A new index to describe joint roughness coefficient (JRC) under cyclic shear

Engineering Geology 212 (2016) 72–85

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A new index to describe joint roughness coefficient (JRC) under cyclic shear Bowen Zheng a,b, Shengwen Qi a,⁎ a b

Key Laboratory of Shale Gas and Geoengineering, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

i n f o

Article history: Received 28 December 2015 Received in revised form 18 June 2016 Accepted 31 July 2016 Available online 3 August 2016 Keywords: Joint roughness coefficient (JRC) Barton standard profiles Shear direction Forward positive shear Backward positive shear

a b s t r a c t Rock joint roughness coefficient (JRC) is an important parameter to determine the shear strength of unfilled hard rock joints. It is of great significance to acquire a representative value of JRC. For irregular undulated joints, the JRC has a strong directivity and surface inclination can reflect the roughness difference under the condition of cyclic shear. Considering Barton's standard joint roughness profiles as an example, the secant angles and tangent angles of surfaces dipping opposite to shear direction were calculated and then the mathematical relationships with JRC were investigated. Based on the results of the study, it can be said that all mathematical relationships of surface inclination of surfaces dipping opposite to shear direction with JRC satisfy the power law equations. The average value of entire secant angles of all surfaces dipping opposite to shear direction (β100%) has a higher sensitivity to evaluate JRC comparing with other parameters. It (β100%) is recommended as a new index to describe JRC considering the directivity. Under cyclic shear, it shows that the JRC of forward positive shear process (JRCfp) is basically larger than that of backward positive shear process (JRCbp) except the first and the fifth standard profile. Moreover, the JRCbp value of the seventh standard profile is smaller than that of the fifth and sixth one. With the digitized sampling interval increasing, the correlation coefficient of the relationships between β100% and α100% (average value of entire tangent angles of surfaces dipping opposite to shear direction) and JRC of Barton standard profiles both present a trend of decreasing and the former is a little larger than the latter under the condition of equal sampling interval. © 2016 Elsevier B.V. All rights reserved.

1. Instruction The joints or discontinuities play a dominant role in the behavior of rock mass which consists of rock block and discontinuities (Hoek, 1983; Guo and Qi, 2015). In the low stress level, the main instability mode of engineering rock mass presents as shear deformation failure along the rock joint (Du, 1995; Du and Guo, 2003). For unfilled hard rock joint, morphological characteristics of joint surface, namely undulation and roughness play a vital role in determining of mechanical properties especially the shear strength of joint (Newland and Allely, 1957; Patton, 1966a, b; Sun, 1988; Bahaaddini et al., 2016); moreover, the effect of undulation and roughness on the shear strength of joint is different under variant direct shear test conditions (constant normal load and constant normal stiffness) and joint sizes (Boulon, 1995; Muralha et al., 2014; Vallier et al., 2010). Patton (1966a, b) conducted direct shear tests on artificial plaster joints with

⁎ Corresponding author. E-mail address: [email protected] (S. Qi).

http://dx.doi.org/10.1016/j.enggeo.2016.07.017 0013-7952/© 2016 Elsevier B.V. All rights reserved.

regular saw-tooth shape, and reached the envelope line of peak shear strength under the condition of low normal stress based on the Newland equation (Newland and Allely, 1957), as is given in Eq. (1):
 S ¼ N tan ϕμ þ i

ð1Þ

where S is total shear resistance, N is total normal force, ϕμ is sliding friction angle, i is included angle of saw-tooth inclined plane and horizontal plane. Barton (1973) divided the surface morphology of natural rock joint into three categories as smooth flat, smooth undulation and roughness undulation, and put forward the concept of joint roughness coefficient (JRC) for the first time. Barton and Choubey (1977) conducted direct shear tests on 136 natural rock joint samples to study JRC. The surface morphology of each joint was measured through three profiles before test. The JRC values were back calculated considering test results and divided into ten groups by the numerical intervals of 0–2, 2–4, 4–6, 6–8, 8–10, 10–12, 12–14, 14–16, 16–18, 18–20. The most typical profile of every group

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

73

Fig. 1. Barton joint standard roughness profiles (after Jang et al., 2014).

was selected to be the joint standard roughness profile, as shown in Fig. 1. The International Society for Rock Mechanics (ISRM) regarded the ten profiles mentioned above as standard profiles for defining of the

joint roughness coefficient values (ISRM, 1978). After comparing with Barton standard profiles, the actual joint profile roughness coefficient value can be finally determined. This method is called the empirical comparison method.

Table 1 Equations of digital measurement methods and statistical parameters of Barton standard profiles roughness coefficient. Digital measurement method and statistical parameter

Equation

Reference

Straight edge method Modified straight edge method Elongation method Direct measurement method θ⁎max

JRC = (450 + 50lgL)Ry/L JRC = 49e6.44/Larctan(8Ry/L) JRC = lg[(Lt − L)/L]/lg1.0910216 JRC = arcos(L/Lt)/lg(JCS/σn) JRC = 3.95[θ⁎max/(C + 1)2d]0.7 − 7.98 JRC=5.3[θ⁎max/(C + 1)2d]0.605 − 9.49 JRC = −1133.6 + 1141.6(D) JRC = −1022.55 + 1023.92(D) JRC = −0.878 + 37.784[(D − 1)/0.0015] − 16.93[(D − 1)/0.0015]2 JRC = 85.2671(D − 1)0.5679{D = lg4/log[2(1 + cos(arctan(2h*/L*)))]} JRC = −256.22/(1 + e(an − 0.9892)/0.00462) + 21.42 JRC = 103.37(D − 1)0.3 − 8.54 JRC = 138.71(an − 1)0.393 − 5.15 JRC = 520.28(D − 1)0.7588 JRC = 2.37 + 70.97(RMS) JRC = 10.9577log(RMS) + 11.5207

Barton and Bandis (1982) Du et al. (1996) Wang (1982) Turk and Dearman (1985) Tatone and Grasselli (2010) Jang et al. (2014) Turk et al. (1987) Carr and Warriner (1989) Lee et al. (1990) Xie and Pariseau (1994) Jang et al. (2006) Jang et al. (2014) Jang et al. (2014) Li and Huang (2015) Tse and Cruden (1979) Li and Zhang (2015)

JRC = 32.2 + 32.47lg(Z2) JRC = 61.79(Z2) − 3.47 JRC = 51.85(Z2)0.6 − 10.37 JRC = 32.69 + 32.98lg(Z2) JRC = 51.16(Z2)0.531 − 11.44 JRC = 55.7376(Z2) − 4.1166 JRC = 37.28 + 16.58lg(SF) JRC = 121.13(SF)0.5 − 3.28 JRC = 37.63 + 16.5lg(SF) JRC = 73.95(SF)0.266 − 11.38 JRC = 137.1739(SF)0.5 − 3.9998 JRC = 411.1(Rp − 1) JRC = 92.07(Rp − 1)0.5 − 3.28 JRC = [0.0336 + 0.00124/ln(Rp)] JRC = 65.9(Rp − 1)0.302 − 9.65 JRC = 229.44Rp − 226.9357

Tse and Cruden (1979) Yu and Vayssade (1991) Yang et al. (2001) Tatone and Grasselli (2010) Jang et al. (2014) Li and Zhang (2015) Tse and Cruden (1979) Yu and Vayssade (1991) Yang et al. (2001) Jang et al. (2014) Li and Zhang (2015) Maerz et al. (1990) Yu and Vayssade (1991) Tatone and Grasselli (2010) Jang et al. (2014) Li and Zhang (2015)

a D ¼ 1  lg Llg lg r

1=2

n−1

RMS ¼ ½1L ∑ ðyiþ1 −yi Þ2 ðxiþ1 −xi Þ i¼0

n−1

Z 2 ¼ ½1L ∑ i¼0

ðyiþ1 −yi Þ2  xiþ1 −xi

1=2

n−1

SF ¼

∑ ðyiþ1 −yi Þ2 ðxiþ1 −xi Þ i¼0

L

n−1

Rp ¼

∑ ½ðxiþ1 −xi Þ2 þ ðyiþ1 −yi Þ2  i¼0

L

1=2

74

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

Fig. 2. Statistical parameter of joint profile (Kulatilake et al., 1995).

According to Barton (1976), the peak shear strength of the rough joints could be approximated:

The Eq. (2) was modified to the generalized form by Barton and Choubey (1977) as Eq. (3) which incorporates different degrees of surface roughness from 0 to 20 for predicting of peak shear strength of joint.

    σc þ 30 τ ¼ σ n tan 20 log10 σn

    JCS þ ϕb τ ¼ σ n tan JRC log10 σn

ð2Þ

ð3Þ

Table 2 Constants of mathematical relationships of JRC with statistical parameters (Jang et al., 2014). Parameter

a b c R2 Parameter

a b c R2

Z2

SF

Rp − 1

Sampling interval (mm)

Sampling interval (mm)

Sampling interval (mm)

0.1

0.5

1

2

0.1

0.5

1

2

0.1

0.5

1

2

54.57 0.394 −19.13 0.962

51.16 0.531 −11.44 0.972

53.15 0.692 −6.32 0.986

54.14 0.65 −6.4 0.99

135.11 0.197 −19.15 0.962

73.95 0.266 −11.38 0.972

53.15 0.346 −6.31 0.986

34.49 0.325 −6.4 0.99

64.37 0.248 −14.35 0.964

65.9 0.302 −9.65 0.973

73.64 0.377 −5.52 0.987

72.85 0.35 −5.69 0.99

θ⁎max/(C + 1)2d

D−1

Sampling interval (mm)

Sampling interval (mm)

0.1

0.5

1

2

0.1

0.5

1

2

6.82 0.538 −12.13 0.971

5.3 0.605 −9.49 0.978

3 0.768 −4.83 0.99

2.78 0.813 −3.98 0.992

103.37 0.3 −8.54 0.99

107.76 0.319 −6.99 0.991

106.74 0.316 −6.63 0.985

96.29 0.276 −8.13 0.97

Table 3 Rock joint peak shear strength in the first cyclic shear test. Sample type

Joint morphological characteristic

Normal stress (MPa)

Peak shear strength of FPSP (MPa)

Peak shear strength of BPSP (MPa)

Reference

Limestone joint Granite joint

Sine shape

2.8

1.2

1.2

Cosine shape

6.5

0.95

0.85

Plaster joint Mortar joint Mortar joint Dolomite joint Granite joint Granite joint Granite joint Marble joint Marble joint Marble joint

Saw-tooth shape Sine shape Sine shape Irregular undulated shape Irregular undulated shape Irregular undulated shape Irregular undulated shape Irregular undulated shape Irregular undulated shape Irregular undulated shape

1 1 4 4.5 1.2 1 0.5 0.5 3 3

1.22 0.43 4.4 5.4 1.25 1.35 0.85 1.125 4.75 4.5

1.17 0.41 4.6 4.4 1.75 1.3 0.81 0.7 2.8 3

Hutson and Dowding (1990) Hutson and Dowding (1990) Huang et al. (1993) Homand et al. (2001) Homand et al. (2001) Huang et al. (1993) Homand et al. (2001) Lee et al. (2001) Lee et al. (2001) Lee et al. (2001) Lee et al. (2001) Lee et al. (2001)

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

Fig. 3. Shear stress-deformation curve of a regular undulated mortar joint (Homand et al., 2001).

where τ is peak shear strength, σn is effective normal stress, JRC is joint roughness coefficient, JCS is joint wall compression strength, which can be accepted as equal to uniaxial compression strength of rock for unweathered joint, ϕb is basic friction angle, and the majority of unweathered rock surfaces with values of ϕb ranging from 25° to 35° (Barton, 1976). Considering the impact of size effect on JRC value, Barton et al. (1985) proposed Eq. (4) as: JRCn ¼ JRC0 ðLn =L0 Þ−0:02JRC0

ð4Þ

where JRCn is representative roughness coefficient value of joint for the study, Ln is the length of engineering dimension, JRC0 is Barton's roughness coefficient value of joint in laboratory scale, for the length of 100 mm as L0. The empirical comparison method to acquire JRC is simple and convenient, and has been widely used in rock mechanics and engineering field. However, to a large extent, the method depends on the surveyor's experience (Beer et al., 2002). In order to reduce subjectivity of the empirical comparison method, the digital measurement methods and statistical parameters such as straight edge method and modified straight edge method (Barton and Bandis, 1982; Du et al., 1996), elongation method and direct measurement method (Wang, 1982; Turk and Dearman, 1985), largest surface inclination (θ⁎max) method (Tatone and Grasselli, 2010; Jang et al., 2014), fractal (D) method (Turk et al., 1987; Carr and Warriner, 1989; Lee et al., 1990; Xie and Pariseau, 1994; Jang et al., 2006; Jang et al., 2014; Li and Huang, 2015), root mean square (RMS) height method

Fig. 4. Shear stress-deformation curve of an irregular undulated granite joint (Homand et al., 2001).

75

(Tse and Cruden, 1979; Li and Zhang, 2015), root mean square first derivative (Z2) method (Tse and Cruden, 1979; Yu and Vayssade, 1991; Yang et al., 2001; Tatone and Grasselli, 2010; Jang et al., 2014; Li and Zhang, 2015), structural equation (SF) method (Tse and Cruden, 1979; Yu and Vayssade, 1991; Yang et al., 2001; Jang et al., 2014; Li and Zhang, 2015), roughness profile index (Rp) method (Maerz et al., 1990; Yu and Vayssade, 1991; Tatone and Grasselli, 2010; Jang et al., 2014; Li and Zhang, 2015) have been presented, as shown in Table 1. In Table 1, L is horizontal projection length of joint profile (mm), Ry is maximal height difference of joint profile (mm), Lt is trace length of joint profile (mm), yi is height at the point of xi of joint profile (mm), as are shown in Fig. 2 (Δx is horizontal spacing between xi and xi + 1, namely sampling interval) (Kulatilake et al., 1995), C is dimensionless parameter, D is fractal dimension, lga is intercept in the logarithm coordinate system, r is division value of joint profile, an is normalization intercept, h* is average base length of joint profile, L* is average height of joint profile. It was known that the statistical parameters mentioned above are closely related to JRC value and the digitized sampling interval of parameter has an effect on the correlation (Yu and Vayssade, 1991; Xia, 1996). Jang et al. (2014) established the mathematical relationships between statistical parameters such as Z2, SF, (Rp − 1), θ⁎max/(C + 1)2d, (D − 1) listed in Table 1 with different sampling intervals and JRC of Barton standard profiles. The results indicated that all mathematical relationships satisfy power law equations as shown in Eq. (5) with the sampling interval varying from 0.1 mm to 2 mm. The correlation coefficients between Z2, SF, (Rp − 1), θ⁎max/(C + 1)2d and JRC increase and the correlation coefficients of (D − 1) with JRC decrease with the increase of the sampling interval. JRC ¼ aðP Þb þ c

ð5Þ

where a, b, c are constants, the values of which are related to sampling interval, P is statistical parameter, as shown in Table 2 (R is correlation coefficient) (Jang et al., 2014). Rock joint dynamic shear behavior is important to understand the dynamic characteristics of rock mass (Qi et al., 2007). Some cyclic shear tests of natural rock joints or artificial joints have been carried

Fig. 5. Barton standard profiles after digital processing.

76

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

out to explore the rock joint dynamic shear behavior i. e. Plesha (1987), Hutson and Dowding (1990), Jing et al. (1992, 1993), Huang et al. (1993), Qiu et al. (1993), Homand et al. (2001), Lee et al. (2001), Jafari et al. (2003, 2004), Liu et al. (2011, 2013) et al. The test results showed that in the first cyclic shear test the difference between peak shear strength of forward positive shear process (FPSP) and peak shear strength of backward positive shear process (BPSP) of regular undulated joints was relatively small, however, the difference between peak shear strength of FPSP and BPSP of irregular undulated joints was relatively large, as shown in Table 3 and Figs. 3 and 4 acquired by Homand et al. (2001) with a computer-controlled 3D-shear apparatus presented by Boulon (1995). Therefore it can be concluded that the peak shear strength of irregular undulated joints is sensitive to its directivity. The reason is that the surfaces dipping opposite to shear direction

play a major role in the shear resistance behavior, but the impact of surfaces dipping towards the shear direction on the shear resistance behavior is relatively small. Under the condition of cyclic shear, the surfaces dipping opposite to shear direction of FPSP turn into surfaces dipping towards the shear direction of BPSP and the surfaces dipping towards the shear direction of FPSP become surfaces dipping opposite to shear direction of BPSP. Due to the differences of morphology characteristics between surfaces dipping opposite to shear direction and surfaces dipping towards the shear direction, the contributions from the surfaces dipping opposite to shear direction and the surfaces dipping towards the shear direction to peak shear strength of FPSP are different from that of BPSP. However, the mathematical relationships of digital measurement methods and statistical parameters with joint roughness coefficient don't consider the directivity of JRC from Table 1, and it cannot

Simulated curve

1.2

1.2

0.7

0.7

y (mm)

y (mm)

JRC=0.4

0.2 -0.3 0

50

100

0.2 -0.3 0

x (mm)

3

3

2

2

1 0 -1

1 0

0

50

100

0 -1

x (mm)

2

2

1.5

1.5

1 0.5 0 -0.5 0

0.5 0

50

100

-0.5 0

x (mm)

50

100

x (mm)

Simulated curve 1.5

1.5

1

y (mm)

1

y (mm)

100

1

JRC=6.7

0.5 0 -0.5 0

50

x (mm)

Simulated curve

y (mm)

y (mm)

JRC=5.8

-1

100

Simulated curve

y (mm)

y (mm)

JRC=2.8

50

x (mm)

50

x (mm)

100

0.5 0 -0.5 0 -1

50

x (mm)

Fig. 6. Comparison of Barton standard profiles and the simulated curves.

100

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

4 3 2 1 0 -1 0 -2

Simulated curve

y (mm)

y (mm)

JRC=9.5

50

100

x (mm)

4 3 2 1 0 -1 0 -2

JRC=10.8

0

50

100

-2

y (mm)

y (mm)

0

x (mm)

0 0

50

100

-2 -4

-6

-6

x (mm)

x (mm)

JRC=12.8

Simulated curve

4

4

2

2

0 0

50

100

y (mm)

y (mm)

100

2

-4

-2 -4

0 0 -4

x (mm)

100

x (mm)

Simulated curve 1

50

100

-3 -5

y (mm)

1 -1 0

50

-2

JRC=14.5

y (mm)

50

Simulated curve

2

-7

77

-1 0

50

100

-3 -5 -7

x (mm)

x (mm)

Fig. 6 (continued).

describe the difference of peak shear strength of FPSP and BPSP of irregular undulated joints under the condition of cyclic shear. Thus it is of particular significance to establish the mathematical relationships of statistical parameters considering shear direction with JRC for irregular undulated joints. The paper took Barton joint standard roughness profiles as typical representatives of irregular undulated joints with varied roughness to study the directivity of joint roughness. Based on the digital processing and discrete Fourier series expansion of Barton standard profiles, the original discrete point coordinates and analytic expression of each Barton standard profile were reached, the secant angles and tangent angles of surfaces dipping opposite to shear direction were calculated and then the mathematical relationships with Barton standard profiles roughness coefficient were investigated. And it has been found that the average

value of entire secant angles of all surfaces dipping opposite to shear direction (β100%) considers the directivity and has a higher sensitivity to evaluate JRC comparing with other parameters, and is recommended as a new index to describe JRC considering the directivity. Meanwhile, the impact of digitized sampling interval on the correlation of secant angles and tangent angles with JRC was researched. 2. Quantitative descriptions of Barton standard profiles Fourier series method has been successfully applied to quantitatively describe the characteristics of Barton standard profiles (Huang, 1992; Yang et al., 2001, 2010). An aperiodic function f(x) in the limited interval of L length can be seen as a cycle of periodic function, the cycle of which is L, which can be expressed as Fourier

78

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

Simulated curve

6

6

4

4

y (mm)

y (mm)

JRC=16.7

2 0

2 0

0 -2

50

100

0 -2

x (mm)

4 3 2 1 0 -1 0 -2

50

100

x (mm)

Simulated curve

y (mm)

y (mm)

JRC=18.7

50

100

x (mm)

4 3 2 1 0 -1 0 -2

50

100

x (mm)

Fig. 6 (continued).

series on the basis of Dirichlet condition. f ðxÞ ¼

    þ∞  a0 X 2πx 2πx þ an cos n þ bn sin n 2 n¼1 L L L

an ¼ 2L ∫ f ðxÞ cosðn

where

0

2πx Þdx ðn ¼ 0; 1; 2; …Þ , L

ð6Þ

bn ¼ 2L

L

2πx Þdx ðn ¼ 1; 2; …Þ. L Similarly, Fourier series can be used to describe the discrete function, as shown in Eq. (7): ∫ f ðxÞ sinðn

0

f ðxÞ ¼

where

    þ∞  a0 X 2πx 2πx þ an cos n þ bn sin n 2 n¼1 L L an ¼ 2L ∑ f ðxi Þ cosn i

2πxi Δx ðn ¼ 0; 1; 2; …Þ , L

ð7Þ

According to the original coordinates of discrete points of each Barton standard profile and the coordinates of corresponding discrete points of the simulated curve, the maximal height difference of standard profiles, parameters Z2, SF and Rp were calculated and the mathematical relationships of Z2, SF, Rp with JRC were established respectively. The power law equations of strong correlation are shown in Fig. 7. Fig. 7a shows that the relationship of JRC of each Barton standard profile with its maximal height difference is not positively correlative strictly, which is consistent with the conclusion of Jang et al. (2014). In consequence, the straight edge method and modified straight edge method of Table 1 are inapplicable to calculate the values of joint roughness coefficient of laboratory scale. Under the condition of equal sampling interval of 0.5 mm, the correlation coefficients of Z2, SF, Rp with JRC (Fig. 7b to g) calculated in this paper are larger than that of Jang et al. (2014) (Table 4), which further verifies the accuracy of digital processing and discrete Fourier series expansion of Barton standard profiles mentioned above.

bn ¼ 2L

2πxi Δx ðn ¼ 1; 2; …Þ. L Eq. (7) can be transformed into Eq. (8):

∑ f ðxi Þ sinn

3. Relationship of joint surface inclination with Barton standard profiles roughness coefficient

  2πx f ðxÞ ¼ A0 þ An cos n þ θn L n¼1

At present, more than ten joint peak shear strength equations are put forward in the rock mechanics and engineering field, and the largest surface inclination of surfaces dipping opposite to shear direction is directly incorporated into them, such as Patton (1966a, b); Plesha (1987), Jing et al. (1993), Homand et al. (2001) and Grasselli and Egger (2003). While in the Barton's strength empirical equation, the contribution of largest surface inclination of surfaces dipping opposite to shear direction to the peak shear strength is included in the JRC. The Barton standard profiles shown in Fig. 5 were established on the basis of one-way shear condition and the default one-way shear direction is from left to right. Under the condition of one-way shear, the largest secant angle (βmax) and largest tangent angle (αmax) of surfaces dipping opposite to shear direction were calculated through the original discrete point coordinates (sampling interval was 0.5 mm) and analytic expression (parameter n was 100) of each Barton standard profile respectively. The largest secant angle (βmax) was determined by substituting the ratio of maximal ordinate difference of adjacent discrete points of surfaces dipping opposite to shear direction to sampling interval

i

þ∞ X

where A0 ¼ a20 , An ¼

ð8Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n a2n þ bn , cosθn ¼ Aann , sinθn ¼ −b An ðn ¼ 1; 2; ⋯Þ,

and An is amplitude, θn is phase item. Each Barton standard profile has a length of about 100 mm. For easy to study, all of the profiles were set as length of 100 mm and conducted by digital processing (see Fig. 5). The length direction of each profile was set as abscissa axis and its normal direction was set as vertical axis and then each profile was discretized by equal sampling interval of 0.5 mm. The values of parameters an, bn were back calculated by coordinates of discrete points. The value of parameter n was set as 100, namely cosine curves and sine curves of top 100 were adopted to simulate the Barton standard profiles. Fig. 6 shows that the simulated curves are very similar to the standard profiles. According to Eqs. (7) or (8), the analytic expressions of Barton standard profiles were acquired when the value of parameter n was 100.

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

a

79

Maximal height difference (mm)

6 5 4 3 2 1 0 2

0

4

6

8

10

12

1 14

16

20

18

JR RC

b

c JRC= J =62 2.6611(Z2)0.95554-22

JRC C=662.86 66(Z Z2)0.99575--2

2

R2=0.9 9 = 9759

R =00.9751

d

f

20

g

JRC=104.13(Rp-1)0.5282-1.5 R2=0.9754

0 8 C=1122.0 SF)0.4788 09(S JRC -2 9 = 9759 R2=0.9

JRC=104.62(Rp-1)0.5295-1.5 R2=0.9769

20 15

JRC

15 JRC

e

F)0.47772-22 JRC= J =121.511(SF 2 0.9751 R =0

10 5

10 5

0

0 0

0.02

0.04

0.06

Rp-1

0

0.02

0.04

0.06

Rp-1

Fig. 7. Mathematical relationships of statistical parameters with JRC (a) Maximal height difference of each Barton standard profile; (b) Relationship of JRC with Z2 of Barton standard profiles; (c) Relationship of simulated curves roughness coefficient with Z2; (d) Relationship of JRC with SF of Barton standard profiles roughness coefficient; (e) Relationship of simulated curves roughness coefficient with SF; (f) Relationship of JRC with (Rp − 1) Barton standard profiles; (g) Relationship of simulated curves roughness coefficient with (Rp − 1).

(0.5 mm) into arctangent function. Taking the derivative of analytic expression (parameter n was 100) of Barton standard profile, the largest tangent angle (αmax) was acquired through substituting the Table 4 Comparison of mathematical relationships of Z2, SF, Rp with JRC. Parameter Z2

SF

Rp − 1

Equation 0.9554

JRC = 62.661(Z2) −2 JRC = 62.866(Z2)0.9575 − 2 JRC = 51.16(Z2)0.531 − 11.44 JRC=121.51(SF)0.4772 − 2 JRC=122.09(SF)0.4788 − 2 JRC=73.95(SF)0.266 − 11.38 JRC=104.13(Rp-1)0.5282 − 1.5 JRC=104.62(Rp-1)0.5295 − 1.5 JRC=65.9(Rp-1)0.302 − 9.65

R2

Reference

0.9751 0.9759 0.972 0.9751 0.9759 0.972 0.9754 0.9769 0.973

This paper This paper Jang et al. (2014) This paper This paper Jang et al. (2014) This paper This paper Jang et al. (2014)

maximal derivative value at the discrete points of surfaces dipping opposite to shear direction into arctangent function. The calculated results indicate that βmax of surfaces dipping opposite to shear direction is increasing with the increase of JRC except the third standard profile (JRC = 5.8), the fifth (JRC = 9.5), the sixth (JRC = 10.8) and the ninth one (JRC = 16.7) (Fig. 8a), and αmax is also increasing with the increase of JRC except the third standard profile (JRC = 5.8), the fifth (JRC = 9.5) and the sixth one (JRC = 10.8) (Fig. 8b). The exceptions may be caused by that all of the surfaces dipping opposite to shear direction with different surface inclinations play a role in the shear resistance behavior before reaching the joint peak shear strength, and the impact of joint on the peak shear strength is not merely reflect in the largest surface inclination of surfaces dipping opposite to shear direction. Therefore, it should consider other surfaces dipping opposite to shear direction besides the one with largest inclination.

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b 60 50 40 30 20 10 0 0

5

10

15

20

Largest tangent angle (o)

Largest secant angle (o)

a

60 50 40 30 20 10 0 0

JRC

5

10

15

20

JRC

Fig. 8. (a) Relationship of Barton standard profiles roughness coefficient with largest secant angle (βmax) of surfaces dipping opposite to shear direction; (b) Relationship of Barton standard profiles roughness coefficient with largest tangent angle (αmax) of surfaces dipping opposite to shear direction. Table 5 Secant angles and tangent angles of surfaces dipping opposite to shear direction of Barton standard profiles under the condition of one-way shear. βmax(°)

Average value of secant angles of surfaces dipping opposite to shear direction βx% β10%(°)

β20%(°)

β30%(°)

β40%(°)

β50%(°)

β60%(°)

β70%(°)

β80%(°)

β90%(°)

β100% (°)

4 14 12 25 22 18 30 34 33 50

3.33333 10.20000 11.18182 19.36364 18.25000 16.00000 24.00000 27.10000 29.80000 36.22222

2.96000 6.83333 9.95455 17.09091 15.64000 14.95000 20.68421 22.47619 28.05000 31.52632

2.63158 5.55556 9.15152 14.67647 13.89474 14.10000 18.00000 19.81250 26.13333 28.24138

2.48000 4.78333 8.45455 12.53333 12.35294 13.37500 15.58974 17.76190 24.20000 25.64103

2.20635 4.20000 7.69091 10.59649 10.85938 12.58824 13.81633 15.75472 22.08000 23.63265

2.00000 3.66667 6.98485 9.19118 9.77632 11.73770 12.45763 13.96875 19.95000 21.86207

1.85393 3.28571 6.29870 8.05063 8.91011 10.74648 11.27536 12.50667 18.15714 19.98529

1.65347 2.94167 5.69318 7.02198 8.15686 9.82716 10.21519 11.35294 16.61250 18.08974

1.46491 2.59559 5.11111 6.26471 7.40870 8.92308 9.25843 10.18750 14.84615 16.36364

1.31496 2.33775 4.60000 5.60526 6.66406 7.97059 8.32323 9.19626 13.41584 14.78571

αmax(°)

Average value of tangent angles of surfaces dipping opposite to shear direction αx% α10%(°)

α20%(°)

α30%(°)

α40%(°)

α50%(°)

α60%(°)

α70%(°)

α80%(°)

α90%(°)

α100% (°)

4 14 12 25 22 18 30 34 38 48

3.33333 10.20000 11.00000 19.63636 18.58333 16.10000 23.90000 26.63636 31.00000 36.66667

2.92000 6.76667 9.90909 17.08696 15.92000 14.95000 20.30000 22.47619 27.95238 32.00000

2.60526 5.51111 9.08824 15.00000 14.07895 14.20000 17.73333 19.81250 25.68750 28.82759

2.45098 4.71666 8.37778 12.78261 12.50000 13.45000 15.37500 17.51163 23.88095 26.38462

2.20313 4.17333 7.64286 10.96491 10.93651 12.62745 13.64000 15.50000 21.77358 24.18367

2.01316 3.64444 6.89552 9.42029 9.75000 11.73770 12.31667 13.75384 19.71785 22.29310

1.86517 3.26667 6.18987 8.26250 8.87640 10.76056 11.21429 12.36842 17.77333 20.29412

1.65686 2.95000 5.58889 7.38889 8.14852 9.79012 10.20000 11.08046 16.14118 18.43590

1.46957 2.62222 5.02970 6.45631 7.38597 8.85714 9.19838 9.96939 14.51042 16.72727

1.32031 2.36000 4.49558 5.78261 6.66142 7.93137 8.21782 8.97248 13.04673 15.10204

All secant angles and tangent angles of surfaces dipping opposite to shear direction were calculated by the ratios of ordinate difference of adjacent discrete points of surfaces dipping opposite to shear direction to sampling interval (0.5 mm) and derivative values at the discrete points of surfaces dipping opposite to shear direction for each Barton standard profile respectively. The entire secant angles and tangent angles of each Barton standard profile were arranged in order from big to small respectively. Assumed βx% and αx% are average values of secant angles and tangent angles of surfaces dipping opposite to shear direction respectively, and subscript x% represents the top percent of angles of surfaces dipping opposite to shear direction, e.g. if there are fifty angles, subscript 10% refers to the top five angles, and β10% is average value of top five entire secant angles. And then we established the mathematical relationships of β10%, β20%, β30%,…, β90%, β100% and α10%, α20%, α30%,…, α90%, α100% with JRC by power law equations respectively. The mathematical relationships of secant angle and tangent angle with JRC are shown in Tables 5 and 6 and Fig. 9. Table 6 shows that if x% is more than fifty percent, βx% (αx%) will increase with the increase of JRC under the condition of one-way shear. The larger the x%, that means the more the sample quantity of tangent angles of surfaces dipping opposite to shear direction are involved in, the stronger the correlation of βx% (αx%) with JRC is reached and β100% has the largest correlation coefficient with JRC. That is all of the surfaces dipping opposite to shear direction with varied surface inclinations play

Table 6 Relationships of secant angle and tangent angle of surfaces dipping opposite to shear direction of Barton standard profiles with JRC under the condition of one-way shear. Secant angle βmax(°) β10%(°) β20%(°) β30%(°) β40%(°) β50%(°) β60%(°) β70%(°) β80%(°) β90%(°) β100%(°) Tangent angle αmax(°) α10%(°) α20%(°) α30%(°) α40%(°) α50%(°) α60%(°) α70%(°) α80%(°) α90%(°) α100%(°)

R2

Equation 1.2728

JRC = 0.1696βmax − 0.4 JRC = 0.2047β10%1.3047 − 0.5 1.0655 JRC = 0.5342β20% − 1.2 JRC = 0.7700β30%0.9902 − 1.5 JRC = 1.0891β40%0.9160 − 2 JRC = 1.4040β50%0.8662 − 2.3 JRC = 1.7503β60%0.8234 − 2.6 JRC = 2.1582β70%0.7837 − 3 JRC = 2.4423β80%0.7674 − 3.1 JRC = 2.6126β90%0.7700 − 3 JRC = 2.9206β100%0.7592 − 3.1

0.8902 0.9453 0.9637 0.9721 0.9769 0.9812 0.9824 0.9833 0.9855 0.9860 0.9864 R2

Equation 1.2238

JRC = 0.1985αmax − 0.5 JRC = 0.2103α10%1.2927 − 0.5 1.0576 JRC = 0.5466α20% − 1.2 JRC = 0.7809α30%0.9829 − 1.5 JRC = 1.1147α40%0.9068 − 2 JRC = 1.3572α50%0.8753 − 2.2 JRC = 1.7562α60%0.8223 − 2.6 JRC = 2.1599α70%0.784 − 3 JRC = 2.2926α80%0.7885 − 2.9 JRC = 2.6035α90%0.7727 − 3 JRC = 2.8309α100%0.772 − 3

0.8965 0.9427 0.9636 0.9696 0.9738 0.9767 0.9787 0.9806 0.9815 0.9829 0.9834

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

a role in the shear resistance behavior before reaching the joint peak shear strength. Establishing the mathematical relationships of β100%, α100% with JRC respectively as shown in Eqs. (9) and (10), it can be found that the former (R2 = 0.9864) has a higher sensitivity than the later (R2 = 0.9834). Therefore β100% is recommended as a new statistical parameter to establish the mathematical relationship with JRC as Eq. (9): JRC ¼ 2:9206β0:7592 100% −3:1

defined as joint roughness coefficients of FPSP with shear direction from left to right and BPSP with shear direction from right to left respectively, as shown in Fig. 5. Therefore the Barton standard profiles roughness coefficient under the condition of one-way shear mentioned above is JRCfp. Adopting the same analytical method as the condition of oneway shear, β100% and α100% of BPSP with JRCfp were researched, as shown in Table 7 and Fig. 10. It can be seen that the correlation of β100% and α100% of BPSP with JRCfp is a little weaker than that of FPSP, the reason of which is β100% and α100% have their directivity, in consequence, JRCfp and JRCbp should be evaluated by β100% or α100% of FPSP and β100% or α100% of BPSP respectively. Substituting β100% of BPSP into Eq. (9), the JRCbp values of Barton standard profiles were calculated, as shown in Table 8 and Fig. 11. It can be found that for each Barton standard profile, the JRCbp has a little

ð9Þ

JRC ¼ 2:8309α 0:772 100% −3

ð10Þ

To study the directivity of JRC, the Barton standard profiles were supposed under the condition of cyclic shear, and JRCfp and JRCbp were

a

JRC

20 JRC

b

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 15 10 5

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0

0

0 0

10

20

30

40

50

JRC

JRC

d

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0 0

5

10

15

20

25

30

35

0

JRC

JRC

0

10

20

30

5

10

15

20

25

Average value of top 60% of entire angles (o)

20

30

5

10

15

20

25

Average value of top 50% of entire angles (o)

JRC

JRC

0

40

Secant angle Tangent angle Secant angle matching Tangent angle matching

0

h

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0

10

25 20 15 10 5 0

Average value of top 40% of entire angles (o)

g

30

Average value of top 30% of entire angles (o)

f

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0

20

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0

Average value of top 20% of entire angles (o)

e

10

Average value of top 10% of entire angles (o)

Largest angle (o)

c

81

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0 0

5

10

15

20

Average value of top 70% of entire angles (o)

Fig. 9. Relationships of secant angle and tangent angle of surfaces dipping opposite to shear direction of Barton standard profiles with JRC under the condition of one-way shear (a) Relationship of largest angle with JRC; (b) Relationship of average value of top 10% of entire angles with JRC; (c) Relationship of average value of top 20% of entire angles with JRC; (d) Relationship of average value of top 30% of entire angles with JRC; (e) Relationship of average value of top 40% of entire angles with JRC; (f) Relationship of average value of top 50% of entire angles with JRC; (g) Relationship of average value of top 60% of entire angles with JRC; (h) Relationship of average value of top 70% of entire angles with JRC; (i) Relationship of average value of top 80% of entire angles with JRC; (j) Relationship of average value of top 90% of entire angles with JRC; (k) Relationship of average value of entire angles with JRC; (l) Relationship of secant angle and tangent angle with JRC.

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B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

j

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0

JRC

JRC

i

0

5

10

15

20

0

Average value of top 80% of entire angles (o)

k

Secant angle Tangent angle Secant angle matching Tangent angle matching

25

l

15 10 5 0 0

5

10

5

10

15

20

Average value of top 90% of entire angles (o)

R2

JRC

20

Secant angle Tangent angle Secant angle matching Tangent angle matching

25 20 15 10 5 0

1 0.98 0.96 0.94 0.92 0.9 0.88

Secant angle Tangent angle

0 10 20 30 40 50 60 70 80 90 100

15

Average value of entire angles (o)

Top percent of angles (%)

Fig. 9 (continued).

Table 7 Relationships of secant angle and tangent angle of surfaces dipping opposite to shear direction of BPSP with JRCfp value. JRCfp value

β100% of BPSP (°)

α100% of BPSP (°)

0.4 2.8 5.8 6.7 9.5 10.8 12.8 14.5 16.7 18.7 Equation R2

1.45872 1.45882 4.10280 4.70339 7.33333 7.69444 6.77478 10.51020 11.33333 12.57547 JRC = 5.5779β100%0.5535 − 5.6 0.9404

1.47706 1.53012 4.12381 4.87179 7.75000 8.00943 6.96296 10.07843 11.83333 12.32381 JRC = 4.9361α100%0.5903 − 5 0.9405

larger difference with the JRCfp, and basically the latter is larger than the former except the first and the fifth standard profile. The difference between JRCbp and JRCfp verifies the close relationship of irregular undulated joint roughness coefficient and shear direction further. In comparison, the mathematical relationships of digital measurement methods and statistical parameters such as Z2, SF, (Rp − 1), (D − 1) (Table 1) don't reflect the directivity of JRC which have limitations to

a

calculate JRC value for irregular undulated joints. Meanwhile, it should be noted that for BPSP, the JRCbp of the seventh standard profile is even smaller than that of the fifth and the sixth one, that means under BPSP condition, the sequence of the Barton standard profile should be rearranged by the increase of JRCbp, and the seventh standard profile should be moved to front of the fifth one as shown in the right column of Fig. 11. 4. Impact of digitized sampling interval on the Barton standard profiles roughness coefficient Previous researches showed that the correlation of Barton standard profiles roughness coefficient with statistical parameters such as Z2, SF, (Rp − 1), θ⁎max/(C + 1)2d, (D − 1) varies with the digitized sampling interval varying, which affects the acquisition of JRC accurately (Jang et al., 2014). The Barton standard profiles after digital processing were discretized by four kinds of equal sampling intervals of 0.5 mm, 1 mm, 2 mm and 4 mm. The correlation coefficients of JRCfp with β100% and α100% of FPSP of different sampling intervals were calculated respectively and then the impact of digitized sampling interval on the correlation was explored, as shown in Tables 9 and 10. Tables 9 and 10 show that the correlation coefficients of JRCfp with β100% and α100% of FPSP decrease with the increase of digitized sampling interval in the ranges of 0.5 mm to 4 mm basically except the abnormal increase of β100% once the interval is 2 mm. The reason is that with

b

20 JRC=5.5779β100%0.5535-5.6

JRC

JRC

15 R2=0.9404 10 5

20

JRC=4.9361α100%0.5903-5

15

R2=0.9405

10 5

0

0 0

5

10

15

Average value of entire secant angles (o)

0

5

10

15

Average value of entire tangent angles (o)

Fig. 10. (a) Relationship of secant angle of surfaces dipping opposite to shear direction of BPSP with JRCfp value; (b) Relationship of tangent angle of surfaces dipping opposite to shear direction of BPSP with JRCfp value.

B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

shear. Based on the digital processing and discrete Fourier series expansion techniques, the paper explored the morphological characteristics of the Barton joint standard roughness profiles under cyclic shear, the surface inclinations of FPSP and BPSP were calculated, and then the relationships with JRC were investigated respectively. The main conclusions are as follows:

Table 8 Barton standard profiles roughness coefficient under the condition of cyclic shear. Profile No.

JRCfp value

Profile No.

1 2 3 4 5 6 7 8 9 10 Equation R2

0.4 2.8 5.8 6.7 9.5 10.8 12.8 14.5 16.7 18.7

1 0.7901 2 0.7903 3 5.429 4 6.362 5 (6) 10.156 6 (7) 10.649 7 (5) 9.382 8 14.321 9 15.348 10 16.863 JRCfp = 2.9206β100%0.7592 − 3.1 0.9864

JRCbp value

Note: the number in parenthesis is the sequence number of Barton standard profiles under BPSP condition.

sampling interval increasing, the sample quantity of secant angles and tangent angles involved in the calculation decreases, which is unable to represent the integral shear effect of surfaces dipping opposite to shear direction entirely. It also can be found that the coefficient of JRCfp with β100% is a little larger than that of α100% with same sampling interval. According to β100% and α100% with the sampling interval of 0.5 mm, the JRCfp values were calculated by Eqs. (9) and (10), as shown in Table 11. It can be seen that the difference between JRCfp values calculated through Eq. (9) and JRC standard values is ranged from 0.10 to 1.86, while the difference of JRC values calculated through the Eq. (10) and JRC standard values is ranged from 0.11 to 2.10. 5. Conclusions The paper gave a deep review on the study carried out on the joint roughness (shear strength), and found that the roughness of irregular undulated joints under the condition of cyclic shear has a strong directivity. In order to reduce subjectivity of selection of representative value of JRC, the statistical parameter related to shear direction should be introduced to determine the JRC value. Surface inclination can reflect the roughness difference of a joint and embody the directivity dependence, so the paper studied the relationship of surface inclination with JRC of irregular undulated joints under cyclic

Profile No. Barton standard profiles

83

JRCfp

(1) All mathematical relationships of βmax (αmax), βx% (αx%) (x% = 10%, 20%, 30%,…, 90%, 100%) of surfaces dipping opposite to shear direction with JRC satisfy the power law equations. With the increase of x%, the correlation coefficient is increasing, β100% has the largest correlation coefficient with JRC, and βmax has the smallest one. This indicates that all of the surfaces dipping opposite to shear direction with varied inclinations play a role in the shear resistance behavior before reaching the joint peak shear strength, and it is not suitable to just use the largest inclination to describe the shear behavior of a joint. (2) Comparing with other statistical parameters, β100% has a higher sensitivity to evaluate JRC, and with the increase of β100%, the JRC value always increases without exception. (3) With Eq. (9), the JRC values of BPSP (JRCbp) were calculated by β100% of BPSP. The calculation results show that the JRC of FPSP (JRCfp) is different from JRCbp for each Barton standard profile, and the former is basically larger than the latter except the first and the fifth standard profile. Moreover, the JRCbp value of the seventh standard profile is smaller than that of the fifth and sixth one, and the sequence of Barton standard profiles should be adjusted under BPSP condition, as shown in the right column of Fig. 11. (4) Comparing with other statistical parameters, the law of impact of digitized sampling intervals on the mathematical relationships of surface inclinations of surfaces dipping opposite to shear direction with JRC of Barton standard profiles is similar to (D-1) and different from Z2, SF, (Rp − 1), θ⁎max/(C + 1)2d (Jang et al., 2014). With sampling interval increasing, the correlation coefficient of the relationships between β100% and α100% and JRC both present a trend of decreasing and the former is a little larger than the latter. (5) β100% can be used as a good index to evaluate the value of JRC with Eq. (9) under cyclic shear. Moreover, it's easy to acquire the discrete point coordinates and β100% according to digital processing of actual joint profile in practice.

Profile No. Barton standard profiles JRCbp

1

0.4

1

0.7901

2

2.8

2

0.7903

3

5.8

3

5.429

4

6.7

4

6.362

5

9.5

5 ( 6)

10.156

6

10.8

6 ( 7)

10.649

7

12.8

7 (5)

9.382

8

14.5

8

14.321

9

16.7

9

15.348

10

18.7

10

16.863

0

100

mm

0

100

mm

Note: the number in parenthesis is the sequence number of Barton standard profiles under BPSP condition. Fig. 11. Barton joint standard roughness profiles under the condition of cyclic shear.

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B. Zheng, S. Qi / Engineering Geology 212 (2016) 72–85

Table 9 Relationships of secant angle of surfaces dipping opposite to shear direction of FPSP with JRCfp value under the condition of different sampling intervals. JRCfp value

0.4 2.8 5.8 6.7 9.5 10.8 12.8 14.5 16.7 18.7 Equation R2

Sampling interval 0.5 mm

1 mm

2 mm

4 mm

β100% (°)

β100% (°)

β100% (°)

β100% (°)

1.31450 2.33775 4.60000 5.60526 6.66406 7.97059 8.32323 9.19626 13.41584 14.78571 JRCfp = 2.9206β100%0.7592 − 3.1 0.9864

1.29688 2.34667 4.50000 5.53571 6.58730 7.75000 7.80769 8.85455 13.07692 14.29787 JRCfp = 2.7658β100%0.7886 − 2.9 0.9839

1.40000 2.28947 4.24138 4.75862 6.09375 7.40741 8.00000 9.34615 10.69231 13.86364 JRCfp = 3.8544β100%0.7001 − 3 0.9944

1.2352941 2.38889 3.85714 4.61538 6.00000 8.25000 8.08333 8.13333 9.42857 14.60000 JRCfp = 3.0396β100%0.7804 − 3.1 0.9787

Table 10 Relationships of tangent angle of surfaces dipping opposite to shear direction of FPSP with JRCfp value under the condition of different sampling intervals. JRCfp value

0.4 2.8 5.8 6.7 9.5 10.8 12.8 14.5 16.7 18.7 Equation R2

Sampling interval 0.5 mm

1 mm

2 mm

4 mm

α100% (°)

α100% (°)

α100% (°)

α100% (°)

1.32031 2.36000 4.49558 5.78261 6.66142 7.93137 8.21782 8.97248 13.04673 15.10204 JRCfp = 2.8309α100%0.772 − 3 0.9834

1.32813 2.36000 4.50877 5.65000 6.69841 8.01961 8.28000 9.05556 12.79630 16.33333 JRCfp = 3.263α100%0.718 − 3.5 0.9828

1.33333 2.40541 4.75000 5.93333 6.87097 8.54167 8.75000 8.71429 14.07407 16.66667 JRCfp = 2.9379α100%0.7397 − 3.1 0.9761

1.43750 2.36842 4.84615 5.46154 6.81250 8.90909 7.41667 10.07692 13.00000 16.41667 JRCfp = 3.5624α100%0.6856 − 4 0.9756

Table 11 Comparison of JRC standard value and JRCfp simulated value of Barton standard profiles. JRC standard value

0.4 2.8 5.8 6.7 9.5 10.8 12.8 14.5 16.7 18.7

Sampling interval 0.5 mm

0.5 mm

Secant angle matching JRCfp value

Tangent angle matching JRCfp value

0.495 2.464 6.203 7.710 9.227 11.022 11.493 12.641 17.868 19.474

0.508 2.493 6.034 7.972 9.238 11.003 11.392 12.402 17.563 20.022

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