quantifying the error associated with the use of triaxial rock … · · 2015-04-24stress state...

Quantifying the error associated with the use of triaxialrock strength criteria in rock stability assessment around

underground openings

by

Roozbeh Roostaei

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied ScienceGraduate Department of Civil Engineering

University of Toronto

c© Copyright 2014 by Roozbeh Roostaei

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

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 knowl-

edge. 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 pa-

tience 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

Contents

1 Introduction 1

2 Peak strength criteria 2

2.1 Triaxial strength criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Effect of intermediate principal stress . . . . . . . . . . . . . . . . 4

2.2 Polyaxial strength criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Ottosen peak strength criterion . . . . . . . . . . . . . . . . . . . 11

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Stability analysis methodology 16

3.1 Degree of polyaxiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Strength factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Estimation of strength factor using triaxial rock strength criteria . 20

3.2.2 Estimation of strength factor using polyaxial rock strength criteria 21

3.2.3 Error in strength factor when using of triaxial criteria . . . . . . . 22

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Effective use of colour for visualization 27

4.1 Typology of colour schemes . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Bivariate colour scheme for rock stability analysis . . . . . . . . . 30

4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Numerical analysis of stress state around an advancing tunnel face 33

5.1 Tunnel geometry and boundary element mesh used for 3D numerical analysis 35

iv

5.1.1 Effect of initial stress field on stability analysis . . . . . . . . . . . 37

5.1.2 Effect of geometry on stability and the error in prediction: elliptical

tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Conclusions 47

6.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography 50

Appendix: MATLAB script 54

v

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 σ2 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 σ2. The application of such criteria in 3-dimensions enables to estimate the

strength factor accurately. Regarding the error associated with the 2-dimensional ap-

proach 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 con-

sequences of an inappropriate approach in assessment of stability.

1

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

Chapter 2. Peak strength criteria 3

failure criterion, i.e. Mohr-Coulomb, the Hoek-Brown succeeded to predict nonlinear in-

creasing effect of confining pressure in rock strength (Figure 2.1) (Labuz and Zang, 2012).

Hoek-Brown (nonlinear)

Mohr-Coulomb (linear)

Normal stress, σn (MPa)

Sh

ea

r str

ess, τ

n (M

Pa

)

50 100 150 200 250 3000

50

100

150

200

Figure 2.1: Comparison of Hoek-Brown and Mohr-Coulomb failure criteria plotted in σn− τnspace 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:

σ′1 = σ′3 +√

(m.σc.σ′3 + s.σ2c ) (2.1)

where:

σ′1 and σ′3 are major and minor principal effective stresses, respectively,

σc is the uniaxial compressive strength,

m and s are dimensionless constants empirically determined.


In terms of material properties, m is correspondent to frictional strength of the rock

(Eberhardt, 2012) and always has a positive value ranges between 0.001-25 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.

20 40 60 80 1000

20

40

60

80

100

120

140

-20

m = 7.5

m = 15

m = 30

Normal stress, σn (MPa)

Sh

ea

r str

ess, τ

n (M

Pa

)

Figure 2.2: Inclination of Hoek-Brown failure envelope as a function of m value plotted inσn − τn 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.

2.1.1 Effect of intermediate principal stress

As discussed in section 2.1, Hoek-Brown failure criterion was introduced in terms of σ1

and σ3. In other words, it is assumed that intermediate principal stress, i.e. σ2, has no

effect on rock strength, or is equal to σ3, and that is why this criterion fits on triaxial

compression test data reasonably perfect. However, many experiments have shown that


when σ2 increases to a larger value than σ3, i.e. polyaxial stress states σ1 > σ2 > σ3,

strength of rock changes.

Studies on effect of intermediate principal stress were started by conducting triaxial

tests in compression (σ2 = σ3) and tensile (σ2 = σ1) stress state by Karman (1911) and

Boker (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, σ2 has

an increasing effect on rock peak strength when it increases from σ3 to σ1.

σ2=σ

3

σ3(MPa)

σ1

(M

Pa

)

0 20015010050

σ2=σ

1

100

200

300

400

500

600

700

von Kármán(1911)

Böker (1915)

Carrara marble

Figure 2.3: Triaxial tests conducted by (Karman, 1911) and (Boker, 1915) in compression(filled circles) and tensile (open circles) (summarized by Murrell, 1963, digitized by Jimenezand 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 tri-

axial compression (σ1 > σ2 = σ3) to triaxial extension (σ1 = σ2 > σ3), the peak strength

of material increases to a maximum value, before it decreases to a value higher than that


200

1000

125

145

σ3= 0 MPa

105

Dunham dolomite

800

600

400

200 5004003001000 600

σ2

(MPa)

σ1

(M

Pa

)

σ2=σ

3

85

65

45

25

Figure 2.4: Polyaxial tests conducted by Mogi (1971) in seven different groups with σ3 varyingin the range of 0 ≤ σ3 ≤ 145 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 σ2 effect into account when fitting to the polyaxial data, is generally done in

σ1 − σ2 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 σ1−σ2 space. Figure 2.6 shows

how triaxial failure criteria behave in σ1 − σ2 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 σ1 − σ2 − σ3 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


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 bound-

ary 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 3.

500

400

300

200

100

0 100 200 300 400 500

500

400

300

200

100

0 100 200 300 400 500

σ2

σ1

(MPa)

(MP

a)

σ 1=σ 2

σ2=σ

3

σ3=0

σ3=60 MPa

σ3=30 MPa

σ3=90 MPa

σ 1=σ 2

σ2=σ

3

σ3=0

σ3=60 MPa

σ3=30 MPa

σ3=90 MPa

σc = 60 MPa

ϕ = 0.6

σc = 60 MPa

m = 16

s = 1

σ2

σ1

(MPa)

(MP

a)

(a) Mohr-Coulomb criterion (b) Hoek-Brown criterion

Figure 2.6: Sensitivity of the triaxial strength (σ1) to the intermediate stress (σ2) in triaxialcriteria (after Colmenares and Zoback, 2002).

Figure 2.7 shows how a point in principal stress space, in the form of P (σ1, σ2, σ3),

can be characterized by distance from the origin of the plane passing through P that


σ1*

σ3*

σ2*

ρ

θP

60°

σ1

σ2

σ3

o

P (σ1, σ

2, σ

3)

o’

ρ

ξ

Deviatoric plane

(σ1 + σ

2 + σ

3 = constant)

Hydrostatic axis

(σ1 = σ

2 = σ

3)

(a) Principal stress space (b) Deviatoric plane

o’

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 (σ1 = σ2 = σ3), 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 σ∗1-axis, which is

the projection of σ1-axis on the π-plane, and varies within the range of 0 < θ < π3

, which

represents the condition σ1 > σ2 > σ3. A triaxial compression state (σ1 > σ2 = σ3) is rep-

resented by θ = 0 , while θ = π3

corresponds to a triaxial extension state (σ1 = σ2 > σ3).

The parameters ξ, ρ and θ may be written in terms of stress invariants as

ξ =I13, ρ =

√2J2, θ =

1

3cos−1(

3√

3

2

J3

J3/22

) (2.2)

where I1 is the first invariant of stress tensor, while J2, and J3 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-


-√3ccotφ

θ=π/3

θ=0

ρc

ρt

ξ

ρ

ρt

ρc

ρ

θ

σ1*

σ3*

σ2*

(a) Deviatoric plane (b) Meridian plane

o’

Figure 2.8: Mohr-Coulomb failure criterion in the deviatoric and the meridian plane (fromLee 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 σ2 makes both criteria perform linearly in π-

plane in the range of 0 ≤ θ ≤ π3

(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).

-√3σcs/m

θ=π/3

θ=0

ρc

ρt

ξ

ρ

ρt

ρc

ρ

θ

σ1*

σ3*

σ2*

(a) Deviatoric plane (b) Meridian plane

o’

Figure 2.9: Hoek-Brown failure criterion in the deviatoric and the meridian plane (from Leeet al., 2012).


2.2 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 σ2 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 ini-

tially being referred here, since they are commonly used in different studies, and also

take Hoek-Brown strength parameters (m, σc), 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 (I1, J2, 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 com-

pression. 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 σ2.

Figure 2.11 summarizes the performance of all mentioned criteria in the deviatoric

plane and in σ1−σ2 space. From these plots, it can be inferred that the only criterion that


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.

2.2.1 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 polyax-

ial 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 men-

tioned failure criteria with conventional Hoek-Brown criterion, and clearly shows unlike

the other criteria, Ottosen does not necessarily reduce to Hoek-Brown in triaxial exten-

sion.

Ottosen peak strength criterion in general form of failure surfaces (Eq. 2.3) is given

as:

AJ2σ2c

+ λ

√J2σc

+BI1σc− 1 = 0 (2.4)

where λ is a function of Lode angle θ and is defined as:

λ = K1 cos(Ψ) (2.5)

and

Ψ =

1

3cos−1(K2 cos 3θ) cos 3θ ≥ 0

π

3− 1

3cos−1(−K2 cos 3θ) cos 3θ < 0

(2.6)

Now it is clear that Ottosen criterion obtains four dimensionless parameters: A,B,K1, K2,


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. σc: Uniaxial compressive strength, where σ1 ≥ σ2 = σ3 = 0;

2. σbc: Biaxial compressive strength, where σ1 = σ2 ≥ σ3 = 0;

3. σt: Uniaxial tensile strength, where σ1 = σ2 = 0 ≤ σ3;

4. An arbitrary peak strength along the compressive meridian, representing by the

point (x, y) in (I1,√J2) space.

J

I

Compressive

Meridian

Meridian

(x, y)

Biaxial

Compressive

Strength

Uniaxial

Compressive

StrengthUniaxial

Tensile

Strength

Tensile

Figure 2.10: Failure states on compressive and tensile meridians used to obtain the fourparameters 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)(σt − σbc) (2.7)

A =−3σ2

c [γ − 3(σtσbc)(y√

3

σc− 1)]

σtσbc[γ − 3y(3y − σc√

3)](2.8)

B =

[γ3y − σc

√3

y√

3− x][

1√3σtσbc − σcy]

σtσbc[γ − 3y(3y − σc√

3)](2.9)


λt =√

3(B +σcσt− A σt

3σc) (2.10)

λc = −√

3(A

3+B − 1) (2.11)

K1 =2√3

√λ2t + λ2c − λtλc (2.12)

K2 = 4(λcK1

)3 − 3λcK1

(2.13)

It appears from this formulations that A ≥ 0, B ≤ 0, K1 ≥ 0, −1 ≤ K2 ≤ 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 inter-

mediate principal stress has an increasing effect on rock peak strength.

Many polyaxial criteria has been proposed, mostly derived based on empirical meth-

ods, 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.


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.


100 300 500 700

300

500

700

σ3=0 MPa

σ3=60

σ3=30

σ3=90

σ2

σ1

(MPa)

(MP

a)

100 300 500 700

300

500

700

σ3=0 MPa

σ3=60

σ3=30

σ3=90

σ2

σ1

(MPa)

(MP

a)

100 300 500 700

300

500

700

σ3=0 MPa

σ3=60

σ3=30

σ3=90

σ2

σ1

(MPa)

(MP

a)

100 300 500 700

300

500

700

σ3=0 MPa

σ3=60

σ3=30

σ3=90

σ2

σ1

(MPa)

(MP

a)

100 300 500 700

300

500

700

σ3=0 MPa

σ3=60

σ3=30

σ3=90

σ2

σ1

(MPa)

(MP

a)

σ1*

σ3*

σ2*

o’

σ1*

σ3*

σ2*

o’

σ1*

σ3*

σ2*

o’

σ1*

σ3*

σ2*

o’

σ1*

σ3*

σ2*

o’

Pan-Hudson

Zhang-Zhu

Jiang-Xie

HB-WW

Ottosen

Figure 2.11: Comparison of different polyaxial criteria with Hoek-Brown in the deviatoricplane (left column) and in σ1−σ2 space (right column). Solid lines represent polyaxial criteria,and dashed lines represent Hoek-Brown failure criterion.

Chapter 3

Stability analysis methodology

Triaxial strength criteria and their shortcomings in prediction of rock peak strength in

polyaxial stress regime (σ1 > σ2 > σ3) were discussed in chapter 2, followed by an outline

of some polyaxial peak strength criteria along with a comparison of triaxial and polyax-

ial 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 open-

ings 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 usu-

ally 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

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(δ1, δ3)

r(3.1)

where :

r =(σ1 − σ3)

2,

δ1 = (σ1 − σ2) ,

δ3 = (σ2 − σ3).


σ

τ

σ1

σ2

σ3

O

δ1

δ3

Figure 3.2: Mohr circle in 3d stress state (after Davis and Selvadurai, 2002).

As shown in Figure 3.2, when δ1 is equal to zero means σ2 = σ1, and thus the

stress regime is triaxial extension (σ1 = σ2 > σ3), degree of polyaxiality is minimum

(α = 0). Likewise, when δ3 is equal to zero and stress condition is triaxial compression

(σ1 > σ2 = σ3), degree of polyaxiality is minimum as zero. On the other hand, maximum

degree of polyaxiality (α = 1) occurs when δ1 = δ3, i.e. σ2 =(σ1 − σ3)

2, and stress state

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.


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 (σv = σ3), 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 ten-

dency 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 withrespect to the tunnel axis

hydrostatictriaxial

extensiontriaxial

compressionpolyaxial

σv σ3 σ3 σ3 σ3σh σ3 σ1 σ1 σ1σa σ3 σ1 σ3 σ2

3.2 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 cri-

terion 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


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.

3.2.1 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 σ1 − σ3 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, σ1 on failure envelope, corresponding to a certain

σ3, indicates rock peak strength. So, Eq.3.2 for conventional Hoek-Brown failure criterion

can be written as:

S.F.2 =σ1,HB

σ1,i(3.3)

where σ1,HB is the peak strength predicted by Hoek-Brown criterion, and σ1,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 σn− τn space to estimate the strength

factor (Corkum, 1997).


σ3

σ1

induced stress

peak strength

Figure 3.3: Estimation of S.F. using Hoek-Brown failure criterion.

3.2.2 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 σ1− σ2− σ3space. 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

ρi(3.4)

where ρOT

is the maximum ρ value in Ottosen π-plane with the same Lode angle as of

induced stress, which has the distance of ρi from the origin of the π-plane. Figure 3.4

illustrates how strength factor is estimated in the π-plane.


σ2*

σ1*

σ1*

induced

stress

peak strength

ρ

Figure 3.4: Estimation of S.F. using Ottosen failure criterion.

3.2.3 Error in strength factor when using of triaxial criteria

Following discussions in sections 3.2.1 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:

% ε = 100× S.F.2 − S.F.3S.F.3

(3.5)

where S.F.2 and S.F.3 are strength factors predicted by the 2d and 3d criteria, respec-

tively. We are therefore able to show in which regions around an opening we may under-

or 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 cri-

terion 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


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 calcu-

late 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. σ1 > σ2 > σ3, 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 fail-

ure. 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.


3.3 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 inves-

tigation 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.


0.0

0

0.7

1

0.5

7

0.4

3

0.2

9

0.1

4

0.8

6

1.0

0

(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

Degree of polyaxiality (α)

Cross section A,

0.5R behind tunnel face

σ3

σ2σ1

Horizontal longitudinal

section, H

Vertical longitudinal

section, V

Figure 3.5: Degree of polyaxiality around a planar end of a circular opening (α = 1 indicatesmaximum polyaxiality)


Cross section A,

0.5R behind tunnel face


section, H


section, V

<1.0

2.6

2.2

1.8

1.4

1.0

3.0

>3.0

Str

en

gth

fa

cto

r σ

3

σ2σ

1

−67

−30

−25

−20

−15

−10

797

Err

or

in p

red

ictio

n o

f str

en

gth

fa

cto

r (%

)

>30

30

25

20

15

10

5

0

-5

-10

-15

-20

-25

-30

<-30

(a) strength factor predicted

using Hoek-Brown criterion.

(b) strength factor predicted

using Ottosen criterion.

(c) error in prediction of S.F.

when the Hoek-Brown is used.

(σ1>σ

2>σ

3) S

tre

ng

th

Un

de

restim

ate

d

(Un

eco

no

mic

de

sig

n)

Str

en

gth

Ove

restim

ate

d

(Un

sa

fe d

esig

n)

Figure 3.6: Preliminary stability analysis using (a) triaxial and (b) polyaxial strength criteriaand (c) the error in prediction.

Chapter 4

Effective use of colour for

visualization

Up until a few decades ago, colours had been rarely used in data representation. Develop-

ments 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 ran-

domly 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 characteris-

tics of each category. Here, a brief review of all is presented. Sequential and diverging

27

Chapter 4. Effective use of colour for visualization 28

Table 4.1: Data categorization and colour schemes (after Brewer, 1994).

perceptual dimension of colour

data categoryand scheme type

hue brightness

qualitativehue steps

(not ordered)constant

brightness

binary(special case for qualitative)

neutrals,one hue

or one hue step

one brightnessstep

sequentialneutrals,

one hue orhue transition

single sequence ofbrightness step

divergingtwo hues,

one hue and neutrals,or two hues transitions

two divergingsequence 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 qualita-

tive 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).


The primary interest of this study on rock stability analysis, as well as many other en-

gineering problems, deals with quantitative data, i.e. sequential and diverging categories.

Sequential data classes, such as degree of polyaxiality and strength factor here, are or-

dered 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 percent-

age 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 comple-

mentary 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 se-

quential 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.


90

80

70

60

50

40

30

20

90

80

70

60

50

40

30

20

(b) spectral scheme with hue steps and constant brightness

(Examine3D)

(a) sequential scheme with single hue and steps of brightness

and saturation

Figure 4.1: Comparison of a sequential and a spectral scheme on a same dataset.

4.1.1 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, bivari-

ate 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 se-

quential/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-


Hue 1

Hue 2Low bri.

High sat.

High bri.

Low sat.

(a) bivariate colour scheme structure.

Hue transition

Brig

htne

ss

tra

nsition

& S

atur

ation

first variable

se

co

nd

va

ria

ble

first variablemin. max.

se

co

nd

va

ria

ble

min.

max.

(b) a bivariate colour scheme with major hues: yellow and magenta.

Figure 4.2: Sequential/sequential scheme with transitional hue mixtures in major diagonaland 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).

445

H1

H2

H3

H4

ze

ro(-) (+)

min.

max.

Lo. bri.

Hi. sat.

Lo. bri.

Hi. sat.

Hi. bri.

Lo. sat.

Hi. bri.

Lo. sat.

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


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 vari-

ables 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 vari-

able, i.e. percentage error.

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

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 com-

puter 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 Chan-

dra, 2013). It has been discussed in Chapter 2 that neglecting the influence of intermedi-

ate 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 tri-

axial 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.


5.1 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 pri-

mary assumptions made by an elastic boundary element calculation is that the structure

being modelled is located in a homogeneous, isotropic, and linearly elastic medium (Cur-

ran 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).

Geometric node

Function node

(a) Constant (b) Linear (c) Quadratic

Figure 5.1: Element library of Examine3D. Accuracy and computation time increases fromleft (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-


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, el-

ements 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, ob-

tain 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).

(a) Mesh generated in the face of the tunnel

(b) Three-dimensional mesh generation

tunnel radius: 2m

# of elements: 1792

# of nodes: 898

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.


Table 5.1: parameters used in Hoek-Brown and Ottosen failure criteria

strength parameter

σc (MPa) σbc (MPa) σt (MPa) m s(x, y) = (I1,

√J2)

along thecompressive meridian

value 40 70 2.2 18 1 (496, 148)

Visualization of the data, in Examine3D, is defined by the user usually in the form

of uniform grids (Figure 5.3). A 100× 100 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

Cross section A,

0.5R behind tunnel faceσ

v

σaσ

h


section, H


section, V

R

2R

2R

(a) (b)

1.5R

1.5R

0.5R0.25R

Figure 5.4: (a) Cross sections ahead and behind tunnel face, and (b) horizontal and verticallongitudinal sections along the main axis of a circular tunnel. In this model, the tunnel radiusis set to R=2m.

5.1.1 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 (σv), horizontal (σh) and axial (σa) relative to the alignment of the tunnel main


axis (see Figure 5.4).

In each case, the minor principal in-situ stress was assumed vertical (σv = σ3), 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.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

500

1000

1500

2000

2500

3000

z =

De

pth

be

low

su

rfa

ce

(m

)

k = σh / σ

v

k = (1500 / z) + 0.5

k = (100 / z) + 0.3

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).


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 (σ1 = σ2 = σ3), and

holding σ3 = 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

1 2 3 4 5 6

hydrostatictriaxial

extensiontriaxial

compressiontriaxial

compressionpolyaxial polyaxial

σv (MPa) 30 30 30 30 30 30

σh (MPa) 30 60 30 60 45 60

σa (MPa) 30 60 60 30 60 45

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 in-

vestigate 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, % ε, 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 bound-

ary (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).


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 σ1 = σ2 = σh, ex-

cavation 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. σ1 > σ2 = σ3, however, changes in the di-

rection of the tunnel affects stress-induced instabilities dramatically. Figures 5.7.c and

5.7.d clearly show that when the tunnel advances along σ2 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 σ1, is not desirable

since it prevents the rock breaks itself (zone D1).

Same as triaxial compression condition, in polyaxial stress regime, i.e. σ1 > σ2 > σ3,

the direction of the tunnel plays an essential role in stability state around the excava-

tion. As shown in Figure 5.7.e the stability of the wall is extremely overestimated by the

triaxial criterion when the tunnel advances along σ2, and in fact Hoek-Brown does not

predict the possible failure in the wall in this case (zone E1). Whereas, the same problem


occurs immediately ahead of the working face when the tunnel axis is along σ1 and per-

pendicular to σ2 (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 σ1 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 σ3 for all cases, and is generally underestimated by

the Hoek-Brown criterion.

5.1.2 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


(c) Horizontal elliptical(b) Circular(a) Vertical elliptical

2.5 m

2.5

m

1.6

m

1.6 m

2.0

m

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 tun-

nels, 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 insta-

bility 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).


Analysis in triaxial compression (Case 3) and polyaxial (Case 5) in-situ stress con-

ditions 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 con-

centration 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.

5.1.3 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 hap-

pened where σ1 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.


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 tun-

nels, 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.


(e) Case 5: polyaxial

(a) Case 1: hydrostatic

(c) Case 3: triaxial compression (d) Case 4: triaxial compression

(f) Case 6: polyaxial

(b) Case 2: triaxial extension

0-5-10

-15

-20

-25

-30 5 10 15 20 25 30 >30

<-30

1.0

3.0

1.8

2.6

2.2

1.4

>3.0

<1.0

Predicted strength factor error, %

Sec

tion

H

Sec

tion

V

Sec

tion

A

Sec

tion

B

Sec

tion

H

Sec

tion

V

Sec

tion

A

Sec

tion

B

Sec

tion

H

Sec

tion

V

Sec

tion

A

Sec

tion

B

C.L.

C.L.

Sec

tion

H

Sec

tion

V

Sec

tion

A

Sec

tion

B

Sec

tion

H

Sec

tion

V

Sec

tion

A

Sec

tion

B

Sec

tion

H

Sec

tion

V

Sec

tion

A

Sec

tion

B

Tunnnel

A1

A3A2 B2

B1

C1 D1

E1

E2

F1

Stre

ngth

fact

or

StrengthUnderestimated

(Uneconomic design)

Strength Overestimated

(Unsafe design) Increasing Stability

Unstable

Figure 5.7: Plots of S.F. and error in prediction in four planes around the tunnel for sixassumed cases.


(a) case 1 - hydrostatic A2

A1

B1

B2

C1 C2C3

D1D2

D3D4

C4

0-5-10

-15

-20

-25

-30 5 10 15 20 25 30 >30

<-30

1.0

3.0

1.8

2.6

2.2

1.4

>3.0

<1.0

Predicted strength factor error, %

Stre

ngth

fact

or

StrengthUnderestimated

(Uneconomic design)

StrengthOverestimated

(Unsafe design) IncreasingStability

Unstable

(b) case 2 - triaxial extension

(c) case 3 - triaxial compression

(d) case 5 - polyaxial

Figure 5.8: Comparison of S.F. and error in prediction in a vertical elliptical, a circular, anda horizontal elliptical excavation.

Chapter 6

Conclusions

The effect of intermediate principal stress on rock strength has been shown to be sig-

nificant by numerous experiments. With the conventional triaxial criteria neglecting the

influence of σ2, several polyaxial criteria has been proposed. None of those criteria, how-

ever, guarantee to perform well in all stress conditions and in different materials.

Five different polyaxial criteria were selected to make a brief comparison, and show

that there are advantages and disadvantages with the use of each. Among those, Ottosen

failure criterion is shown to perform more accurately, since it does not reduce to the

Hoek-Brown in triaxial extension regime, as suggested by polyaxial test data. However,

complication of determining parameters can make it difficult to use.

Prediction of stress-induced instabilities around an underground excavation is usually

done by determining the ratio of ultimate allowable stress to current induced stress, i.e.

strength factor. The value of strength factor is conventionally calculated in 2-dimensional

space using a simple triaxial criterion, while it does not seem to be an appropriate method

in a 3-dimensional model.

To investigate the error in prediction of strength factor associated with the use of

a triaxial criterion, the ratio of peak strength to induced stress must be assessed in 3-

dimensional space, and a polyaxial criterion in needed as a true indicator of rock strength.

47

Chapter 6. Conclusions 48

So, not only the failure criterion must be chosen appropriately, but also the state of stress

has to be assessed in σ1 − σ2 − σ3 space.

A bivariate colour scheme was used to effectively show the accurate estimation of

strength factor and the error of a triaxial criterion at the same time. This technique is

shown to be very successful to draw the attention to unstable regions that have also large

error.

The effect of in-situ stress state was examined assuming six different conditions. Ma-

jor changes are observed when the stress state deviates from hydrostatic to triaxial and

polyaxial regimes. The substantial influence of tunnel direction with respect to the prin-

cipal stress state were addressed, and a significant overestimation in the wall is shown to

be related to the direction of σ1 in triaxial compression and polyaxial stress states. This

zone of overestimation, that may result in an unsafe design, shows a significant reduction

in a horizontal elliptical tunnel.

The error in prediction is shown to be always significant in the proximity of the

boundary where the stress is extremely disturbed. Keeping in mind that those are the

regions with the lowest strength factor, substantial problems may arise from use of tri-

axial criteria.

6.1 Recommendations

While a limited number of situations were examined here, infinite various conditions can

be analyzed drawing different conclusions, but the influence of the intermediate principal

stress in analysis is undeniable. This study clearly shows that using conventional 2d cri-

teria is an important shortcoming of numerical modelling software. A simple approach

needs to be introduced to incorporate the polyaxial state of stress into numerical methods.

Chapter 6. Conclusions 49

A great number of polyaxial criteria have been proposed, while no one meet all nec-

essary conditions and have the ease of applicability. Development of a criterion that

can be generally accepted in rock engineering requires better understanding of material

characteristics and failure mechanism.

It is also suggested that a systematic data visualization is a necessity in developments

of software suits. In this study, it has been shown that how an appropriate colour scheme

can improve the visualization of data and thus, results in a sound interpretation. These

techniques, however, have not been developed for the particular purpose of geomechanics

and thus, have a limited application.

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50

BIBLIOGRAPHY 51

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Appendix: MATLAB script

1 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

2 %−−−−− This code loads the data obtained from EXAMINE3D, −−−−%

3 %−−−− c a l c u l a t e s the s t a b i l i t y parameters and v i s u a l i z e s −−−−%

4 %−−−−− the r e s u l t s , us ing a b i v a r i a t e co l our scheme . −−−−−−−−%

5 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

6 %−−−−−−−−−−−−−−− By : Roozbeh Roostae i 2014 −−−−−−−−−−−−−−−−−−%

7 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

8

9 load sigma 1 ; % load data s h e e t s from d i r e c t o r y

10 load sigma 2 ;

11 load sigma 3 ;

12

13 s1 = sigma 1 ; % save the data in to matr i ce s

14 s2 = sigma 2 ;

15 s3 = sigma 3 ;

16

17 % s t r u c t u r e o f s t r e s s matr i ce s with 5 columns :

18 %∗|−−−|−−−−−−−−−−−−|−−−−−−−−−−−−|−−−−−−−−−−−−|−−−−−−−−−|

19 %∗ | # | North (X) | Up (Y) | East (Z) | value |

20 %∗|−−−|−−−−−−−−−−−−|−−−−−−−−−−−−|−−−−−−−−−−− |−−−−−−−−−|

21

22 m = 18 ; % HB parameter m

23 s = 1 ; % HB parameter s

24 sc = 40 ; % u n i a x i a l compress ive s t r ength

25 sbc = 70 ; % b i a x i a l compress ive s t r ength

26 s t = (−0.5)∗ sc ∗(m−s q r t ( (mˆ2)+4) ) ; % t e n s i l e s t r ength

27 s3xy = 80 ; % a r b i t r a r y sigma 3 on comp . meridian ( Ottosen )

54

APPENDIX 55

28 s1xy = 336 ; % a r b i t r a r y sigma 1 on comp . meridian ( Ottosen )

29 numu = 101 ; % # of p i x e l s (H d i r = # of c e l l s in u d i r + 1 )

30 numv = 102 ; % # of p i x e l s (V d i r = # of c e l l s in v d i r + 1 )

31 N = 7 ; % # of c o l o u r s on each a x i s

32 %%

33 %−−−−TRIAXIAL−−−−%

34 s1 hb = s q r t ( (m .∗ sc .∗ ( s3 ( : , 5) ) + s ∗ sc ˆ2) ) + s3 ( : , 5) ; %st r ength −− HB

35 s t rength hb2 ( : , 1 : 4 ) = s1 ( : , 1 : 4 ) ; %s e t other columns

36 s t rength hb2 ( : , 5) = ( s1 hb ) . / ( s1 ( : , 5) ) ; %s t r ength f a c t o r −− HB

37 s t rength hb2 ( s t rength hb2 == − i n f ) = 0 ; %p i x e l s in excavat ion area

38 s t rength hb2 ( s t rength hb2 == i n f ) = 0 ; %p i x e l s in excavat ion area

39

40 %−−−−POLYAXIAL−−−−%

41 I1 = s1 ( : , 5)+s2 ( : , 5)+s3 ( : , 5) ; % s t r e s s i n v a r i a n t s

42 I2 = s1 ( : , 5) .∗ s2 ( : , 5) + s2 ( : , 5) .∗ s3 ( : , 5) + s3 ( : , 5) .∗ s1 ( : , 5) ;

43 I3 = s1 ( : , 5) .∗ s2 ( : , 5) .∗ s3 ( : , 5) ;

44 J2 = (1/6) ∗ ( ( s1 ( : , 5) − s2 ( : , 5) ) . ˆ2 + ( s2 ( : , 5) − s3 ( : , 5) ) . ˆ2 + ( s3 ( : , 5)

− s1 ( : , 5) ) . ˆ 2 ) ;

45 J3 = ((2/27) ∗ I1 . ˆ 3 ) − ( ( 1/3 ) ∗( I1 .∗ I2 ) ) + I3 ;

46 rho ex = s q r t (2∗ J2 ) ; % dstnce o f induced s t r e s s from o r i g i n o f pi−plane

47

48 x = s1xy+ (2∗ s3xy ) ; % I1 f o r a r b i t r a r y s t r e s s s t a t e

49 y = ( s1xy − s3xy ) /( s q r t (3 ) ) ; % J2 f o r a r b i t r a r y s t r e s s s t a t e

50

51 gamma = ( ( y∗ s q r t (3 ) )−x ) ∗( st−sbc ) ; % Ottosen parameters

52 A = ((−3∗( sc ˆ2) ) ∗(gamma−(3∗ s t ∗ sbc ∗ ( ( ( y∗ s q r t (3 ) ) / sc )−1) ) ) ) /( s t ∗ sbc ∗(gamma

+(3∗y ∗ ( ( sc ∗ s q r t (3 ) )−(3∗y ) ) ) ) ) ;

53 B = ( (gamma/( ( y∗ s q r t (3 ) )−(x ) ) ) ∗ ( ( sc ∗y ∗ ( ( sc ∗ s q r t (3 ) )−(3∗y ) ) )−( s t ∗ sbc ∗( sc−(y∗

s q r t (3 ) ) ) ) ) ) /( s t ∗ sbc ∗(gamma+(3∗y ∗ ( ( sc ∗ s q r t (3 ) )−(3∗y ) ) ) ) ) ;

54 lamda t = ( s q r t (3 ) ) ∗(B+(sc / s t )−((A∗ s t ) /(3∗ sc ) ) ) ;

55 lamda c = (− s q r t (3 ) ) ∗ ( (A/3)+B−1) ;

56 K1 = (2/ s q r t (3 ) ) ∗( s q r t ( ( lamda c ˆ2)−(lamda c∗ lamda t )+(lamda t ˆ2) ) ) ;

57 K2 = ( 4∗ ( ( lamda t /K1) ˆ3) )−(3∗( lamda t /K1) ) ;

58

59 c o s 3 t he ta = cos (3∗ theta ) ;

60 lamda = ze ro s ( l ength ( co s 3 the ta ) ,1 ) ;

APPENDIX 56

61 f o r n = 1 : l ength ( c o s 3 t he ta )

62 i f c o s 3 t he ta (n) >= 0

63 lamda (n , 1 ) = K1∗ cos ( (1/3 ) ∗ acos ((−K2∗ c o s 3 the ta (n) ) ) ) ;

64 e l s e

65 lamda (n , 1 ) = K1∗ cos ( ( p i /3) −((1/3)∗ acos (K2∗ c o s 3 the ta (n) ) ) ) ;

66 end

67 end

68

69 s q r t J 2 = ( ( sc ˆ2) /(2∗A) ) ∗((− lamda/ sc )+s q r t ( ( ( lamda/ sc ) . ˆ 2 ) −(4∗(A/( sc ˆ2) )

. ∗ ( ( (B.∗ I1 ) / sc )−1) ) ) ) ;

70 rho OT = s q r t (2 ) ∗ s q r t J 2 ; % dstnce o f peak s t r ength from o r i g i n o f pi−plane

71

72 s t r e n g t h o t ( : , 1 : 4 ) = s1 ( : , 1 : 4 ) ;

73 s t r e n g t h o t ( : , 5) = abs ( rho OT ) . / abs ( rho ex ) ; % st r ength f a c t o r −− OT

74 s t r e n g t h o t ( any ( i snan ( s t r e n g t h o t ) ,2 ) , 5 ) =0;

75

76 %Error in p r e d i c t i o n

77 e r r ( : , 5) = ( ( s t rength hb2 ( : , 5) − s t r e n g t h o t ( : , 5) ) ∗ (100) ) . / s t r e n g t h o t

( : , 5) ; % c a l c u l a t e the e r r o r

78 e r r ( : , 1 : 4 ) = s1 ( : , 1 : 4 ) ;

79 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

80 % ∗ i f a c a r t e s i a n coord inate system (X,Y, Z) , then X=NORTH,Y=UP, Z=EAST

81 % ∗ numu = number o f plane c e l l s in the U d i r e c t i o n (P0−>P1)

82 % ∗ numv = number o f plane c e l l s in the V d i r e c t i o n (P0−>P3)

83 % ∗ ˆ V

84 % ∗ |

85 % ∗

86 % ∗ 3−−−−2

87 % ∗ | |

88 % ∗ | |

89 % ∗ 0 1 −> U

90 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

91 % ∗ PLANE DEFINITION

92 % ∗|−−−−−−−−−−−−−|−−−−−|−−−−−|

93 % ∗ | P0 P1 P2 P3 |numu |numv |

94 % ∗|−−−−−−−−−−−−−|−−−−−|−−−−−|

APPENDIX 57

95 % 0 0 19

96 % 6 0 19

97 % 6 6 19

98 % 0 6 19

99 % 100 101

100 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

101

102 % Plane A

103 e r r o r 1 = e r r (3∗numu∗numv+1:4∗numu∗numv , 5) ; % e r r o r va lue s f o r plane A

104 e r r o r 1 = reshape ( e r ro r 1 , numu, numv) ; % reshape the matrix to

image s i z e (h∗v )

105 SF3 1 = s t r e n g t h o t (3∗numu∗numv+1:4∗numu∗num, 5) ; % SF va lue s f o r plane A

106 SF3 1 = reshape ( SF3 1 , numu, numv) ;

107 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

108 % Plane B

109 e r r o r 2 = e r r (2∗numu∗numv+1:3∗numu∗numv , 5) ;

110 e r r o r 2 = reshape ( e r ro r 2 , numu, numv) ;

111 SF3 2 = s t r e n g t h o t (2∗numu∗numv+1:3∗numu∗numv , 5) ;


113 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

114 % Plane H

115 e r r o r 3 = e r r ( 1 :numu∗numv , 5) ;


117 SF3 3 = s t r e n g t h o t ( 1 :numu∗numv , 5) ;


119 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

120 % Plane V

121 e r r o r 4 = e r r (numu∗numv+1:2∗numu∗numv , 5) ;


123 SF3 4 = s t r e n g t h o t (numu∗numv+1:2∗numu∗numv , 5) ;


125 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

126 % d e f i n e ranges and c l a s s e s o f data

127 e r r r n g = [ min ( e r r ( : , 5 ) ) max( e r r ( : , 5 ) ) ] ; % range o f e r r o r

128 i n t 1 = ( q u a n t i l e ( e r r ( : , 5 ) , 0 . 9 8 ) − 0) /(N−1) ; % s e t 0 .02 f o r out o f

range − d iv id e p o s i t i v e e r r o r range

APPENDIX 58

129 i n t 2 = (0 − q u a n t i l e ( e r r ( : , 5 ) , 0 . 0 2 ) ) /(N−1) ; % s e t 0 .02 f o r out o f

range − d iv id e negat ive e r r o r range

130 y = ze ro s (2 ,N+1) ; % array o f d iv ided range f o r e r r o r

131 y (1 , 1 ) = e r r r n g (1 ) ; % minimum e r r o r

132 y (1 , 2 ) = q u a n t i l e ( e r r ( : , 5 ) , 0 . 0 2 ) ; % s t a r t po int o f the range

133 y (1 ,N+1) = 0 ; % mid−range zero

134

135 f o r i =2:N;

136 y (1 , i +1) = y (1 , i )+in t2 ; % s e t other va lue s with in the range − negat ive

137 end

138

139 f o r i=N+1:2∗N−1;

140 y (1 , i +1) = y (1 , i )+in t1 ; % s e t other va lue s with in the range − p o s i t i v e

141 end

142

143 y (1 ,2∗N+1) = e r r r n g (2 ) ; % end po int o f the range

144

145 SF3 rng = [ min ( s t r e n g t h o t ( : , 5 ) ) max( s t r e n g t h o t ( : , 5 ) ) ] ; % range o f SF

146

147 i n t 3 = ( q u a n t i l e ( s t r e n g t h o t ( : , 5 ) , 0 . 9 8 )−1)/(N−1) ; % d iv id e the range

148 x = ze ro s (1 ,N+1) ; % array o f d iv ided range f o r SF

149

150 x (1 , 1 ) = SF3 rng (1 ) ; % s t a r t po int o f the range

151 x (1 , 2 ) =1; % s e t 1 as c r i t i c a l po int

152

153 f o r i =2:N;

154 x (1 , i +1) = x (1 , i )+in t3 ; % s e t other va lue s with in the range

155 end

156

157 %−−−−−−−−−−−−−−−−−−−−−− COLOUR SCHEME −−−−−−−−−−−−−−−−−−−−−−−%

158 subplot ( 2 , 3 , [ 1 , 6 ] )

159 rgb1= twovar (N,50 , 240 ) ;

160

161 f o r i =1:3

162 rgb1 ( : , : , i ) = imrotate ( rgb1 ( : , : , i ) ,−90) ;

163 end

APPENDIX 59

164 %

165 f o r i =1:3

166 rgb1 ( : , : , i ) = f l i p l r ( rgb1 ( : , : , i ) ) ;

167 end

168

169 rgb2= twovar RR (N, 5 0 , 0 ) ;

170

171 f o r i =1:3

172 rgb2 ( : , : , i ) = f l i p u d ( rgb2 ( : , : , i ) ) ;

173 end

174

175 rgb = ze ro s (2∗N, N, 3) ;

176

177 f o r i =1:3

178 rgb ( 1 :N, 1 :N, i ) = rgb1 ( 1 :N, 1 :N, i ) ;

179 rgb (N+1:2∗N, 1 :N, i ) = rgb2 ( 1 :N, 1 :N, i ) ;

180 end

181

182 rgb ( : ,N+1 , :) =0.5 ;

183

184 [ IND, rgb ] = rgb2ind ( rgb , ( 2∗N∗N)+1) ;

185 colormap ( rgb ) ;

186

187 image (IND)

188

189 s e t ( gca , ’ xLim ’ , ( [ . 5 N+0.5 ] ) )

190 a x i s on

191

192 f o r i =1:2∗N+1

193 t ex t (0 , 0.5+ i −1, num2str ( y (1 , i ) , ’ %2.0 f ’ ) ) ;

194 end

195

196 f o r i =1:N+1

197 t ex t (0.5+ i −1, 2∗N+1.7 , num2str ( x (1 , i ) , ’ %2.2 f ’ ) , ’ r o t a t i o n ’ , 90) ;

198 end

199

APPENDIX 60

200 s e t ( gca , ’ YTick ’ , [ ] , ’ XTick ’ , [ ] ) ;

201

202

203 %−−−−−−−−−−−−−−−−−−−−−−−−− PLANE 1 −−−−−−−−−−−−−−−−−−−−−−−−−−%

204 f i g u r e ( ’Name ’ , ’ Error−SR3 ’ )

205 subplot ( 2 , 3 , 1 )

206

207 img1 = ze ro s (numu, numv , ’ u int8 ’ ) ; % c r e a t e an 8−b i t image −

numu∗numv p i x e l s

208

209 f o r i = 1 :N

210 f o r j = 1 :2∗N

211 conds = SF3 1>=x (1 , i ) & e r ro r 1>=y (1 , j ) ; % a s s e s s each p i x e l f o r

SF an e r r o r

212 img1 ( conds ) = IND ( j , i ) ; % a s s i g n the r e l a t e d co l our

213 end

214 end

215 cond0 = e r r o r 1==0 & SF3 1==0; % f i n d excavat ion area

216 img1 ( cond0 ) = 0 ; % a s s i g n grey to excavat ion area

217

218 %Convert IND to RGB

219 img1 = ind2rgb ( img1 , rgb ) ; %Convert IND to RGB

220 img1 = imrotate ( img1 , 9 0 ) ; %r o t a t e image 90 CCW

221 imshow ( img1 )

222 t i t l e ( s p r i n t f ( ’ Cutting plane 1/2 rad iu s \n back from tunne l f a c e ’ ) , ’

FontSize ’ , 8)

223

224 %−−−−−−−−−−−−−−−−−−−−−−−−− PLANE 2 −−−−−−−−−−−−−−−−−−−−−−−−−−%

225 subplot ( 2 , 3 , 2 )

226

227 img2 = ze ro s (numu, numv , ’ u int8 ’ ) ;

228

229 x (2 , 1 ) = SF3 rng (1 ) ;

230

231 f o r i = 1 :N

232 f o r j = 1 :2∗N

APPENDIX 61

233 conds = SF3 2>=x (1 , i ) & e r ro r 2>=y (1 , j ) ;

234 img2 ( conds ) = IND ( j , i ) ;

235 end

236 end

237 cond0 = e r r o r 2==0 & SF3 2==0;

238 img2 ( cond0 ) = 0 ;

239

240 img2 = ind2rgb ( img2 , rgb ) ;


242 imshow ( img2 )

243 t i t l e ( s p r i n t f ( ’ Cutting plane 1/4 rad iu s \n in to unexcavated rock ’ ) , ’

FontSize ’ , 8)

244

245 %−−−−−−−−−−−−−−−−−−−−−−−−− PLANE 3 −−−−−−−−−−−−−−−−−−−−−−−−−−%

246 subplot ( 2 , 3 , 4 )


248 y (2 , 1 ) = e r r r n g (1 ) ;

249 x (2 , 1 ) = SF3 rng (1 ) ;

250

251 f o r i = 1 :N

252 f o r j = 1 :2∗N


254 img3 ( conds ) = IND ( j , i ) ;

255 end

256 end

257 cond0 = e r r o r 3==0 & SF3 3==0;

258 img3 ( cond0 ) = 0 ;

259



262 imshow ( img3 )

263 t i t l e ( s p r i n t f ( ’Top view ’ ) , ’ FontSize ’ , 8)

264

265 %−−−−−−−−−−−−−−−−−−−−−−−−− PLANE 4 −−−−−−−−−−−−−−−−−−−−−−−−−−%

266 subplot ( 2 , 3 , 5 )


APPENDIX 62

268 y (2 , 1 ) = e r r r n g (1 ) ;

269 f o r i =1:N;

270 y (2 , i +1) = y (2 , i )+in t1 ;

271 end

272

273 x (2 , 1 ) = SF3 rng (1 ) ;

274

275 conds = ze ro s (N,N) ;

276

277 f o r i = 1 :N

278 f o r j = 1 :2∗N


280 img4 ( conds ) = IND ( j , i ) ;

281 end

282 end

283 cond0 = e r r o r 4==0 & SF3 4==0;

284 img4 ( cond0 ) = 0 ;



287 imshow ( img4 )

288 t i t l e ( s p r i n t f ( ’ S ide view ’ ) , ’ FontSize ’ , 8)

APPENDIX 63

1 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

2 %−−−−−− This func t i on produces b i v a r i a t e co l our scheme −−−−−−%

3 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

4 %−−−−−−−−−−−−−− Modif ied a f t e r J .P. Harr i son −−−−−−−−−−−−−−−−−%

5 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

6

7 f unc t i on rgb = twovar (N, hs ,hm)

8

9 n = N; % number o f c o l o u r s

10 h=ze ro s (n , n) ; % hue

11 s=ze ro s (n , n) ; % s a t u r a t i o n

12 v=ze ro s (n , n) ; % b r i g h t n e s s

13 r=ze ro s (n , n) ; % red

14 g=ze ro s (n , n) ; % green

15 b=ze ro s (n , n) ; % blue

16 RGB=ze ro s (n∗n , 3 ) ;

17 rgb=ze ro s (n , n , 3 ) ;

18

19 l i n e a r = 1 ;

20 hspread = hs /360 ;

21 hspread = mod( hspread , 0 . 5 ) ; % upper l i m i t i s 1/2 o f e n t i r e hue range

22 hmiddle = hm/360 ;

23

24 i f ( l i n e a r ==1)

25 % compute l i n e a r hues based on d i s t anc e from l ead ing d iagona l

26 t = hspread /(n−1) ;

27 f o r j =1:n

28 f o r i =1:n

29 h( i , j ) = mod( 1 . 0 + hmiddle + t ∗( i−j ) , 1 ) ;

30 end

31 end

32 e l s e

33 % compute s i n u s o i d a l hues based on d i s t anc e from l ead ing d iagona l

34 s c a l e = 2 .0∗ atan ( hspread ) /(n−1) ;

35 f o r j =1:n

36 f o r i =1:n

APPENDIX 64

37 t = s i n ( s c a l e ∗( i−j ) ) ;

38 h( i , j ) = mod(1.0+ hmiddle+(t / 6 . 0 ) , 1 . 0 ) ;

39 end

40 end

41 end

42

43

44 % compute s a t u r a t i o n us ing l oga r i thmi c p r o g r e s s i o n

45 satmin = 0 . 0 5 ; % min f o r l e ad ing d iagona l

46 satmax = 0 . 7 0 ; % max f o r l e ad ing d iagona l

47 s (1 , n ) = 0 . 8 0 ; % max f o r i n d i v i d u a l v a r i a b l e s

48 t = log ( satmin ) ; % bu i ld l e ad ing d iagona l

49 i n c = ( log ( satmax )−t ) /(2∗n−2) ;

50 f o r i =1:n

51 s ( i , i ) = exp ( t + inc ∗(2∗ i −2) ) ;

52 end

53 i n c = ( log ( s (1 , n ) )−t ) /(2∗n−2) ; % bu i ld top row

54 f o r j =1:n

55 s (1 , j ) = exp ( t + inc ∗(2∗ j−2) ) ;

56 end

57 f o r j =3:n % bu i ld each column

58 t = log ( s (1 , j ) ) ;

59 i n c = ( log ( s ( j , j ) )−t ) /( j−1) ;

60 f o r i =1: j

61 s ( i , j )= exp ( t + inc ∗( i −1) ) ;

62 s ( j , i ) = s ( i , j ) ; % r e f l e c t f o r other h a l f

63 end

64 end

65 s ( 2 , 1 ) = s (1 , 2 ) ; % copy as c o l 2 not analysed

66

67 % compute b r i g h t n e s s us ing l i n e a r p r o g r e s s i o n

68 % in d i r e c t i o n o f l e ad ing d iagona l

69 brimax = 1 . 0 0 ; % value f o r b (1 , 1 )

70 brimin = 0 . 7 5 ; % value f o r b(n , n)

71 i n c = ( brimax−brimin ) /(2∗n−2) ;

72 f o r j =1:n

APPENDIX 65

73 f o r i=j : n

74 v ( i , j ) = brimax−i n c ∗( i+j−2) ;

75 v ( j , i ) = v ( i , j ) ;

76 end

77 end

78

79 h = reshape (h , n∗n , 1 ) ;

80 s = reshape ( s , n∗n , 1 ) ;

81 v = reshape (v , n∗n , 1 ) ;

82

83 RGB = hsv2rgb ( [ h , s , v ] ) ;

84

85 h = reshape (h , n , n) ;

86 s = reshape ( s , n , n ) ;

87 v = reshape (v , n , n) ;

88

89 r = reshape (RGB( : , 1 ) ,n , n ) ;

90 g = reshape (RGB( : , 2 ) ,n , n ) ;

91 b = reshape (RGB( : , 3 ) ,n , n ) ;

92

93 r = rot90 ( r , 2 ) ;

94 g = rot90 ( g , 2 ) ;

95 b = rot90 (b , 2 ) ;

96

97 rgb ( : , : , 1 )=r ( : , : ) ;

98 rgb ( : , : , 2 )=g ( : , : ) ;

99 rgb ( : , : , 3 )=b ( : , : ) ;

100

101 rgb ( : , : , 1 )=rgb ( : , : , 1 ) ;

102 rgb ( : , : , 2 )=rgb ( : , : , 2 ) ;

103 rgb ( : , : , 3 )=rgb ( : , : , 3 ) ;

104

105

106 re turn

quantifying the error associated with the use of triaxial rock … · · 2015-04-24stress state...

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