SIZE OF EFFECT RECTANGULAR PRISM ON DIRECT SHEAR …

SIZE OF EFFECT RECTANGULAR PRISM ON DIRECT SHEAR STRENGTH IN PHRA WIHAN SANDSTONE

Miss. Kanya Kraipru D6110444 Mr. Nattapon Sukjaroen M6110840

A Project of Advance Rock Mechanics Institute of civil, Transportation and

Geo-resources engineering Suranaree University of Technology

Academic Year 2018

TABLE OF CONTENTS

CHAPTER Page

I INTRODUCTION 1

1.1 Background and rationale 1

1.2 Research objectives 1

1.3 Scope and limitations 1

1.4 Research methodology 1

1.4.1 Literature review 2

1.4.2 Sample preparation 2

1.4.3 Laboratory testing 3

1.5 Report contents 3

II INTRODUCTION 4

2.1 Shear strength criteria 4

2.2 Shear strength parameter 5

2.2.1 Joint roughness coefficient (JRC) 5

2.2.2 Normal Stress 6

2.2.3 Cohesion 6

2.2.4 Friction Angle of Rock Joint 7

2.3 Shear strength in Pra Wihan sandstone 7

2.4 Size effects on shear strength parameter 8

2.4.1 Effect of scale on peak shear displacement 9

2.4.2 Effect on scale on JRC 9

III SAMPLE PREPARATION 10

3.1 Introduction 10

3.2 Sample preparation 10

II

IV LABORATORY TESTING 14

4.1 Direct Shear Tests on Rough Surfaces 14

4.2 Test results 19

4.3 Summary 22

IV Discussions, conclusions and future studies 23

5.1 Discussions and conclusions 23

5.2 Recommendations for future studies 23

LIST OF FIGURES

Figure Page

1.1 Research methodology 2

2.1 Alternative method for estimating JRC form measurements of surface roughness amplitude forms a straight edge (Barton 1982) 5

2.2 Shear strength – shear displacement of different size rock 8

3.1 Splitting tensile fractures of specimens prepared for direct shear testing 10

3.2 Roughness coefficient of fracture in (a) Area 26.29 cm2, (b) Area 49.95 cm2 and (c) Area 103.34 cm2 specimens before test 12

4.1 Direct shear device SBEL DR44 used in this study 14

4.2 Shows The specimens were different rock size 14

4.3 Pre and post-test some PWSS specimens for area 26.29 cm2 15



4.6 Roughness coefficient of fracture in (a) Area 26.29 cm2, (b) Area 49.95 cm2 and (c) Area 103.34 cm2 specimens After test 19

4.7 Shear stresses (τ) as a function of shear displacement (ds) of tension induced fractures for PWSS 20

4.8 Normal displacement (dn) as a function of shear displacement (ds) for PWSS 20

4.9 Normal displacement (dn) as a function of shear displacement (ds) rate and perpendicular stress of PWSS samples 21

4.10 Peak Shear Strength (τp) and residual shear strength (τr) as a function of normal stress (σn) of rough surfaces for PWSS 21

4.11 Model of joint roughness in same scale on different size 22

LIST OF FIGURES

Table Page

3.1 Physical properties of specimen for direct shear tests on rough joint surfaces 13

4.1 Summary cohesion and friction angle for PWSS different rock size 22

CHAPTER I

INTRODUCTION

1.1 Background and rationale

Direct shear testing (e.g.,ASTM D5607-08) has widely been used to determine the shear strength properties of a rock specimen that is an important aspect in the design of structures such as rock slopes, dam foundations, tunnels, shafts, waste repositories, caverns for storage, and other purposes. In the past, there has been a study on the shear stress in the laboratory, a series of Phra Wihan sandstone. But there are different rock size in use today. There has never been seriously studied and tangible enough about the effect of the varying rock size. Therefore, there is the determination about the effects of different rock size in the shear stress test series of Phra Wihan sandstone in the rock conditions at ambient temperature.

1.2 Research objectives

The objectives of this study are to determine the effects of in different rock size on direct shear testing the sandstone specimens belong to the Phra Wihan formations. The shear stress is determined under various normal load under the ambient temperature conditions.

1.3 Scope and limitations

The scope and limitations of the research include as follows.

1) Laboratory experiments are conducted on rocks,a series of Phra Wihan sandstone 2) Laboratory testing made under various normal load from 0.5, 1, 1.5 to 2 MPa with

rate 0.01mm/s. 3) Different rock size are 26.29 cm2, 49.95 cm2, and 103.34 cm2. 4) All test rocks are prepared in the laboratory. 5) Up to 4 samples are tested for each rock size. 6) No field testing is conducted.

1.4 Research methodology

As shown in Figure 1.1, the research methodology comprises 5 steps’ including literature review, sample collection and preparation, laboratory testing (direct shear test), discussions and conclusions and report writing.

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Literature review

Sample Collection and

Preparation

Direct Shear Test

Discussions and Conclusions

Report writing

Figure 1.1 Research methodology

1.4.1 Literature review

Literature review has been carried out to summary and explanation of the complete and current state of knowledge about shear strength criteria, shear strength parameter, shear strength properties of Phra Wihan sandstone and the effects of size on shear strength. The sources of information are from text books, journals, technical reports and conference papers. A summary of the literature review is given in chapter two.

1.4.2 Sample preparation

The specimens used for direct shear tests are prepare from the Pra Wihan sandstone. They are cut to square with different rock size are 26.29 cm2 ,49.95 cm2 and 103.34 cm2. A line load is applied to obtain a tension-induced fracture diagonally across the sandstone block. The asperity amplitudes on the fracture planes are measured from the laser-scanned profiles along the shear direction. The maximum and minimum

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amplitudes are used to estimate the joint roughness coefficients (JRC) of each fracture based on Barton’s chart (Barton, 1982).

1.4.3 Laboratory testing

The direct shear test methods follow the ASTM (D5607) standard practice. The constant normal stresses on the fracture are varied from 0.5, 1, 1.5 and 2 MPa

1.5 Report contents

This report is devices into five chapters. The first chapter includes background and rationale, scope and limitations, and research methodology. Chapter II present results of the literature review to summary and explanation of the complete and current state of knowledge of size effect rectangular prism on direct shear strength in Pra Wihan sandstone. Chapter III describes sample collection and preparation. Chapter IV describes the laboratory testing. Chapter V is discussions, conclusions and future studies.

CHAPTER II

LITERLATURE REVIEW

This chapter summarizes the results of literature review on the topics relevant to this research, including shear strength criteria, shear strength parameter, shear strength in Pra Wihan sandstone and size effects on shear strength parameter.

2.1 Shear strength criteria

The mechanical rock properties are one of the most important parameters that will be used in the analysis and design of any engineering structures in rock mass. If structural mapping identifies discontinuities in rock mass on which shear type failures may take place, it will be necessary to determine the fiction angle and cohesion of the discontinuity surface in order to carry out stability analyses and design remedial work. Several criteria have been proposed in the past to identify the strength of a rough rock joint. These criteria delineate the state of stress that separates pre-sliding and post sliding of the joint. The simplest peak-shear strength model for rock joints is perhaps Patton’s model (Patton, 1966). Based on the Coulomb friction law, this model characterizes the joint behavior of a single surface parameter that is the average roughness angle. More complicated joint models appeared later, accompanying the development of numerical methods. Notable among them are Ladanyi’s empirical model (Ladanyi and Archambault, 1970) and Barton’s empirical model (Barton, 1973).

Coulomb criterion represents the relationship between the peak shear strength and normal stress by

τ = c + σn tan ∅

τ is joint shear strength, σn is normal stress, c is the cohesive strength, and ∅ is angle of friction. Patton (1966) performed a series of constant load stress direct shear tests with regular teeth inclination (i) at varying normal stresses. From these tests, he established a bilinear failure envelope - failure from an asperity sliding and asperity shearing mode.

τ = σn tan (∅b + i) τ is joint shear strength, σn is normal stress, ∅b is basic friction angle, and (i) is regular teeth inclination.

Ladanyi and Archambault (1970) suggest the shear strength of the material adjacent to the discontinuity surfaces,

τ=σ(1−𝑎𝑎𝑠𝑠)(𝑣𝑣+𝑡𝑡𝑎𝑎𝑡𝑡φ)+𝑎𝑎𝑠𝑠.τ𝑟𝑟1−(1−𝑎𝑎𝑠𝑠)𝑣𝑣.𝑡𝑡𝑎𝑎𝑡𝑡φ

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τ is joint shear strength, ∅B is basic friction angle, as is the proportion of the discontinuity surface which is sheared through projections of intact material, V is the dilation rate (dv/du) at peak shear strength, and τr is the shear strength of the intact material. Barton (1973) has studied the behavior of natural rock joints and proposed a criterion that is modified from Patton. It can be re-written as

τ = σn tan {∅b + JRC Log 10 (JCS / σn)}

τ is joint shear strength, ∅B is basic friction angle, σn is normal stress, JRC is the joint roughness coefficient, and JCS is the joint wall compressive strength

2.2 Shear strength parameter

2.2.1 Joint roughness coefficient (JRC)

Barton (1972, 1973, 1976), and Barton and Choubey (1977) have proposed a joint roughness coefficient (JRC) to describe the surface roughness scaled from 0 to 20. Typical roughness profiles for entire JRC range are represented in Figure 2.1 the measurements of this index are estimated by using the Alternative method for estimating JRC form measurements of surface roughness amplitude form a straight edge (Barton 1982)

Figure 2.1 Alternative method for estimating JRC form measurements of surface roughness amplitude forms a straight edge (Barton 1982)

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Coulson (1970) investigate the shear strength of a discontinuity, or joint, in a rock mass that derived from two components: (1) the frictional component offered by two flat surfaces sliding relative to one another (2) the geometrical component derived from the necessity of interlocking surface irregularities to be overridden or sheared off as sliding commences. The chief purpose of this investigation has been to determine the manner in which surface preparation affects the frictional component of shear strength

N. Fardin (2001) investigated roughness parameters of rock joints are scale-dependent and their descriptors change with scale, a systematic investigation has been carried out to understand the effect of scale on the surface roughness of rock joints. This scale-dependency is limited to a certain size, defined as the stationarity threshold, and for sampling windows larger than the stationarity threshold, the estimated parameters remain almost constant. It is concluded that, for surface roughness to be accurately characterized on a laboratory scale or in the field, samples need to be equal to or larger than the stationarity limit.

Dong Hyun Kim et al (2013) suggested surface roughness of rock discontinuities is an important factor that determines the strength characteristics of rock mass. Joint roughness coefficient (JRC), there are three different methods such as manual measurements, photogrammetry, and tilt tests were employed to determine. The obtained results showed some discrepancy in the values of JRC obtained from these three different measurements. In particular, the JRC obtained using the Barton’s comb had slightly higher values compared to those determined through the photogrammetry method while the tilt test results tended to yield overestimated values of JRC.

2.2.2 Normal Stress

M. F. Linker (1992) investigate the effects of variable normal stress on frictional resistance by performing quasi‐static sliding experiments. The observed response to a change in normal stress mimics that which occurs in response to a change in slip velocity.

Vasarhelyi (1999) has investigated the dependence of the constant normal load on the rate of the dilation. The results show that the measured dilation angle decreases with increasing normal force. Proper modeling of the shear behavior of rock discontinuities must take into account of the conditions imposed by the rock mass rigidity. In this situation, the normal stress path should be properly understood for the accurate prediction of shear strength. Such behavior is more realistically represented in the controlled normal stiffness shear test rather than in the conventional constant normal load test. Ladany and Archambault (1970) equation is correct for the cases when the Patton (1966) and Haberfield and Johnston (1994) equations fail. This means that Ladany and Archambault (1970) equation is a more general equation and it should be valid until the irregularities are not sheared off.

2.2.3 Cohesion

Wyllie (1998) The cohesion develops discontinuity surfaces in many conditions because a small cohesive strength can have significant effect on the shear strength of 12 rock. It is important that this rock strength parameter is properly accounted. The following are some of the conditions in which cohesion is developed on sliding surface. For intact rock and jointed, strong rock masses with no through-going discontinuities parallel to the sliding surface, the cohesion will usually have values of several hundred

https://agupubs.onlinelibrary.wiley.com/action/doSearch?ContribAuthorStored=Linker%2C+M+F

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kilopascals and at this high strength there little risk of shear failure. For rough rock surfaces, an apparent cohesion is developed as the asperities are sheared off when movement occurs. The magnitude of the apparent cohesion is the intercept on the shear stress axis of the tangent to the curve shear strength envelope on a Mohr diagram. The apparent cohesion will increase with increasing normal stress until residual strength of the surface is reached. For discontinuity containing infillings, the cohesion will depend on both the characteristics and thickness of the infilling.

2.2.4 Friction Angle of Rock Joint

E.Z.Lajtai (1969) studied the influence of a single plane of weakness on shear strength in direct shear loading. It is shown that total shear strength is determined by fundamental shear strength (cohesion) and internal friction in solid bridges and by joint friction along the separated parts of the weakness plane. If neither of the frictional resistances is mobilized before total loss of cohesion occurs, maximum strength is controlled solely by the tensile strength of the solid bridges

Both angles of basic (∅b) and residual (∅r) friction angle represent minimum shear resistance. Conceptually, ∅b refers to smooth, planar surface in fresh rock and can be considered as a material constant. ∅r refers to the residual condition of natural joint surface, which is attained after large shear displacement. If the natural joint surface is un-weathered, ∅r can be taken equal to ∅b.

The measurement is difficult due to the very large shear displacements requirement (Xu and de Freitas, 1990). In an indirect approach, ∅r may be obtained by allowing for the dilatation in the measurement of shear stresses during testing under very low normal stress, which also have been corrected (Hencher, 1987)

2.3 Shear strength in Pra Wihan sandstone

Piyanat Kapang (2012) studied true triaxial shear tests have been performed to determine the peak shear strengths of tension-induced fractures in three Thai sandstones. And suggested the lateral stress that is parallel to the fracture plane (sp) can significantly reduce the peak shear strength of the fractures. Under the same normal stress (sn) the fractures under high sp dilate more than those under low sp. According to the Coulomb criterion, the friction angle decreases exponentially with increasing sp/ so ratio and the cohesion decreases with increasing sp. The lateral stress sp has insignificant effect on the basic friction angle of the smooth saw-cut surfaces.

Fuenkajorn (2015) investigated the fracture shear strengths in Phra Wihan and Sao Krua sandstones and marl specimens under constant normal load (CNL) and constant normal stiffness (CNS), and compare the results by performing numerical simulations of roof and sidewall of underground openings. And suggested the fractures in Phra Wihan sandstone show greatest shearing resistance compared to those in the other two rock types. This is due to that the Phra Wihan sandstone is stronger than the other two.

https://www.sciencedirect.com/science/article/pii/0148906269900163#!

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2.4 Size effects on shear strength parameter

Bandis (1981) studied the effect of scale on the shear behavior of joint by performing direct shear on different size. The result show significant scale effect on both the shear strength and deformation characteristics. The peak shear strength was a strongly scale-dependent variable. The key factor behind the changing behavior and strength with increasing scale is the involvement of different size of asperities in controlling the peak behavior of different lengths of joint in Figure 2.2, it seems of shear strength – shear displacement of different size rock.

Figure 2.2 shear strength – shear displacement of different size rock

Amy B. Cerato1 and Alan J. Lutenegger (2006) studied the direct shear box tests on sands that there are different size shear boxes in use today and the effect of the varying specimen size on the resulting friction angle to determine the friction angle, or shear strength of the sand for engineering design purposes. Results of the direct shear tests show that the friction angle can be dependent on specimen size and that the influence of specimen size is also a function of sand type and relative density. The tests indicate that for well- graded, angular sands, decreases as box size increases and that the influence of box size is dependent onrelative density.

Johansson and Stille (2014) suggests how roughness and matedness at different scales influences the peak shear strength for fresh, rough and unweathered joint. And Johansson (2015) investigate the ability of the conceptual model to estimate the peak shear strength at different degrees of matedness and scales. A series of direct shear test were carried out at two different scales and two different degrees of matedness. The peak shear strength from the tests was compared to the peak shear strength calculated with the conceptual model. The results showed that the model can predict the peak shear strength for both the perfectly mated and the unmated joints. No scale effect was observed in the shear tests, which is in line with the predictions using the model.

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2.4.1 Effect of scale on peak shear displacement

Bandis (1981) suggested the type of surface roughness has a decisive influence on the variation of peak shear displacement (dhp) with increasing block size. Peak shear displacement is effectively a measure of the distance a joint has to travel until effective contactis made between the asperities controlling its peak resistance. The displacement-scale effect clearly implies that under the same level of normal stress the peak behaviour of different joint lengths is controlled by irregularities of different size or base-length.

2.4.2 Effect on scale on JRC

Bandis (1981) suggested the mobilization of asperities of different base length means that the value of the joint roughness coefficient (JRC) for a particular joint or joint set will depend on scale. A joint with small steep asperities controlling peak behaviour would have a higher JRC value than a longer profile of the same joint whose behaviour was dominated by larger and less steeply inclined surface features.

CHAPTER III

SAMPLE PREPARATION

3.1 Introduction

This chapter describes the sample preparation for the direct shear tests. the sample rocks using is Phra Wihan sandstones (PWSS) formations, preliminary Specimens were prepared to obtain octagonal prism with Area 26.29, 49.95 and 103.34 cm2. the samples are chosen at the perfect, homogeneous, no crack and fresh rocks.

3.2 Sample preparation

The specimens used for the direct shear test are prepares from the Phra Wihan sandstones. Specimens were prepared to obtain octagonal prism with 3 Area (26.29, 49.95, 103.34 cm2) for 4 normal stress of each area. The rough surfaces are prepared by line load is applied at the mid-section of the sandstone octagonal prism until splitting tensile failure occurs in rocks. There are four pairs of specimens are prepared for each Area. The asperity amplitudes on the fracture planes are measured from the laser-scanned profiles. Table 3.1 shows physical properties of specimen for direct shear tests on rough joint. Shows roughness coefficient of fracture in rocks (Figure 3.2) The tension-induced fractures are in the range between three and nine.

Figure 3.1 Splitting tensile fractures of specimens prepared for

direct shear testi

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(a)

(b)

12

(c)

Figure 3.2 roughness coefficient of fracture in (a) Area 26.29 cm2, (b) Area 49.95 cm2 and (c) Area 103.34 cm2 specimens before test.

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Table 3.1 Physical properties of specimen for direct shear tests on rough joint surfaces.

specimen no. Area (cm3) Weight (g) Density (g/cc) JRC

DS5-01 26.93 630.53 2.29 3

DS5-02 26.78 653.53 2.37 8

DS5-03 25.84 621.59 2.36 4

DS5-04 25.59 623.97 2.38 8

DS7-01 49.36 1158.24 2.28 5

DS7-02 50.41 1368.78 2.31 7

DS7-03 50.42 1203.65 2.37 5

DS7-04 49.59 1198.49 2.35 5

DS10-01 104.64 2506.00 2.33 9

DS10-02 103.02 2500.00 2.36 4

DS10-03 103.58 2383.0 2.24 4

DS10-04 102.10 2500.10 2.36 4

CHAPTER IV

LABORATORY TESTING

The primary objectives of the laboratory tests are shear strengths of rough joint surfaces under different rock size.

4.1 Direct Shear Tests on Rough Surfaces

A series of direct shear strength tests on rough joints have been conducted on two types. The sample preparation and test procedure follow the applicable ASTM D5607 standard practice. Selected normal stresses are 0.5, 1, 1.5 and 2 MPa. Post-test observation on the sheared off area indicates that the asperity areas that have been sheared off are small for all specimens, Figure 4.1 show the laboratory arrangement of direct shear test. Figure 4.2 The different rock size are 26.29 cm2, 49.95 cm2, 10.34 cm2 and 138.58 cm2. Figure 4.3 – 4.5 shows the pre and post-test fractures of PWSS specimens for all conditions. And Figure 4.6 shows roughness coefficient of fracture in PWSS specimens after test.

Figure 4.1 Direct shear device SBEL DR44 used in this study

Figure 4.2 Shows The specimens were different rock size

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Figure 4.3 Pre and post-test some PWSS specimens for area 26.29 cm2

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17


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(a)

(b)

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(c)

Figure 4.6 roughness coefficient of fracture in (a) Area 26.29 cm2, (b) Area 49.95 cm2 and (c) Area 103.34 cm2 specimens After test.

4.2 Test results

Direct shear test According to the coulomb criteria the shear stress (τ) can be represented by

τ = c + σn tanφ

Where σn is the normal stress, c is the cohesion and φ is the friction angle.

From coulomb criteria the shear stress depends on normal stress. According to the results of PWSS specimens for different rock size. Figure 4.7 shows shear stresses (τ) as a function of shear displacement (ds) and figure 4.10 shows normal displacement as a function of shear displacement of tension induced fractures only.

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Figure 4.7 Shear stresses (τ) as a function of shear displacement (ds) of tension induced fractures for PWSS.

Figure 4.8 Normal displacement (dn) as a function of shear displacement (ds) for PWSS.

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Figure 4.9 Normal displacement (dn) as a function of shear displacement (ds) rate and perpendicular stress of PWSS samples.

The peak shear strength (τp) and residual shear strength (τr) are calculated by the equations;

τp = Pp/A

τr = Pr/A

For vary area τp = (1.079σn) + (-0.005 A+0.786) (MPa)

τr = (1.053σn) + (0.860 e(-0.024A)) (MPa)

Where τp is the maximum shear force, τr is the residual shear force, and A is the contact area between both specimens.

Figure 4.10 Peak Shear Strength (τp) and residual shear strength (τr) as a function of normal stress (σn) of rough surfaces for PWSS.

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Table 4.1 Summary cohesion and friction angle for PWSS different rock size.

Shear stress (MPa)

PWSS different rock size condition. 26.29 cm2 49.95 cm2 103.34 cm2

c (MPa) φ (degrees)

c (MPa)

φ (degrees)

c (MPa)

φ (degrees)

τp 0.59 49 0.53 47 0.36 46

τr 0.40 48 0.28 46 0.09 46

4.3 Summary Shear strength relate to Area, small area has shear strength more than large area. Because in same joint roughness scale the small area use too much force to make joint roughness is peak shear strength show on Figure 4.11. Therefore, Shear area of small size have percent of loss area more than large size.

Figure 4.11 Model of joint roughness in same scale on different size

CHAPTER V

Discussions, conclusions and future studies.

5.1 Discussions and conclusions

The test results indicate that the difference sizes effect on the shear strength of rock. The effect of sizes depend on many factor such as joint roughness coefficient (JRC) and dilation. Under the same level of normal stress, the large area decrease shear strength. Therefore size area of rock is also reflected as a reduction of cohesion and friction angle. Otherwise, scale effect decreases with increasing normal stress because both the peak dilation and the joint roughness coefficient of the samples decrease.

5.2 Recommendations for future studies

More testing is required on a asperity of rocks have reflect how to shear strength including size and distribution of contact area on small and large joint samples. This also suggests that test results under vary normal load should be obtained.

REFERENCES

Bandis, S., Lumsden, A. C., & Barton, N. R. (1981, February). Experimental studies of scale effects on the shear behaviour of rock joints. In International journal of rock mechanics and mining sciences & geomechanics abstracts (Vol. 18, No. 1, pp. 1-21). Pergamon.

Barton, N. (2013). Shear strength criteria for rock, rock joints, rockfill and rock masses: Problems and some solutions. Journal of Rock Mechanics and Geotechnical Engineering, 5(4), 249-261.

Barton, N., & Bandis, S. (1982, January). Effects of block size on the shear behavior of jointed rock. In The 23rd US symposium on rock mechanics (USRMS). American Rock Mechanics Association.

Cerato, A. B., & Lutenegger, A. J. (2006). Specimen size and scale effects of direct shear box tests of sands. Geotechnical Testing Journal, 29(6), 507-516.

Coulson, J. H. (1970). The Effects of Surface Roughness on the Shear Strength of Joints in Rock. ILLINOIS UNIV URBANA.

Fardin, N., Stephansson, O., & Jing, L. (2001). The scale dependence of rock joint surface roughness. International Journal of Rock Mechanics and Mining Sciences, 38(5), 659-669.

Kapang, P., Walsri, C., Sriapai, T., & Fuenkajorn, K. (2013). Shear strengths of sandstone fractures under true triaxial stresses. Journal of Structural Geology, 48, 57-71.

Kim, D. H., Gratchev, I., & Balasubramaniam, A. (2013). Determination of joint roughness coefficient (JRC) for slope stability analysis: a case study from the Gold Coast area, Australia. Landslides, 10(5), 657-664.

Lajtai, E. Z. (1969, September). Shear strength of weakness planes in rock. In International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts (Vol. 6, No. 5, pp. 499-515). Pergamon.

Linker, M. F., & Dieterich, J. H. (1992). Effects of variable normal stress on rock friction: Observations and constitutive equations. Journal of Geophysical Research: Solid Earth, 97(B4), 4923-4940.

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