Evaluation of influence from matedness on the peak shear strength of natural rock joints

IN

DEGREE PROJECT CIVIL ENGINEERING AND URBAN MANAGEMENT,

SECOND CYCLE, 30 CREDITS ,

STOCKHOLM SWEDEN 2019

Evaluation of influence from

matedness on the peak shear

strength of natural rock joints

EMIL ANDERSSON

Evaluation of influence from matedness

on the peak shear strength of natural

rock joints

By

EMILANDERSSON

Master thesis in Rock Mechanics, 30 credits

KTH ROYALINSTITUTE OF TECHNOLOGY INSTOCKHOLML

KTH Royal Institute of Technology

School of Architecture and the Built Enviroment Department of Civil and Architectural Engineering Division of Soil and Rock Mechanics

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In Sweden, the rock mass is commonly used for construction of tunnels and caverns. The rock mass is also used as a foundation for large structures such as bridge abutments and dams. For these structures, the understanding of the rock mechanical properties play a key role for reaching an acceptable safety level and minimizing cost. One of the properties that has a high uncertainty is the shear strength of rock joints. These rock joints constitute the weakest link in the rock mass and often govern it´s strength. The uncertainty lies in the amount of factors that affect the shear strength such as the degree of weathering, the matedness, the roughness of the surface and the scale. Various authors have tried to develop a failure criterion that can predict the peak shear strength of rock joints and takes into account the influence of the various factors.

The aim of this thesis is to evaluate the ability of the newly developed Casagrande et al. criterion to determine the peak shear strength for perfectly mated and natural rock joints with different degrees of matedness. All samples analyzed in this thesis have been scanned and customized to run in the programmed version of the Casagrande et al. criterion. This iterative process will stop as the application reach the apparent dip angle where the total shearing force is smaller than the total sliding force. This angle combined with the basic friction angle gives the peak friction angle for calculations of the peak shear strength.

The result show that the Casagrande et al. criterion can predict the peak shear strength for perfectly mated joint. However, for the natural rock joint, as the degree of matedness decreases, the accuracy of the prediction of the peak shear strength decreases. The conclusion of this study is that the Casagrande’s criterion cannot determine the peak shear strength of natural rock joints and that further development of the Casagrande et al. criterion is needed taking this parameter into account.

Keywords

Rock joints, Peak shear strength, Matedness, Casagrande et al. criterion, Peak friction angle, Shear strength

S

I Sverige är berg ett vanligt material för byggande av tunnlar och bergrum. För dessa konstruktioner spelar bergegenskaperna en nyckelroll för att nå en acceptabel säkerhetsnivå och minimera kostnaden. En av de egenskaper som har stor osäkerhet är skjuvhållfastheten för bergsprickor. Osäkerheten ligger i de många faktorer som påverkar skjuvhållfastheten, såsom graden av vittring, passning, ytans råhet och skala. Olika författare har försökt att anpassa ett brottkriterium för bergsprickor som tar hänsyn till faktorernas inflytande och som kan användas till att uppskatta den maximala skjuvhållfastheten.

Syftet med detta examensarbete är att utvärdera förmågan hos det nyligen utvecklade brottkriteriet av Casagrande et al. att bestämma den maximala skjuvhållfastheten för perfekt passade sprickor och naturliga sprickor med olika grad av passning. Alla prover i detta arbete har skannats in och anpassats för att köras i den programmerade algoritmen som beräknar den maximala skjuvhållfastheten enligt kriteriet av Casagrande et al.. Kriteriet använder sig av en iterativ process som pågår tills algoritmen når den vinkel där den totala skjuvkraften är mindre än den totala glidkraften. Denna vinkel kombinerad med sprickans basfriktionsvinkeln ger den maximala friktionsvinkeln för beräkning av skjuvhållfastheten.

Resultaten visar att Casagrande et al. kan förutspå den maximala skjuvhållfastheten för perfekt passade sprickor. När passningsgraden minskar för naturliga bergsprickor minskar kri-teriets förmåga att prediktera den maximala skjuvhållfastheten. Slutsatsen från detta arbete är att kriteriet av Casagrande et al. kan prediktera skjuvhållfastheten för perfekt passade sprickor men saknar förmågan att beakta inverkan från passning, vilket leder till att skjuvhållfastheten överskattas om kriteriet användas på naturliga sprickor som inte är perfekt passade. Fortsatt forskning krävs för att vidareutveckla kriteriet så att graden av passning kan beaktas.

Nyckelord

Bergssprickor, Maximal skjuvhållfasthet, Passning, Casagrande et al. kriterium, Friktionsvinkel, Skjuvhållfasthet

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The research presented in this master thesis was carried out from January to June 2019 at the Division of Soil and Rock Mechanics, Department of Civil and Architectural Engineering, at KTH Royal Institute of Technology in Stockholm, Sweden. The project was intiated by Assoc Prof Fredrik Johansson and supervised by Ph.D. student Francisco Ríos-Bayona.

First of all, I would like to express my gratitude and thankfulness towards Assoc Prof Fredrik Johansson and Ph.d Student Francisco Ríos-Bayona for their guidance, advice, support and valuable discussions throughout the project.

I would also like to thank to Martin Stigsson at SKB, for providing the algorithm for the Casagrande criterion application, his advice and valuable discussions of the outcome of the application.

Finally, I would to thank my sister Ida for helping me with proof reading the thesis.

Stockholm, June 2019

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2 Literature study: Peak shear strength of rock joints 5 2.1 Introduction . . . 5

2.2 Patton . . . 6

2.2.1 Introduction . . . 6

2.2.2 Experimental test . . . 6

2.2.3 Influence from geological composition . . . 8

2.2.4 Bi-linear shear failure envelope . . . 8

2.3 Barton and Choubey . . . 9

2.3.1 Shear failure criterion . . . 9

2.3.2 Joint roughness profiles . . . 10

2.3.3 Scale dependency . . . 12

2.3.4 Joint matching coefficient . . . 12

2.4 Grasselli . . . 12

2.4.1 Introduction . . . 12

2.4.2 Asperity Geometry . . . 13

2.4.3 Contact area . . . 13

2.4.4 Shear failure criterion . . . 14

2.5 Johansson & Stille . . . 15

2.5.1 Introduction . . . 15

2.5.2 Shear failure criterion . . . 16

TABLE OF CONTENTS

3 Method, Design and Implementation 21

3.1 Introduction . . . 21

3.2 Prestudy of the Casagrande et al. criterion . . . 21

3.2.1 Introduction . . . 21

3.2.2 Shear failure criterion . . . 21

3.3 Implementation . . . 27

3.3.1 Rock joint samples . . . 27

3.3.2 Scanning . . . 28 3.3.3 Shear test . . . 29 3.3.4 Model setup . . . 29 4 Results 33 4.1 Introduction . . . 33 4.2 Scanning . . . 33 4.3 Input Data . . . 36

4.4 Results: Peak friction angle . . . 36

5 Discussion 41

6 Conclusions and suggestions for future work 43

References 45

A Natural Rock Joint Samples 49 B Calculations Rock Properties 53

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Greek Letters

α Inclination of the saw tooth irregularities by Euler [deg] Patton (1966) βa p p_i Apparent dip angle [deg] (Casagrande et al., 2018)

β∗ _{Final apparent dip angle [deg] (Casagrande et al., 2018)} φb Basic friction angle [deg] (Patton, 1966)

φf Intact friction angle of intact rock (Mohr-Coulomb) [deg] φR Residual friction angle [deg]

φ0

r Residual friction angle standard displacement of 5 mm [deg] µ Friction coefficient [-]

σc Compressive strength [MPa] σl ocal_i Local vertical normal stress [MPa] σn Effective normal stress [MPa] σt Tensile strength of intact rock [MPa] σti Flexural strength intact rock

σu.c Unconfined compression strength of unweathered rock [MPa] τ Shear strength [MPa]

τp Peak shear strength

Θ Dip angle [deg] (Grasselli, 2001) Θ∗

max Maximum apparent dip angle with respect to shear direction [deg] (Grasselli, 2001) Θ∗ _{Apparent dip angle [deg] (Grasselli, 2001)}

Θ∗

LIST OF SYMBOLS

Roman Letters

A0 The maximum potential contact area ratio for specified direction ratio [-] (Grasselli, 2001) Ac Potential contact ratio [-] (Grasselli, 2001)

Ac.p True contact area [-] Johansson (2016)

Ai Active faces area [m2] (Casagrande et al., 2018)

A_{i p} Projected active faces area [m2] (Casagrande et al., 2018) a0, b0, x0 Surface best fitting coefficient [-] (Grasselli, 2001)

B Roughness parameter [-] (Grasselli, 2001)

C Roughness parameter (Grasseli (Grasselli, 2001)) c Cohesion of intact rock [MPa]

c_x Cohesion after the teeth has been sheared of (Patton, 1966)[N] ˆ

d Shearing resistance vector

Fmacro Total vertical force [N] (Casagrande et al., 2018) f_{l ocal_i} Local vertical force [N] (Casagrande et al., 2018) fpeak Peak shear force [N] (Casagrande et al., 2018) f_residual Residual shear force [N] (Casagrande et al., 2018)

fshear_i Local asperity shearing force [N] (Casagrande et al., 2018) f_{sl id in g_i} Local asperity sliding force [N] (Casagrande et al., 2018) has p Height of an asperity [m] (Johansson and Stille, 2014)

H Hurst exponent [-]

i Inclination angle of the asperities by Patton (Patton, 1966) [deg] in Dilation angle [deg] (Johansson, 2016)

JCS Joint wall compressive strength [MPa] (Barton and Choubey, 1977) J MC Joint matching coefficient [-] (Zhao, 1997)

JRC Joint roughness coefficient [-] (Barton and Choubey, 1977) k Empirical constant (Johansson and Stille, 2014) [-]

L Joint length [m]

L0 Specimen length [m] (Barton and Bandis, 1982) Las p Length of an asperity [m] (Johansson and Stille, 2014) L_c Critical length of joint [m](Barton, 1973)

Ln Block length along joint [m] (Barton and Bandis, 1982)

N Normal force [N]

N_{c f} Total number of contributing faces [-] ˆ

n_i Unit vector normal to face i

S Shear force [N]

ˆs Unit vector indicating the shear direction in the discontinuity plane. ˆt Shear direction vector

x, y, z Coordinates [m] ˆ

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1

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1.1

Background

Many of Sweden’s large infrastructures have rock as a foundation or are constructed within it. Example of structures are hydropower dams for producing electricity, the metro-tunnels in Stockholm or the caverns for disposal of the nuclear waste. These structures have a high required safety level to avoid failure, since a failure could lead to catastrophic impacts on the social infrastructure and in worst case fatalities. To reach an acceptable safety for these structures, different types of rock supports have to be added to avoid failure. The support constitute a large cost in the overall cost for many projects. The client wants to minimize the costs in order to increase the profit without lowering the safety level of the design. A way to minimize the amount of support can be to reduce the uncertainties of the rock mechanical parameters, which leads to a more optimized design.

The rock mass is compared with many other construction materials not a changeably material. It is not possible to modify the material components to increase it’s strength. The Swedish bedrock consists of several rock types where the oldest bedrock is approximately 2,8 billions years old and the youngest about 55 millions years old (SGU, 2018). The one thing these bedrocks have in common is that nature has set its maximum limit, both during its creation as a bedrock and experienced geological events. An effect of the geological events is the natural rock joints in the bedrock, which plays a key-role for deciding the shear strength of the overall rock mass.

The shear strength for a joint has a direct relationship with possible infilling material, the normal stress, the degree of weathering, the uniaxial compressive strength of the joint surface, the matedness of the joint and the scale of it. This means that several parameters needs to be considered to determine the shear strength, a bi-effect is uncertainties within the parameters (Palmström and Stille, 2015). For the structures mentioned earlier this uncertainty in the level of

CHAPTER 1. INTRODUCTION

shear resistance is important. Since an overestimated shear strength can lead to a costly or a too conservative design, while an underestimation can lead to a potential unsafe design.

Today, there are several ways to determine the shear strength: analytical, numerical or testing of rock joint samples (lab or in-situ). There exist several peak shear strength criteria. However, every criterion has its advantage and disadvantage over each other – no criterion yet is able to account for all parameters that affect the peak shear strength of rock joints. In 2018, a new method was presented for determining the shear strength in natural rock joints, the Casagrande et al. method (Casagrande et al., 2018). This method is a semi-analytical stochastic approach for predicting the peak shear strength of rock joints. The problem with the Casagrande et al. criterion is that it assumes perfect matedness of the rock joint. For natural rock joints that are usually not perfectly mated, this can give an incorrect level of the peak shear strength.

1.2

Aim of thesis

This thesis aims at investigating the capability of the Casagrande et al. criterion to estimate the peak shear strength of perfectly mated and natural rock joints.

1.3

Outline of thesis

This thesis starts in the second chapter with a literature-study of different failure criteria for determining the peak shear strength of rock joints, where advantages and disadvantages of each criterion is discussed in the end of the chapter.

The third chapter describes the methodology used for characterizing the joint surfaces of the spec-imens and the procedure to calculate the peak shear strength using the method by Casagrande et al..

In the fourth chapter results of the calculations are presented. This chapter includes a comparison between the obtained result with the criterion by Casagrande et al. and results from performed shear test. A discussion regarding the results is presented in fifth chapter.

The conclusion of the thesis is presented in the last chapter. This chapter also contains suggestions for further development of the peak shear strength criterion by Casagrande et al..

1.4. LIMITATIONS

1.4

Limitations

As stated earlier in the aims of the thesis, this thesis will only consider the Casagrande et al. criterion for determining the peak shear strength for perfectly mated joints and natural rock joints.

The limitations for this thesis are as following:

• The two parts of a specimen (top and bottom), will be analyzed separately and the mean value of the two parts will represent the specimen.

• Only good quality hard crystalline rock material will be used for this evaluation, that is typical for Sweden.

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This chapter will begin with a description of the basic theory behind the peak shear strength of rock joints. The remaining part of the chapter will include a description of several important peak shear strength criteria for rock joints from Patton (1966), Barton and Choubey (1977), Grasselli (2001) and Johansson and Stille (2014). The chapter will end with a discussion between the criteria and their advantages and disadvantages.

2.1

Introduction

The studies of friction dates back as early as in the sixteen century by Leonardo da Vinci. He defined that the relationship for friction resistance between two sliding surfaces is directly proportionally to the normal force. In this relationship Da Vinci pointed out that his theory was independent of the contact areas of the participating sliding surfaces. It took nearly two hundred years before this theory was further adopted and used by the French Academy, after confirmation by the French physicist Charles-Augustin de Coulomb (Britannica, 2018a) in 1785. Later it became used in both engineering and science. Coulomb expressed the relationship as:

S =µ· N (2.1)

In equation 2.1, S represent the needed shear force required to keep a block from sliding at an inclined plane. This force is proportionally to the normal force (N) multiplied by a friction coefficient µ, which is a material dependent parameter of the surface roughness. Coulomb observed during his inclination test that at a certain angle the block started to slide, and the friction angle of the material was reached. He labeled this the basic friction angle (φb) (Grasselli, 2001). Coulomb expressed the friction coefficient in terms of the friction angle as (Patton, 1966):

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS

When inserting equation 2.2 into 2.1 you get the more commonly used Coulomb shear failure criterion:

τ=σntan(φb) (2.3)

Another early idea how to formulate the relationship between friction, shear- and normal force was presented by Leonhard Euler in 1750. Euler based his theory that a joint has a saw toothed appearance that governs the shear strength, he expressed it as:

τ=σntan(_αE) (2.4)

the angleαis the inclination of the saw tooth irregularities. This concept is illustrated in figure 2.1.

Figure 2.1: Euler’s proposed model of a saw tooth specimen for sliding friction.

2.2

Patton

2.2.1 Introduction

In 1966 Patton presented his criterion for determining the shear strength of rock fractures (Patton, 1966), which has similarities to Euler´s idea of the contributing parts in shearing. Patton studied the effect of a "saw tooth" surface. The base in his method came from experimental test on specimens formed in plaster of Paris1. The purpose of the plaster test was that Patton wanted to find the analogy between the plaster and rock, to either prove or disprove that this occurs for both intact rock samples and in-situ tests of rock. Based on the plaster specimens tests Patton made several conclusions regarding generalized rock joints.

2.2.2 Experimental test

In one of his test concerning the impact of the teeth inclination, he found that the peak shear strength varied between the different inclination. This is illustrated in Figure 2.2, where the units has been converted to the metric system. The graph displays three shear failure envelopes

1_{Plaster of Paris is so called because its found near Paris (Britannica, 2018b).}

2.2. PATTON

of two teeth specimen with three different inclination, line A 25°, line B 35°and line C 45°. Line D represent the residual angle2,φR, of the plaster specimen. The graph shows that the steeper the inclination is, the less shear resistance in terms of normal load the specimen has before shearing occur of the tooth. Patton explained that this behaviour relates to the base area of the tooth, where the specimen with inclination 25°has a bigger area to shear off than a specimen with inclination 45°.

Figure 2.2: Shear strength envelope different inclination of the teeth (Patton, 1966, p. 133)

The inclination of the asperities is one of the parameters that Patton pointed out that affected the shear failure envelope shape for the specimen. Other parameters that influence the envelope are: 2_{Best fitting line that can be drawn with the remaining shear resistance values from line A,B and C series (Patton,}

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS

• If the applied load is in direction of the inclination of the tooth.

• The intact area of the teeth along the shear plane.

• Internal properties of the specimen.

• Value of the residual shearing resistance.

• The normal load applied.

In tests with specimens with more than two teeth, he saw that when he doubled a specimen from two to four teeth with the same inclination on the teeth, the envelope was slightly higher but the shape almost the same. The thing Patton noted was that the effective mean or effective width of the teeth, displacement or "strain" in the specimen had a bigger influence for the shear strength than the amount of teeth.

2.2.3 Influence from geological composition

In order to fit the obtained results from the experimental testing of plaster of gypsum, Patton analyzed the irregularities of rock joints with respect to their geological constitution. The studies were performed both for the microscopic and macroscopic structure´s perceptive of the joint surfaces and the intact rock. Patton wanted to mimic the properties from the rock joint specimens to the plaster of Paris specimen to get the best base for his failure theory. He classified the two perspective as:

• Microscopic irregularities, which is the mean size of the individual elements in a material (usually the size of grains or minerals).

• Macroscopic irregularities, which are all irregularities over the microscopic definition.

The microscopic irregularities depend on the heterogeneity of the rock and are associated to the failure mode of the intact rock. In other words the shear strength of the intact rock. The macroscopic irregularities describes the asperities on the rock joint surface. The shear failure of the joint will occur in the macroscopic irregularities. Patton pointed out that failure of a rock joint is not necessarily caused by shearing of the asperities, it may also occur by sliding along the discontinuity.

2.2.4 Bi-linear shear failure envelope

From the tests of the specimens and his definition of the geological structures of a joint, he presented his shear failure envelope in 1966. The envelope he presented is a bi-linear approxima-tion of a curved envelope that contains two stages. A simple descripapproxima-tion of the stages is that the first expression describes the envelope before peak shear strength is reached, when sliding over

2.3. BARTON AND CHOUBEY

the asperities occur. The second part describes the condition after shearing of the asperities has occurred. The first part of the failure envelope is expressed as:

τ=σntan¡

φb+ i¢ (2.5)

whereφb is the basic friction angle of the rock and i is the inclination angle of the asperities on the surface. The second part of the envelope is expressed:

τ= cx+σntan¡ φR¢

(2.6)

where cxis cohesion after the teeth has been sheared of at their base. The new angleφRis the residual friction angle and refers to the shear resistance of an initially intact material. The envelope is presented in Figure 2.3 (Patton, 1966).

Figure 2.3: Patton’s bilinear failure envelope ( From Johansson (2009) based on Patton (1966))

2.3

Barton and Choubey

2.3.1 Shear failure criterion

In 1977 Barton and Choubey presented a new shear failure criterion. The criterion was based on empirical relationship, and made it possible to fit and extrapolate data, and predict the peak shear strength of a rock joint. The Barton and Choubey criterion has the basic friction angle in common with the Coulomb and Patton. In their criterion they introduced two new parameters and the following expression was derived:

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS τ=σn· tan · JRC · log10 µ_JCS σn ¶ +φr ¸ (2.7) where

τ Peak shear strength

σn Effective normal stress JRC Joint roughness coefficient JCS Joint wall compressive strength φr Residual friction angle

For obtaining the joint wall compressive strength (JCS) Barton and Choubey (1977) per-formed Schmidt hammer test on their specimens. They applied the test directly on the exposed joint wall in the direction of shearing. A notation is that they observed that for a unweathered rock, the JCS is equal to the uniaxial compressive strength of the intact rock (σci), but as rock joints in general are weathered to some extent JCS can be smaller than_σci.

The third parameter in equation 2.7 is the joint roughness coefficient (JRC). This coefficient describes the roughness of the surface, and can be compared with the inclination angle in Patton’s bi-linear failure criterion. Barton and Choubey (1977) performed several shear tests and with the results they did a back analysis to obtain the JRC coefficient according to the following equation:

JRC = arctan³_στ n ´ −φb log₁₀³JCS_σ n ´ (2.8)

2.3.2 Joint roughness profiles

Barton and Choubey (1977) presented a guide with ten typical profiles for a 100 millimeters trace of a joint surface. The values range between zero to twenty. The purpose with the guide was to provide a possibility to estimate the JRC where direct determination from tests of the parameter is not possible. The joint roughness profiles suggested by Barton and Choubey (1977) is illustrated in Figure 2.4.

2.3. BARTON AND CHOUBEY

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS

2.3.3 Scale dependency

In 1973 Barton proposed that JCS might be scale dependent, based on the outcome of compressive strength tests from Pratt et al. (1972). In 1977 Barton and Choubey performed scale tests where they tested a large block of a rock joint and afterwards the block was sawn into eighteen samples with the dimensions 4.9 by 9.8 cm. The result from the tests showed that the difference between the small and the large tests could not only be explained by a reduction of JCS. They came to the conclusion that the JRC must have a significant impact of the scale effect. They determined that the scale effect onJCS and JRC are related, at least qualitatively. They explained that the scale effect is approximately in proportion to the joint length (L) up to a critical length (L_c).

In 1982 Barton and Bandis presented two new equations to adjust the JRC and JCS value due to the scale effect. They based their equations on previous data (Barton and Bandis, 1982).

JRCn= JRC0 µ_L n L₀ ¶−0.02·JRC0 (2.9) JCSn= JCS0 µ_L n L0 ¶−0.03·JRC0 (2.10)

The index letter (n) correspond to the in-situ scale and (0) is the laboratory scale and L refer to the length of the sample.

2.3.4 Joint matching coefficient

An adjustment to Barton and Choubeys´s criterion was done by Zhao Zhao (1997) to include the "pattern matching" of a rock joint. Zhao introduced the parameter joint matching coefficient (J MC). It is an independent geometrical parameter that is not influenced or related to JRC. The range of the parameter goes from zero to one, where close to zero is a totally miss-matched joint with a minimum contact area and one is a perfect matching of the joint with a high contact area. Zhao modified Barton and Choubey´s criterion with his parameter as:

τ=σn· tan · JRC · JMC · log10 µ_JCS σn ¶ +φb ¸ (2.11)

2.4

Grasselli

The shear failure criteria that both Patton (1966) and Barton and Choubey (1977) presented is missing the information on the effects of the whole rock joint area that participate in the shearing process. In 2001 Grasselli presented a new shear failure criterion where the contact area of an asperity of a joint in 3D was taken into account (Grasselli, 2001).

2.4. GRASSELLI

2.4.2 Asperity Geometry

During the shearing of the joint, the shearing direction plays a key role for identifying active asperities in Grasselli´s method. He simplified the asperities to have a triangle face with an accompanying area. The faces that will be in contact with the bottom/top part of the joint are the ones with the steepest inclination. Grasselli pointed out the necessity of classifying the sliding area of a joint and the area facing the shear direction. An illustration of the geometrical structure of an asperity as a triangle can been seen in Figure 2.5. For every triangle there is a true dip vector, which is calculated from the center of the triangle. The inclination angle of the triangle is important for determining the contact area that is active during shearing.

Figure 2.5: Illustration of the geometrical interpretation of a rock surface with apparent dip angles against the shearing direction (from Grasselli (2001))

2.4.3 Contact area

In Grasselli’s theory the potential contact area Ac and the threshold of the apparent dip angle Θ∗_{plays a key role in the equation. The apparent dip angle is in turn depending on the true dip} angle and the azimuth angle of the surface of the asperity. The azimuth is the angle between the projection of the dip vector on the shear-plane ( ˆw) and the shear vector¡_{ˆt¢ (shear direction). The} angle is measured clockwise from¡_{ˆt¢. The apparent dip angle is given by the following expression:}

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS

The asperities that contribute to the shear resistance according to Grasselli are the surfaces facing the shear direction and are steeper than the threshold inclination¡

Θ∗

cr¢. He also pointed out that areas with inclination equal to or larger than¡

Θ∗

cr¢ are in contact with the other surface. If the inclination is steeper than¡

Θ∗

cr¢ the area will be deformed, sheared or crushed. Grasselli presented three equations for the sum up of the potential contact area.

Ac= A0· e(−B·Θ

∗₎

(2.13)

where A0maximum possible contact area, B "roughness" parameters.

Ac= a0· e µ −1 2· ³_Θ∗−x0 b0 ´2¶ (2.14)

where a0, b0, x0 is surface best fitting coefficient.

Ac= A0· µΘ∗ max−Θ∗ Θ∗ max ¶C (2.15)

whereΘ∗_maxis the maximum apparent dip angle of the surface and C is a roughness param-eter (obtained by best-fit regression function). Both paramparam-eters are dependent on the specific shear direction. Of the three equations, Grasselli means that equation 2.15 is more desirable to use since theΘ∗ is delimited between 0 toΘ∗_max. For Acthe value range between zero to A0. For equation 2.13 and 2.14 the apparent dip angle has a range between 0 and infinite, which makes a unrealistic upper bound.

2.4.4 Shear failure criterion

In 2001 Grasselli presented his criterion for shear failure, and the following expression was proposed: τp=σn· tan(φ0r) · · 1 + e −Θ∗max A0·C ·σnσc ¸ (2.16) where:

τp Peak shear strength

σn Applied average normal stress

σc Uniaxial compressive strength of the rock φ0

r Residual friction angle at a standard displacement of 5 mm A0 The maximum potential contact area for specified direction Θ∗

max Maximum apparent dip angle with respect to shear direction

When the exponential term turns to zero, the bracket part will be equal to two. This will occur ifΘ∗

maxorσngoes to zero. Tests performed by Grasselli showed that in practice theΘ∗maxwill be in the range between 20°and 90°.

2.5. JOHANSSON & STILLE

The minimum value of_σnoccurs when there is no applied load, and the normal stress is only represented by the specimens own weight. As this happens the peak shear angle is defined as:

lim σn→0 µτp σn ¶ = 2 · tan(φ0_{r) = tan(φp) →φp= 65}◦_{∼ 80}◦ _(2.17) In equation 2.16 the governing strength is the compressive strength of the intact rock material. However, Grasselli observed from several rock surfaces where asperities have been sheared off, that a tensile failure rather than a compressive failure occurred. From that he draw the conclusion that the tensile strength is a more important parameter for determination of the peak shear resistance.

The shear failure model that Grasselli presented in 2001 was then modified by substituting theσc with the tensile strength of the intact rock,σt. Despite the substituting ofσt, the equation underestimated the peak shear strength compared to obtained experimental data. To deal with this problem Grasselli introduced a new parameter P to the exponential expression.

τp=σn· tan(φ0r) · ·

1 + e−Θ∗maxP·A0·C·σnσc ¸

(2.18)

The specific value of P was set to 9.0, after performing multiple least-square regression of data from 39 rock specimen. The exponential part can be replaced with g and regards the surface morphology of the rock.

g = e−Θ∗max9·A0·C·σnσc _(2.19) The final expression of the failure criterion suggested by Grasselli thereby becomes:

τp=σn· tan(φ0r) · [1 + g] (2.20) The aim of Grasselli’s studies was to: "develop a simple method for estimating joint shear strength, equation 2.18 is not entirely satisfactory because it requires an estimate of the residual friction angleφ0_rfor each sample". Therefore Grasselli presented an expression forφ0_r, with the notation to indicate that this is a rough approximation.

φ0 r=φb+ µ C · A1.50 ·Θ∗max µ 1 − A 1 C 0 ¶¶cosα (2.21)

2.5

Johansson & Stille

2.5.1 Introduction

In 2014 Johansson and Stille presented a new criterion for describing how the interaction between matedness, scale and roughness contributes to the shear strength. The model that they proposed was for fresh rough rock joints without infilling and weathering effects.

For the rough surface Johansson and Stille describes that the peak friction angle consists of two parts, a microscopic- and a macroscopic part. The microscopic part describes the basic

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS

friction angle, where according to adhesion theory a smooth surface is at microscopic level rough. The macroscopic part is the dilation angle i of the joint due to roughness. The inclination of the asperities in the joint surface will govern which failure that is possible to occur. Johansson and Stille (2014) point out three modes that can occur for an asperity:

(i) Sliding

(ii) Shearing or Crushing

(iii) Rotational tensile failure

It is observed that the shearing and tensile failure mainly occur at high inclination of the asperities and at lower inclination sliding is the dominating failure mode.

2.5.2 Shear failure criterion

In the Johansson and Stille (2014) failure criterion the dilation plays an important part for determine the peak shear strength. The peak friction angle for the peak shear strength is defined as:

φp=φb+ in (2.22)

where the_φ0

b is the basic friction angle and inthe dilation angle for full sized joints. It can be observed there is a similarity with Patton’s criterion from 1966, the difference is the additional term where Patton stated that i is the inclination of the asperities and for Johansson and Stille this is the dilation angle of a joint. For the dilation angle they point out several relationships in a joint regarding the matedness, roughness and scale for deriving their final expressions.

Johansson and Stille describes that the contact area for a joint (Ac) as a quotient between the uniaxial compressive strength of the intact rock material_σci and the effective normal stress_σ0

n multiplied with the area of the sample.

Ac= A ·σ 0 n

σci (2.23)

To account for roughness of the joint surface, a self-affine fractal model may be used. Johans-son and Stille (2014) present a power function that describes the relation between different base length of the asperities Las pand the height of the asperity has p, see figure 2.6.

h_{as p= a · L}H_{as p} (2.24)

The two new parameters introduced in equation 2.25 are the a (amplitude constant) and H (Hurst exponent).

2.5. JOHANSSON & STILLE

Figure 2.6: Illustration of an asperity in two dimension, displaying the has p, Las p and i of a asperity (Johansson and Stille, 2014)

To deal with the changes of contact area as the joint size increases or the matedness changes, Johansson and Stille (2014) explained that during a constant normal stress, the true area of contact will be constant and the amount of contacts points will change with an increased joint size. At the same time, for a decreased matedness, the contact points becomes larger. Based on this reasoning, the following expression for the length of the asperities at contact was

Las p.n= Las p.n· µ_L n L_g ¶k (2.25)

Las p Base length of asperities (g=grain size of rock, n=full sized joints) L Length of sample (g=grain size of rock, n=full sized joints)

k _{Empirical constant, degree of matedness, range between 0 → 1}

By combining the equations 2.22, 2.23, 2.24, 2.25, Grasselli’s potential contact ratio 2.15 and force equilibrium the dilation angle could be expressed as:

i_{n= arctan}    2a ·     0, 5a−1  tan  Θ∗_max− 10

log_{σci −log A0}σ0n C _·Θ∗ max       1 H−1 · µ_L n Lg ¶k    H−1    (2.26)

equation 2.26 is simplified into:

in= arctan  tan



Θ∗_max− 10

log_{σci −log A0}σ0n C _·Θ∗ max  · µ_L n Lg ¶kH−k   (2.27) .

CHAPTER 2. LITERATURE STUDY: PEAK SHEAR STRENGTH OF ROCK JOINTS

2.6

Discussion

The ideas that Coulomb and Euler presented in the 18th century have had huge advantages in the developing of peak shear failure criteria, but in different ways. The advantage in Coulomb´s criterion is the description of the relationship for obtaining the basic friction angle for any rock type. The disadvantage is that it only obtain a friction angle that describes a smooth surface of a joint. In reality, a natural rock joint will have asperities which increase the shear resistance. In equation 2.3, this information is missing, which leads to an underestimation of the peak shear strength. However, this is what Euler´s theory cover. In his theory he describes the importance of the geometry of the asperities for the shearing resistance. The problem in his theory comes with how to estimate the common inclination in equation 2.4.

The advantage in Patton’s (1966) criterion that he proposed was that it described the whole process, before and after the peak shear strength had occurred. The strength with Patton´s bi-linear envelope, see Figure 2.3 is its simplicity, how it explains the transition of the process from sliding over the asperities to shearing through them. It is also easy to see the importance of the included parameters. The downside in his criterion is that it is to simplified with respect to the roughness of the joint. In addition, it does not consider the matedness of the joint and which inclination angle that shall be used in equation 2.5.

In comparison with Patton’s criterion (1966), the criterion proposed by Barton and Choubey (1977) included the effect of roughness of the joint together with the compressive strength of the joint wall surface. An advantage with the Barton and Choubey (1977) criterion is that it takes into account the real roughness of the joint compared with Patton´s ideal saw-toothed surface. Methods to determine the JRC-value are also suggested, by a predefined roughness profile or by back-analyzing the JRC value from tilt tests. The advantage of back-analyzing the JRC-value is that it gives a more correct value, since it indirectly account for the matedness of the joint. A disadvantage of this method is that it sometimes is not possible to obtain a joint sample for performing tilt tests. The advantage with the predefined roughness profiles is that they provide a rapid way to estimate the JRC-value of the sample surface. Though a downside with this guide is that it is based on a subjective estimation made by the observer, where the estimation of JRC can vary between different persons. Another drawback with the predefined roughness profiles is that they only applies for a 100 mm sample. To account for the scale effect, Barton and Bandis presented in 1982 two equations that reduces this problem. However, one of the requirements that can be hard to estimate is the length of the block along from where the rock sample is collected. This can be due to its location in the ground and if it is possibly to reach it. A positive parameter in their criterion is how the normal stress affect the joint wall during the shearing process.

A big advantage in the criterion suggested by Grasselli (2001) is that it identify the asperities that contribute to the shearing resistance. By doing this, the information that would be lost in a 2D-model is now represented, and therefore gives a more accurate peak shear strength. Another

2.6. DISCUSSION

positive effect with Grasselli’s criterion is that it minimizes the influences on the including parameters from the observer own opinion, since the parameters is determined from the scanned surface. However, in his final equation 2.18 there is uncertainties with the value 9 of parameter P. The uncertainty lies in the analyzing process where the obtained value is accurate since it is only based on six types of rock. A negative perspective with the criterion Grasselli suggested is that it does not explain how to deal with the scale effect.

The benefit with the criterion suggested by Johansson and Stille is that it accounts for roughness of the joint, the matedness of the joint and the scale. In their criterion, the matedness is taken into account compared with the other criteria. This give the opportunity to estimate the peak shear strength both for a perfectly mated joint and a total unmated joint. A drawback with the criterion is that it is be difficult to estimate the matedness constant k for natural weathered rock joints. Another negative perspective with the criterion is that the criterion is rather complex and it could be difficult to follow how the influence of the parameter will affect the final value of the dilation angle in equation 2.27. As for the Grasselli criterion, the surface needs to scanned to obtain some of the parameters describing surface roughness. Based on this literature study, it is clear that none of the criteria studied in this chapter manage to capture all of the mechanisms that affects the peak shear strength of natural rock joints. The criterion closest to obtain this, is the criterion suggested by Johansson and Stille (2014).

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3.1

Introduction

This chapter is divided into two sections. In the first section, the criterion proposed by Casagrande et al. (2018) is first presented. Here the necessary assumptions and how to implement the criterion are presented. The rest of the chapter explains the methods used for analyzing the specimens and necessary information for running the algorithm according to the Casagrande et al. criterion. The test method for the direct shear tests is also described.

3.2

Prestudy of the Casagrande et al. criterion

3.2.1 Introduction

A new peak shear criterion was presented by Casagrande et al. (2018) They used the available two dimensional information from fracture traces to create full scale synthetic surfaces in a three dimensional model. Casagrande et al. (2018) used a Monte-Carlo strategy and semi-analytical calculations to determine the peak shear strength, where the random model relies on statistical data. As for all other criteria presented in chapter 2, the roughness of the joint surface plays a key role, and is obtained by examining the joints. This is not always achievable since the joints can be located within the rock mass and only some traces could be possible to observe.

3.2.2 Shear failure criterion

Casagrande et al. wanted to find the key facets of the asperities that contribute to the shearing resistance. The method mobilizes the asperities that during the shearing process is "active"/"contribute" to the shearing resistance. They mention that during shearing the joint will

CHAPTER 3. METHOD, DESIGN AND IMPLEMENTATION

dilate and this will occur along the steepest asperities. In Casagrande et al., the concept of active asperities is described as follows: as the asperities is sheared off, the load will redistribute to other asperities. This process will be repeated until no shearing resistance is left.

Casagrande et al. set up their model in a x yz-grid based on the surface-data and assumes that the joint matedness is perfect. In the model, the shearing occurs along the x y-plane and z describes the height from the x y-plane. They decided that the lowest point in the model will be z=0. The criterion also requires input data about the material strength parameters, the value of the normal stress and the shearing direction in the (x y-plane). The authors suggests using the Mohr-Coulomb criterion to describe the strength of the intact asperities, which includes the parameters cohesion and friction angle for the intact rock. They also point out that it is possible to use the Hoek-Brown criterion to determine the cohesion and friction angle of the intact rock. Casagrande et al. presented a flow chart with important steps for deciding the shear strength, which is illustrated in Figure 3.1. As mentioned above, the Casagrande et al. method intend to identify the active faces in the shearing process. The first step in their method is to divide the joint surface into surface triangles. Suitable distances between the data points is needed to get a good approximation of different asperities. For their method, they used the definition of an apparent dip angleβa p p_i, according to Grasselli (2001). to indicate how steep the facet (βa p p_i) of the asperities against the shear direction is, as defined in equation 3.1.

βa p pii= a · cos( ¯ni· ¯s) − 90 (3.1)

¯

n_i Unit vector normal to facet i.

¯s Unit vector indicating the shear direction in the discontinuity plane.

3.2. PRESTUDY OF THE CASAGRANDE ET AL. CRITERION

Figure 3.1: Flow chart representing the key steps of the semi-analytical Casagrande et al. criterion for shear strength (from Casagrande et al. (2018))

By computing the iterative model described in the flowchart in Figure 3.1 it can be defined at which value of the apparent dip angle (β∗) the facets still are considered active and can be sheared off. This value has to be higher or equal toβ∗. The constructed loop model is decreasing theβ∗ from maximum to zero by a chosen decrement. For their model, they assumed that all the active facets are in contact, irrespective of their_β∗_{. The outcome of the assumption is that the} vertical force acting on the whole discontinuity (Fmacro), can be redistributed uniformly on all the active facets exposed by local vertical force ( fl ocal_i). This is illustrated in Figure 3.2. The authors state that the model does not cover for any displacements to be able to describe which facets that are active. The relationship betweenβ∗and fl ocali is illustrated in equation 3.2:

fl ocali=

Fmacro Nc f

(3.2)

where:

CHAPTER 3. METHOD, DESIGN AND IMPLEMENTATION

Figure 3.2: Illustration of two active faces in contact (see encircled area) (from Casagrande et al. (2018))

For the local facets the vertical normal stresses is expressed as:

σl ocal_i= fl ocal_i

A_{i p} (3.3)

where:

Ai pIs the projected area (Ai) on the x y-plane, see Figure 3.3.

Figure 3.3: Active facets being slided upon or sheared at their base (from Casagrande et al. (2018))

Since joints consist of asperities with different shapes they contribute differently to the shear-ing resistance. Casagrande et al. derived expressions for the required slidshear-ing forces( fsl id in g_i), equation 3.4 and shearing forces ( f_{shear_i}), equation 3.5 for an individual active facet. The fsl id in g_i is determined horizontally along the active facets area Ai. For the fshear_i the authors assumed that the asperities will be sheared of at the bottom of the active facets along the projected area A_{i p}along the horizontal plane. The two scenarios are illustrated in Figure 3.3.

3.2. PRESTUDY OF THE CASAGRANDE ET AL. CRITERION

f_{sl id in g_i= fl ocal_i· tan(φb+βi}) (3.4) fshear_i= Ai p· (c +σl ocal_i· tan(φf) (3.5)

βa p p_i Apparent dip of facet

φf Internal friction angle of intact rock (Mohr-Coulomb) φb Basic friction angle

c Cohesion of intact rock

Ai p The the projected area (Ai) on the x y-plane, see Figure 3.3. σl ocal_i The vertical normal stress acting on facet i along z axis.

The shearing of a local asperity will only happen as long as the shearing resistance is lower than the sliding resistance, fshear_i≤ fsl id in g_i. The reason for this is that in the beginning of the shearing the faces will have enough steep inclination to counteract shearing. But as the process progress more and more the facets will be sheared of and the apparent dip will be lowered until sliding resistance becomes bigger and the joint starts to slide. At this point the method can predict the peak shear strength.

As the geometry of the joints include facets with differentβa p pitheir model wanted to find β∗_{, where shearing transitions to sliding. To findβ}∗ _{Casagrande et al. performs a iterative} process where highestβa p piwill be the start value and lowered with a chosen decrement until β∗_{is reached, see Figures 3.4 to 3.8. As the iteration progress the steeper facets will be sheared} off and surface will smoothed and at the final step have the shape that is illustrated in Figure 3.8.

CHAPTER 3. METHOD, DESIGN AND IMPLEMENTATION

Figure 3.5: Iteration step 2 method to findβ∗(cont.)

Figure 3.6: Iteration step 3 method to findβ∗(cont.)

Figure 3.7: Iteration step 4 method to find_β∗_(cont.)

3.3. IMPLEMENTATION

Figure 3.8: Iteration step 5, final step.

They expressed the peak shear force ( fpeak) as following:

fpeak= Nc f X i=1 fsl id in g_i= Nc f X i=1

fl ocal_i· tan(φb+βa p p_i) (3.6) Equation 3.6 provides the necessary fpeak that is required to predict the peak and resid-ual shear strength respectively for the specified joint according to the criterion presented by Casagrande et al. (2018). The predicted shear strength is given by dividing fpeakwith the area for the joint or the tested specimen area.

τp−predicted= fpeak

Atot

(3.7)

For the residual shear strength the authors consider the effect from the sheared off asperities. This cohesive part have been pulled off from the f_peakto give the required residual force.

f_{residual= fpeak− c · Nc f· Ai p} (3.8) Now the f_residual is divided by same area as for the peak shear strength and predicted is given as following: τres−predicted= fresidual Atot (3.9)

3.3

Implementation

3.3.1 Rock joint samples

In this thesis, three sample groups are used: one group with perfectly mated joints and two groups with varying level of matedness. For each group, a tilting test on sawn surfaces is performed to obtain their basic friction angle. The perfectly mated joint is obtained in laboratory

CHAPTER 3. METHOD, DESIGN AND IMPLEMENTATION

by tensile splitting an intact rock specimen. The specimen is drilled with holes on each side and inserted with wedges that is hammered until the rock samples is cracked into two separate pieces. Figure 3.9 show an illustration when the wedges is hammered in and the crack increases.

Figure 3.9: Schematic illustration of the tensile induced splitting of the intact rock

As a result of this method, the rock will have fresh, rough and unweathered surfaces with a perfect matedness. The natural rock joint samples comes from drilling over an existing joint at Storfinnforsen and Långbjörn hydro power plants. All samples are gathered from Swedish rock and described below:

• The perfectly mated joint sample is an average- to coarse granite from the Flivik quarry. From this quarry, six samples is used in the analysis, three with the dimension of 60 × 60 mm and three with 200 × 200 mm (Johansson, 2016).

• The first natural rock joint samples is from Storfinnforsen at Faxälven located in Väster-norrland county. The rock consists of a grey coarse grained granite and eight samples were tested. All samples were drilled from the same joint, and are only slightly weathered (Ríos-Bayona et al., 2019).

• The joint samples from Långbjörn has been affected by weathering the most and the back-calculated JRC value is 7.0 for LS6 and 3.4 for LS7. The rock is from Långbjörn at Ångermanälven in Jämtland and consists of grey coarse grained granite and some small fractions of pegmatite. From Långbjörn 2 two specimen was taken, one with dimension 125 × 125 mm and one with 240 × 240 mm (Johansson, 2009).

Images of the samples are presented in Appendix A for Långbjörn and Storfinnforsen.

3.3.2 Scanning

The 16 rock joint samples were scanned by the optical scanning system ATOS III. As the joint consist of two parts, a lower and upper part, these are scanned separately. Every part was scanned several times from different angels to cover the geometry of all the asperities. Before the scanning was carried out circular reference points were placed on the parts. This is done to

3.3. IMPLEMENTATION

give the collected data points a localization on the object. When the joints parts are scanned they are put together and scanned again, giving the sample coordinates in a global system. By doing the combined scanning of the parts, the opportunity is given to determine to which degree the sample is mated. The huge amount of collected data point of the surface will be re-generated in MATLAB (2018) with a chosen grid.

3.3.3 Shear test

The shear tests were performed on the samples with a direct shear test with constant normal load. The shear test is performed according to the ISRM suggested methodology (Muralha et al., 2014;47(1). The test were conducted in a servo controlled shear machine with normal load and shear load capacity of 300 kN and 500 kN respectively at Luleå University of Technology (LTU).

Before the shear test, the joint samples were carefully cut and placed in steel molds. In the steel molds the parts were casted with concrete. This ensured that the joint parts stayed horizontal. When the concrete have achieved full strength, the specimens were placed in the shear machine with full contact between the lower and upper part and the tests were performed. The normal stress used in the shearing test were approximately 1 MPa, the exact values are presented in the result. The shearing of the samples was conducted until a shear displacement of 5 mm was reached and the shearing rate was 0.1 mm/min.

3.3.4 Model setup

The Casagrande et al. criterion is as mentioned earlier an iterative process. Therefore, the criterion has been programmed in Visual Studios1 (Microsoft, 2017). This criterion application software was programmed by Martin Stigsson from Swedish Nuclear Fuel and Waste Manage-ment Co (SKB) in 2018. The application software includes one program for the criterion and two subparts applications, one part has the possibility for creating a synthetic surface and the second one for extracting parts of a chosen joint file. However, this thesis will only use the extracting-and criterion application program to test the surfaces.

In MATLAB, the data points provided from the scanning is re-generated to a grid to give a more representative image of the asperities. The grid sizing is defined by earlier studies on the three sample groups by Johansson (2009), Johansson and Stille (2014) and Ríos-Bayona et al. (2019). Their grid size relates to grain length according to Johansson (2016) which is assumed to give the best representation of the asperities taking an active part in the shearing process. To give the created x y-grid information in the z direction, interpolation is done with the closets scanned data points around the particular grid point and the resulting value is assigned to that point.

The final surface image has before running Stigsson´s application been delimited from non-value points and non-values describing the surrounding concrete in the steel modules. Another

CHAPTER 3. METHOD, DESIGN AND IMPLEMENTATION

important step in the delimitation is to adjust the lower and upper part to the same starting and ending coordinates in the x y-grid. This was done since the upper and lower parts will be analyzed separately and they have to be sheared at the same coordinates. An important notation is that the grid length must be the same in x and y, since the application only handles a data matrix with the dimension n by n. This is done in MATLAB where it is simple to see which values that has to be delimited. The sub-application that Stigsson created for doing the same thing is only used to convert the data file to correct data format to run the criterion.

The criterion application is constructed so the shearing will be in the positive x-direction. Therefore, every part has to be controlled so the that the shearing resistance ( ˆd) is perpendicular

against the shearing direction (ˆt).By controlling the indicated shear direction arrow that is marked on tested samples. If needed, the samples matrix is rotated in MATLAB to obtain this. The Casagrande et al. criterion requires few specific material parameters for each sample group. They are: the basic friction_φb, the friction angle (_φf) and the cohesion of intact rock (c_i). The parameters (φf) and (ci) are dependent on the intact rock compressive and tensile strength. For the Flivik granite the compressive and tensile strength was decided by the Swedish National Testing and Research Institute. For the natural rock, the Schmidt Hammer index was used to obtain the uniaxial compressed strength as suggested by Barton and Choubey (1977). Neither article by Johansson (2016) and Ríos-Bayona et al. (2019) includes the tensile strength so an assumption is made that the same relationship for the compressive and tensile strength in Flivik granite applies for Långbjörn and Storfinnforsen. This relationship is described according to equation 3.10 and the constant given is then multiplied with the compressive strength.

D =_σσti

ci (3.10)

D Constant relationship between compressive- and tensile strength σci Compressive strength of the intact rock

σti Tensile strength of the intact rock

To obtain φb pieces of each group has been sawn so the get smooth surface and thereafter performing tilt tests. To decide the cohesion the following equation was used.

c = 2 ∗σti (3.11)

The method used for determine the (_φf) is by using the equation for the uniaxial compressive strength.

σci=¡2 · c · cos¡φf ¢¢ ¡1 − sin¡φf

¢¢ (3.12)

The normal stress that is used in the criterion will be the same as in the shear test in order to be able to do a comparison. The criterion program that Stigsson made start with identifying the

3.3. IMPLEMENTATION

steepest inclination of the asperities of the entered joint file_β∗_{. As this value is given the iterative} process start and β∗ is lowered with chosen decrement until PNc f

i=1fshear_i≤ PNc f

i=1fsl id in g_i is reached. To ensure that the decrement is not too big, a mismatch output is built in and gives in percent how close it is to the total shearing stress³PNc f

i=1fshear_i/Atot ´

and the total sliding stress ³

PNc f

i=1fsl id in g_i/Atot ´

occurs in the same point as the iteration stops. If the mismatch is smaller than one percent, the decrement is within acceptable limit. In Figure 3.10, an illustration of this iterative process is showed explaining this mismatch.

Figure 3.10: Illustration of mismatch at last iteration in criterion application

The finalβ∗together withφbgives the peak friction angleφp. This angle inserted in equation 3.14, which gives the peak shear strengthτpeak

φp=β∗+φb (3.13)

τpeak=σn· tan¡φpeak¢ (3.14) At the present stage the program can only handle one part of the joint at a time for performing the criterion. The results from the lower and upper part will be put together and the average value of this part will represent the joint.

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4.1

Introduction

The chapter will begin with a presentation of the scanned surfaces, input data for the direct shear test and finally a comparison between shear test and results by the Casagrande et al. criterion. In the result the joints will have the sample name as in the articles describing the shear test. Though, the first capital letter in the joint name will be the index from where the sample is gathered, F for Flivik, L for Långbjörn and S for Storfinnforsen.

4.2

Scanning

Figures 4.1 and 4.2 illustrate a 3D visualization of the lower and upper parts of joint F1, respectively. The two parts are also presented in 2D to illustrate the cropped surface in x y-plane, see Figure 4.3 and 4.4 respectively. As the remaining samples will have a similar appearance, only one sample is displayed.

CHAPTER 4. RESULTS

Figure 4.1: 3D-representation of sample F1, lower part

Figure 4.2: 3D-representation of sample F1, upper part

4.2. SCANNING

Figure 4.3: Delimited 2D-representation of sample F1, lower part

CHAPTER 4. RESULTS

4.3

Input Data

In Table 4.1 the compressive strength, σci, tensile strength, σti, cohesion, ci, basic friction angle,φb, and internal friction angle,φf, are presented for the three sample groups. Completed calculations for the parameters are presented in appendix A.

Table 4.1: Input material parameters for Casagrande Criterion

Flivik Långbjörn Storfinnforsen σci[MPa] 197 140 110 σti[MPa] 16.4 11.7 9.2 ci[MPa] 32.8 23.3 18.3 φb[deg] 31 31 31 φf[deg] 53.2 53.2 53.2

4.4

Results: Peak friction angle

The normal stress used in the evaluation of the two samples from Långbjörn varied, for LS6 the normal stress was 0.85 MPa and 0.9 MPa for LL7. Both samples have the same grid size 5 · 10−4 m and the lowering decrement ofβ∗was 0.01 degrees. The mismatch between the sliding force and shearing force is less than 0.1 % and shows that the chosen decrement is within acceptable limits. The results are presented in Table 4.2.

Table 4.2: Calculated and measured values of_φpeakfor sample LS6 and LL7 mm

Sample β∗[deg] φpeak[deg] φpeak[deg]

[Calculated] [Measured] LS6 lower part 23.91 54.91 LS6 upper part 25.89 56.89 Average 24.90 55.90 44.6 LL7 lower part 26.01 57.01 LL7 upper part 28.38 59.38 Average 27.19 58.19 42.4 36

4.4. RESULTS: PEAK FRICTION ANGLE

The normal stress used in the evaluation of samples from Flivik was 1 MPa. The grid size for the sample is 5 · 10−4m and the lowering decrement ofβ∗was 0.01 degrees. The mismatch between the sliding force and shearing force is less than 0.2 % and shows that the chosen decrement is within acceptable limits. Results for the 60 mm by 60 mm samples are presented in Table 4.3 and for the 200 mm by 200 mm samples in Table 4.4.

Table 4.3: Calculated and measured values ofφ_peakfor the Flivik specimens 60 mm by 60 mm (perfectly mated).

Sample β∗[deg] φpeak[deg] φpeak[deg]

[Calculated] [Measured] F1 lower part 36.69 67.69 F1 upper part 34.35 65.39 Average 35.52 66.52 65.0 F2 lower part 36.09 67.09 F2 upper part 34.66 65.66 Average 35.38 66.38 68.7 F3 lower part 32.24 63.24 F3 upper part 35.66 66.66 Average 33.95 64.95 66.1

Table 4.4: Calculated and measured values ofφpeakfor the Flivik specimens 200 mm by 200 mm (perfectly mated)

Sample β∗[deg] φpeak[deg] φpeak[deg]

[Calculated] [Measured] F12 lower part 34.47 65.47 F12 upper part 34.29 65.29 Average 34.38 65.38 67.6 F14 lower part 32.30 63.30 F14 upper part 33.53 64.53 Average 32.92 63.92 69.2 F15 lower part 33.18 64.18 F15 upper part 32.91 63.91 Average 33.05 64.05 64.6

CHAPTER 4. RESULTS

The normal stress used in the evaluation of samples from Storfinnforsen was 1 MPa. The grid size for the sample is 3 · 10−4m and the lowering decrement ofβ∗was 0.0001 degrees. The mismatch between the sliding force and shearing force is lesser than 0.001 % and shows that the chosen decrement is within acceptable limits. The results are presented in Table 4.5.

Table 4.5: Calculated and measured values ofφpeakfor specimen from Storfinnforsen with varying dimensions.

Sample β∗[deg] φpeak[deg] φpeak[deg]

[Calculated] [Measured] S1 lower part 32.49 63.49 S1 upper part 36.14 67.14 Average 34.32 65.32 62.1 S2 lower part 34.02 65.03 S2 upper part 30.84 61.84 Average 32.43 63.43 50.5 S3 lower part 34.18 65.18 S3 upper part 34.54 65.54 Average 34.36 65.36 71.8 S4 lower part 33.41 64.41 S4 upper part 31.14 62.14 Average 32.27 63.27 63.1 S5 lower part 31.09 62.09 S5 upper part 34.15 65.15 Average 32.62 63.62 59.9 S6 lower part 31.84 62.84 S6 upper part 31.63 62.63 Average 31.73 62.73 53.5 S7 lower part 33.29 64.29 S7 upper part 22.20 53.20 Average 27.74 58.74 55.3 S8 lower part 29.95 60.95 S8 upper part 31.67 62.67 Average 30.81 61.81 58.1 38

4.4. RESULTS: PEAK FRICTION ANGLE

In Figure 4.5 all the samples are plotted in a graph which compares the calculated and measured peak friction angle. The straight line is an idealization where the calculated and measured present the same results for the peak friction angle.

30 35 40 45 50 55 60 65 70 75 30 35 40 45 50 55 60 65 70 75

Measured peak friction angle in shear test [degrees]

Calulated

peak

friction

angle

[degrees]

Ideal peak friction angle measured vs calculated Flivik granite (Perfect matedness) Storfinnforsen granite (Natural joint)

Långbjörn granite (Natural joint)

Figure 4.5: Comparison between calculated peak friction angle with the Casagrande et al. criterion (2018) and measured peak friction angles from shear test

C

5

D

The results of the perfectly mated joints from Flivik confirm that the criterion from Casagrande et al. (2018) has the capability to predict the peak friction angle for perfectly mated joints, see Figure 4.5. A notation are the small deviations from the ideal peak friction angle line for the calculated values. Though these values have been considered to be in acceptable distance from the line to affirm the criterion.

During the execution of this thesis a concern has been that the Casagrande et al. criterion overestimates the peak friction angle, especially for natural rock joints. A justification of this concern is observable in Figure 4.5 where the majority of the natural rock joint values shows an overestimation of the calculated value using the Casagrande et al. criterion versus the measured values. Though, for the perfectly mated joints five out six calculated values are slightly below the ideal line between the measured and calculated peak friction angles, see Table 4.3 and 4.4.

From the results of the natural rock joint samples it is noticeable that with an increasing effect from weathering and decreasing matedness the calculated values deviates more from the measured ones. This indicates the limitations the Casagrande et al. criterion has with respect to the influence from the level of weathering and matedness. Even if some results from Storfinnforsen are within acceptable deviations from the ideal peak friction angle line, several values have larger deviations from the ideal peak friction line, which show that the criterion are not able to capture the effect from matedness.

Tables 4.2-4.5 displays the problem with analyzing the parts separately. This is because the two parts show difference between each other. For the natural rock joints the general difference are more than two degrees. This indicates that different amount of asperities area contributing to the shearing resistance and that they not necessary are in contact at all points during shearing, otherwise the results would be similar to the measured.