.0 >30 30 σ 1 σ 3 σ 2 (σ 1 >σ 2 >σ 3 ) (c) error in prediction of S.F. when the Hoek-Brown is used. Strength Overestimated (Unsafe design) Strength Underestimated (Uneconomic design) Error in prediction of strength factor (%) <-30 Figure 3.6: Preliminary stability analysis using (a) triaxial and (b) polyaxial strength criteria and (c) the error in prediction.

32 Chapter 4 Effective use of colour for visualization Up until a few decades ago, colours had been rarely used in data representation. Developments in modern software and increasing use of electronic sources have made it possible to easily use colours to represent data more efficiently. However, choosing colours randomly is likely to confuse the reader. In this chapter, producing colours systematically is being discussed briefly. 4.1 Typology of colour schemes Brewer (1994) presented a comprehensive guideline of use of colours for visualization, particularly for implementation in cartography. In geomechanics, however, as well as many other engineering fields, there has not been such a great effort on the use of colours appropriately. Hue, saturation, and brightness are three dimensions in HSB colour space, which are used to produce colour schemes. In order to provide an instruction to generate colour schemes appropriately, Brewer (1994) classified data types into four primary categories: qualitative, binary, sequential, and diverging. Table 4.1 presents perceptual characteristics of each category. Here, a brief review of all is presented. Sequential and diverging 27

33 Chapter 4. Effective use of colour for visualization 28 Table 4.1: Data categorization and colour schemes (after Brewer, 1994). data category and scheme type perceptual dimension of colour hue brightness qualitative binary (special case for qualitative) hue steps (not ordered) neutrals, one hue or one hue step constant brightness one brightness step sequential neutrals, one hue or hue transition single sequence of brightness step diverging two hues, one hue and neutrals, or two hues transitions two diverging sequence of brightness steps schemes are being used in this study, and the reader is referred to Brewer (1994) and Brewer et al. (2003) for detailed explanation and examples of qualitative and binary data types. Colours in a qualitative colour scheme, e.g. rock type classification, have different hue steps, without implying an order (Harrower and Brewer, 2003). A small difference in brightness, sometimes, makes it easier to differentiate without drawing attention to a particular class (Brewer, 1994). Hues, however, must be elected carefully to help the reader. For instance, classes with greater similarity are better to be presented by hues closer on the hue circle. In this categorization, binary colour scheme is presented as a special case of qualitative data which has only two classes. The main difference between producing qualitative and binary colour schemes is that we are able to use brightness steps, with holding hue constant, to imply importance of a class comparing to another (Brewer, 1994).

34 Chapter 4. Effective use of colour for visualization 29 The primary interest of this study on rock stability analysis, as well as many other engineering problems, deals with quantitative data, i.e. sequential and diverging categories. Sequential data classes, such as degree of polyaxiality and strength factor here, are ordered from a minimum to a maximum value, and thus could be dominated by brightness steps (Harrower and Brewer, 2003), although small changes of hue and saturation might help making a better contrast between colours (Brewer, 1994). In a sequential scheme, the darkest colour conventionally represents the highest value, e.g. degree of polyaxiality (see Figure 3.5). However, depending on the case, the lowest value might be supposed to draw the reader s attention. Strength factor is an example of such data types (see Figures 3.6.a & 3.6.b), in which lower values must be emphasized as a possible situation of failure. A quantitative data emphasizing a mid-range point, such as mean, median, zero point, etc., can be conveniently represented by a diverging colour scheme. This category enables us to effectively show deviation below or above a critical point by systematic regression of hue, brightness, and saturation (Brewer, 1994). An already seen example here is percentage error of triaxial criteria in prediction of strength factor (see Figure 3.6.c). Diverging colour scheme is sometimes described as two separate sequential schemes, with complementary hues at two ends that converge on a shared colour or a neutral at a critical midpoint (Brewer, 1994). Use of a spectral scheme to visualize a sequential data is not recommended, since it does not inherently convey the ordinal information to the reader (Light and Bartlein, 2004). Nonetheless, modern software, such as Examine3D which is used in this research for stress analysis, use this method for data visualization. Figure 4.1 compares a sequential and a spectral scheme to visualize a quantitative data set, and indicates that an appropriate sequential scheme effectively carries the magnitude message and enables the reader to receive the overall information even without looking at the colour scheme.

35 Chapter 4. Effective use of colour for visualization (a) sequential scheme with single hue and steps of brightness and saturation (b) spectral scheme with hue steps and constant brightness (Examine3D) Figure 4.1: Comparison of a sequential and a spectral scheme on a same dataset Bivariate colour scheme for rock stability analysis To provide a convenient representation that allows comparison of two variables at the same time, and prevent the reader going back and forth on two different figures, bivariate colour schemes are recommended. We can produce a bivariate colour scheme by a systematic combination of two one-variable schemes (Brewer, 1994). We have already seen some preliminary analyses in Chapter 3, and discussed the appropriate techniques of data visualization. Parameters being analyzed in this study are categorized into sequential and diverging types. Combination of those produces sequential/sequential, sequential/diverging, and diverging/diverging colour schemes. More combinations can be generated, correspondingly, with other categories which is discussed in detail by Brewer (1994). A Sequential/sequential colour scheme is produced by cross of two sets of one-variable colours, logically mixed to make all combinations of two sequential data sets. Thus, the scheme is built with two major hues at opposite corners with transitional colours in between, and systematic brightness and saturation differences throughout the scheme. Figure 4.2 shows the structure and a 4 4 example of sequential/sequential colour scheme. The main structure of a sequential/diverging scheme is similar to that of a sequen-

36 Chapter 4. Effective use of colour for visualization 31 Hue 1 High bri. Low sat. max. second variable Hue transition Brightness transition & Saturation second variable Low bri. High sat. Hue 2 min. first variable min. first variable max. (a) bivariate colour scheme structure. (b) a bivariate colour scheme with major hues: yellow and magenta. Figure 4.2: Sequential/sequential scheme with transitional hue mixtures in major diagonal and steps of brightness and saturation in minor diagonal tial/sequential, which is built on two sides of a mid-range point transitioning to two opposite hues. This bivariate scheme can conveniently show the critical mid-value of diverging data set, as well as extreme values of both variables (Figure 4.3). max. H1 Hi. bri. Hi. bri. Lo. sat. Lo. sat. H3 min. Lo. bri. Hi. sat. H2 H4 Lo. bri. Hi. sat. (-) (+) zero Figure 4.3: Sequential/diverging scheme with a mid-range value of zero 4.2 Summary A brief introduction on colour scheme categorization is presented in this chapter. It is shown that how a meaningful colour scheme can help the reader to understand the

37 Chapter 4. Effective use of colour for visualization 32 information, even in a simple data set. An appropriate use of colours has barely been considered in data visualization in geomechanics, even in modern software suits, and there is a substantial need for use of such techniques in developments. Moreover, a bivariate colour scheme is introduced that enables to visualize two variables at the same time. According to the discussion in Chapter 3, a bivariate colour scheme can effectively visualize a sequential, i.e. strength factor, and a diverging variable, i.e. percentage error.

38 Chapter 5 Numerical analysis of stress state around an advancing tunnel face Deep in the earth, excavation of the rock disturbs the original field stresses and results in redistribution of primary stress field. Changes in stress magnitudes specifically in the proximity of the excavation boundary play a controlling rule in rock instabilities by direct influence in stress concentration and rock strength degradation. The analysis of induced stresses around an excavation, thus, has become a common practice in design of the tunnel and support (Eberhardt, 2001). In the past, the analysis of stress redistribution around the excavation was limited to 2-dimensions. One of the assumptions made by a 2-dimensional stress analysis is infinite out-of-plane length of the excavation, i.e. plane strain analysis. This assumption makes the analysis shows exaggerated results near the working face, or when the length of the tunnel normal to the cross section becomes close to that of the cross-sectional dimensions (Rocscience Inc., 2009). 33

39 Chapter 5. Numerical analysis 34 As the complexity of the excavation and geological environment increases, the 2- dimensional analysis appears even more inadequate. In the case of an advancing tunnel, it becomes more necessary to extend the analysis near and also ahead of the tunnel face (Eberhardt, 2001), whereas the 2-dimensional analysis restricts us to the planes normal to the tunnel axis and far from the end of the tunnel. Recognizing many deficiencies of 2-dimensional models in practice, 3-dimensional analysis has become more common in engineering practice. With respect to the induced stress concentration in proximity of the ends and edges of an excavation, a 3-dimensional analysis allows for a more careful examination (Eberhardt, 2001). With the numerical methods integrating to the classic approaches, such as analytical and empirical techniques, and the necessity of 3-dimensional analysis, numerous computer programs have been commercially used by geotechnical engineers. The numerical software applications, indeed, have the advantage of inherent ease-of-use over the classic methods in complex problems (Scussel and Chandra, 2013). However, software packages available for commercial purposes in geomechanics, even the 3-dimensional programs, utilize the conventional failure criteria (Scussel and Chandra, 2013). It has been discussed in Chapter 2 that neglecting the influence of intermediate principal stress may result in a poor estimation of peak strength and state of stability of the rock. This chapter sets the target of investigating the error associated with the use of triaxial criteria in a 3-dimensional analysis. For this purpose, induced stresses obtained from a boundary element program, i.e. Examine3D, are used to investigate the stability of the rock using both triaxial and polyaxial criteria. Extended analysis is carried out using MATLAB.

40 Chapter 5. Numerical analysis Tunnel geometry and boundary element mesh used for 3D numerical analysis In this study, the elastic boundary element program, Examine3D, is used to undertake a series of analysis to determine induced stresses around an underground opening. The primary assumptions made by an elastic boundary element calculation is that the structure being modelled is located in a homogeneous, isotropic, and linearly elastic medium (Curran and Corkum, 2000). Keeping in mind that the rock masses do not usually possess all of the assumed properties, the results need to be cautiously looked at with respect to the deviation of actual rock mass properties from the assumptions (Rocscience Inc., 2009). Unlike Finite Element Method (FEM) and Finite Difference Method (FDM), stresses in a Boundary Element Method (BEM) program can be calculated at any point within the surrounding rock mass. Thus, only the boundary of the excavation needs to be discretized, and the location of stresses to be calculated is defined by the visualization system. The latter provides more flexibility in visualizing the data in a boundary element environment (Corkum, 1997). (a) Constant (b) Linear (c) Quadratic Geometric node Function node Figure 5.1: Element library of Examine3D. Accuracy and computation time increases from left (a) to right (c) (from Rocscience Inc., 2000). The surfaces of the excavation, in Examine3D, are discretized by triangular elements. Figure 5.1 shows three element types available within the software, which differs in accu-

41 Chapter 5. Numerical analysis 36 racy and computation time. The name of each type, i.e. constant, linear, and quadratic, implies the mode of displacement over the element surface. In the present analysis, elements are set to linear type, by which the displacement of the element varies linearly (Curran and Corkum, 2000). This allows to, due to simple geometry of the tunnel, obtain sufficient accuracy in a reasonably short computation time. Note that, in linear triangular discretization, number of elements are more than nodes, because each node is shared by neighbouring elements (Curran and Corkum, 2000). tunnel radius: 2m # of elements: 1792 # of nodes: 898 (a) Mesh generated in the face of the tunnel (b) Three-dimensional mesh generation Figure 5.2: Boundary element mesh in the surfaces of the tunnel. In this research, a long circular tunnel in an elastic ground is modelled. The analysis takes place near the planar end of the tunnel. Figure 5.2 shows the boundary element mesh in the face and around the tunnel. Strength parameters of rock which is used in this analysis are listed in Table 5.1. Figure 5.3: Uniform grid used for visualization of the data.

42 Chapter 5. Numerical analysis 37 Table 5.1: parameters used in Hoek-Brown and Ottosen failure criteria strength parameter σ c (MPa) σ bc (MPa) σ t (MPa) m s (x, y) = (I 1, J 2 ) along the compressive meridian value (496, 148) Visualization of the data, in Examine3D, is defined by the user usually in the form of uniform grids (Figure 5.3). A uniform grid cell is defined in each of the four cutting planes around the end of the tunnel, which is illustrated in Figure 5.4 along with the geometry of the tunnel. Cross section B, 0.25R ahead of tunnel face Vertical longitudinal section, V 2R σ h σ v σ a Cross section A, 0.5R behind tunnel face 0.5R 0.25R R 1.5R 1.5R 2R Horizontal longitudinal section, H (a) (b) Figure 5.4: (a) Cross sections ahead and behind tunnel face, and (b) horizontal and vertical longitudinal sections along the main axis of a circular tunnel. In this model, the tunnel radius is set to R=2m Effect of initial stress field on stability analysis Six different in-situ stress conditions are assumed to investigate the effect of initial field stress state on analysis. In all cases, directions of principal in-situ stresses were assumed vertical (σ v ), horizontal (σ h ) and axial (σ a ) relative to the alignment of the tunnel main

43 Chapter 5. Numerical analysis 38 axis (see Figure 5.4). In each case, the minor principal in-situ stress was assumed vertical (σ v = σ 3 ), with a constant magnitude equal to the overburden load. The magnitudes of horizontal and axial in-situ stresses, however, vary for each case. For this purpose, different k values (i.e. the ratio of horizontal to vertical stress) are assumed to produce different stress conditions. 0 z = Depth below surface (m) k = (1500 / z) k = (100 / z) k = σ h / σ v Figure 5.5: Variation of k value with depth below ground surface (from Scussel and Chandra, 2013, data collected and published by Hoek and Brown, 1980b). In the data collected and published by Hoek and Brown (1980b), it is shown that k value can manifest a wide range at low depth, i.e. up to 1000m below the ground surface (Figure 5.5). This is mainly caused by many factors affecting the magnitude of horizontal stress in upper levels of earth crust, of which tectonic stresses, gravitational force and superficial morphology are shown to have more significant influences (Scussel and Chandra, 2013).

44 Chapter 5. Numerical analysis 39 For the six different stress conditions of this analysis, k is assumed to take values of 1, 1.5 and 2. Setting k = 1 results in hydrostatic stress condition (σ 1 = σ 2 = σ 3 ), and holding σ 3 = 30 MPa, with k = 1.5 and k = 2, the magnitudes of intermediate and major principal stresses increase to 45 and 60 MPa. Table 5.2 summarizes the in-situ stress magnitudes and orientations, with respect to the tunnel axis, for each case. Table 5.2: in-situ stress conditions assumed in analysis with respect to the tunnel axis analysis case hydrostatic triaxial extension triaxial compression triaxial compression polyaxial polyaxial σ v (MPa) σ h (MPa) σ a (MPa) The primary analysis shown in Chapter 3 are extended for 6 in situ stress conditions listed in Table 5.2, and visualized in 4 cross sections which are shown in Figure 5.4 to investigate the error in prediction of instabilities using a triaxial criterion, behind and ahead of the tunnel face. A bivariate colour scheme, i.e. sequential/diverging, is used to display the error in prediction, % ɛ, crossed with an accurate estimation of strength factor, S.F. 3. Figure 5.7 shows the results of stability analysis around a circular opening for cases 1 to 6. Of all assumed cases, the first one with hydrostatic in-situ stress state (Figure 5.7.a) exposes a very different behaviour. In fact, this is the only condition in which a very high S.F. is extremely underestimated by the triaxial criterion far from the tunnel boundary (zone A1). About one tunnel radius ahead and around the excavation boundary (zone A2), however, S.F. is highly overestimated and an unsafe design is likely to happen, especially for a thin layer around the tunnel where S.F < 1 (zone A3).

45 Chapter 5. Numerical analysis 40 A very high S.F. in the regions where the stress state tends to initial field stress, i.e. hydrostatic, could be justified by the methodology of estimating S.F. in π-plane (see Figure 3.4), where a point is characterized by distance from the origin. Noting that the π-plane is perpendicular to the hydrostatic axis (see Figure 2.7), it is now clear that when the state of stress at a point is close to hydrostatic condition, the distance is maximized from the failure envelope. As a result, regardless of the magnitude, as the stress regime tends to hydrostatic, strength factor increases. This results in a very higher value of S.F. when a polyaxial criterion is used. Assuming a triaxial extension in-situ stress (Figure 5.7.b), as σ 1 = σ 2 = σ h, excavation direction does not affect the stress distribution as long as the tunnel axis is horizontal. In this condition a severe overestimation is observed in the proximity of the wall, which results in an unsafe design, specifically close to the boundary of excavation where S.F < 1 (zone B1). However, S.F. is underestimated in the crown, leading to a conservative design (zone B2). In triaxial compression stress state, i.e. σ 1 > σ 2 = σ 3, however, changes in the direction of the tunnel affects stress-induced instabilities dramatically. Figures 5.7.c and 5.7.d clearly show that when the tunnel advances along σ 2 the Hoek-Brown criterion significantly overestimates S.F in the wall, where in a thin layer close to the boundary of excavation an unpredicted instability may occur. However beyond the instant zone of failure, a very high S.F. guarantees the safety of the wall (zone C1). Whereas, high S.F. ahead of the advancement direction, when the tunnel axis is along σ 1, is not desirable since it prevents the rock breaks itself (zone D1). Same as triaxial compression condition, in polyaxial stress regime, i.e. σ 1 > σ 2 > σ 3, the direction of the tunnel plays an essential role in stability state around the excavation. As shown in Figure 5.7.e the stability of the wall is extremely overestimated by the triaxial criterion when the tunnel advances along σ 2, and in fact Hoek-Brown does not predict the possible failure in the wall in this case (zone E1). Whereas, the same problem

46 Chapter 5. Numerical analysis 41 occurs immediately ahead of the working face when the tunnel axis is along σ 1 and perpendicular to σ 2 (zone F1). A very high underestimation in S.F. is observed, when using Hoek-Brown, about one tunnel radius far from the wall, in case 5, which is not likely to cause a serious problem due to high S.F., but may result in an over-conservative design where in fact there is not need to support. (zone E2). In practice, an overestimation of S.F. in the wall and crown of the tunnel may cause more serious problems in an advancing tunnel, as long as failure in the working face and ahead of the advancement is desirable and under control. Excluding case 1, error in prediction reduces in general, as the distance from the boundary increases. In cases 3 to 6, where the magnitude of axial stress differs from that of horizontal stress, stability of the wall is lower when σ 1 acts normal to the main axis, i.e. cases 3 and 5, and thus, requires more supporting force in design. Stability of the roof does not vary considerably due to constant σ 3 for all cases, and is generally underestimated by the Hoek-Brown criterion Effect of geometry on stability and the error in prediction: elliptical tunnels As a part of this study, effect of geometry on stability and the error associated with use of triaxial criteria is investigated. For this purpose, two elliptical tunnel, horizontal and vertical, are modelled to compare with the circular tunnel which has been discussed. Figure 5.6 shows the section of each tunnel that are being analyzed here. Note that all three sections have the same area, thus the volume of excavated rock remains constant by changing the shape of the tunnel. Of the six field stress conditions listed in Table 5.2, four of them, are analyzed here, since the direction of excavation in triaxial compression and polyaxial stress regimes is

47 Chapter 5. Numerical analysis m 1.6 m 2.0 m 1.6 m 2.5 m (a) Vertical elliptical (b) Circular (c) Horizontal elliptical Figure 5.6: Three different tunnel sections to assess the effect of geometry. not of particular interest to this part. Other assumptions remain unchanged. With cases 4 and 6 being omitted, the conditions that are being analyzed in this part is same as listed in Table 5.2 for cases 1, 2, 3 and 5. Analysis in all conditions is plotted in Plane A located 0.5R behind the tunnel face (see Figure 5.4). Figure 5.8 compares the results of stability analysis in three different shapes of tunnels, i.e. two elliptical and one circular. It is evident that, from these plots, effects of the tunnel geometry is more significant on the error rather than stability. Although, in all vertical elliptical tunnels, regardless of in-situ stress conditions, the zone of instability in the wall is slightly larger than that of in horizontal elliptical and circular tunnels. In case 1, where the in-situ stress state is hydrostatic and the distribution of induced stress is uniform around the circular tunnel, the zone of overestimation is smaller and the contours are closer together at the end points of major axes of both ellipses (Zone A1 and A2). An exactly opposite phenomenon is happened at the ends of minor axes. This might need to be considered when the zone of disturbance in being assessed. In triaxial extension condition (case 2), instability zone in the wall of the vertical ellipse is larger, and needs to be considered due to an overestimation that occurs in this region (Zone B1). Changing the geometry from circular to horizontal elliptical, however, results in a more extensive conservative design in the roof (Zone B2).

48 Chapter 5. Numerical analysis 43 Analysis in triaxial compression (Case 3) and polyaxial (Case 5) in-situ stress conditions results in almost similar conclusions. In both, the overestimation in the wall significantly decreases from the vertical elliptical to the circular and is minimized in the horizontal elliptical tunnel (Zone C1 to C3 and D1 to D3). A minor change in stability is also observed in the roof, where a slightly less extensive unstable region in the circular tunnel is observed (zone C4 and D4). To sum up, in hydrostatic (Case 1) and triaxial extension (Case 2) conditions, there is not a significant advantage in elliptical tunnels over the circular one. In fact, stress concentration around the elliptical tunnels result in propagation of instability, particularly in the vertical shape of tunnel. In triaxial compression (Case 3) and polyaxial (Case 5) stress states, however, the overestimation zone of S.F. decreases in horizontal elliptical tunnel. Although, the stability around the tunnel still shows a better condition in circular shapes Conclusions Results of stability analysis were presented in this chapter. Using a bivariate colour scheme which allows effectively visualize two variables simultaneously, some essential conclusion may be drawn. The key issue to be considered in an advancing tunnel is that degradation of the rock in working face is desirable, keeping in mind that an overestimation of S.F. can result in uncontrolled failures and may cause some problems. However, an overestimation of S.F. in the wall and the roof is an absolute danger. A clear example of this phenomenon is in triaxial compression and polyaxial stress states where the direction of the tunnel significantly changes the predictions. It is shown that an overestimation of S.F. is happened where σ 1 is horizontally perpendicular to the tunnel axis, i.e. case 3 and 5, and an analysis using a triaxial criterion leads to an unsafe design for support.

49 Chapter 5. Numerical analysis 44 Another important conclusion, that would be valuable in design of the tunnels in complex stress conditions, can be drawn by comparison of different shapes of the tunnels, particularly in triaxial compression and polyaxial stress states. It is shown that the zone of overestimation in the wall significantly decreases when a horizontal elliptical tunnel is excavated. Thus two factors play a key role in design of tunnels in those stress conditions: the direction of the tunnel with respect to the principal stress components, and shape of the tunnel. In triaxial extension regime, however, changes in shape of the tunnel does not have a significant effect, except for an overestimation that is concentrated in the wall when the vertical elliptical tunnel is excavated. In hydrostatic stress state also, assuming that a uniform stress distribution is easier to deal with, there is perhaps not an advantage in changing the circular tunnel to elliptical.

50 Chapter 5. Numerical analysis 45 A1 Section H Section V (a) Case 1: hydrostatic (b) Case 2: triaxial extension Section H Section V (c) Case 3: triaxial compression (d) Case 4: triaxial compression Section A Section B Section A Section B Section A Section B C.L. Section H Section V Section A Section B A3 A2 Tunnnel C.L. B1 B2 Section H Section V Section A Section B C1 D1 Section H Section V Section A Section B E1 E2 (e) Case 5: polyaxial Strength factor > <1.0 <-30 Strength Underestimated (Uneconomic design) Section H F1 Strength Overestimated (Unsafe design) >30 Predicted strength factor error, % Section V (f) Case 6: polyaxial Increasing Stability Unstable Figure 5.7: Plots of S.F. and error in prediction in four planes around the tunnel for six assumed cases.

51 Chapter 5. Numerical analysis 46 A1 A2 (a) case 1 - hydrostatic B1 B2 (b) case 2 - triaxial extension C1 C4 C2 C3 (c) case 3 - triaxial compression D1 D4 D2 D3 (d) case 5 - polyaxial Strength factor > <1.0 <-30 Strength Underestimated (Uneconomic design) Strength Overestimated (Unsafe design) Predicted strength factor error, % >30 Increasing Stability Unstable Figure 5.8: Comparison of S.F. and error in prediction in a vertical elliptical, a circular, and a horizontal elliptical excavation.