Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface Guangcheng Zhang, Murat Karakus, Huiming Tang, Yunfeng Ge & Qiangqiang Jiang Rock Mechanics and Rock Engineering ISSN 0723-2632 Rock Mech Rock Eng DOI 10.1007/s00603-017-1264-5
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Author's personal copy Rock Mech Rock Eng DOI 10.1007/s00603-017-1264-5
TECHNICAL NOTE
Estimation of Joint Roughness Coefficient from ThreeDimensional Discontinuity Surface Guangcheng Zhang1,2 • Murat Karakus2 Qiangqiang Jiang1
•
Huiming Tang1 • Yunfeng Ge1
•
Received: 11 September 2016 / Accepted: 8 June 2017 Ó Springer-Verlag GmbH Austria 2017
Keywords 3D joint roughness coefficient The modified root mean square method Peak shear strength Rock joints Empirical approach List of symbols k Roughness index Rp Roughness profile Z20 The modified root mean square value Rp Roughness profile index D Fractal dimension h* The inclination of the individual line segments Dr1d 9 Kv A parameter of capturing the overall roughness characteristics of natural rock joints well by fractal techniques Rs Roughness coefficients B A parameter relate to JRC H The mean height of surface asperities in the mean line L The length of the 2D-profile M The divided number of discontinuity surface T The peak shear strength of rock joints rn The effective normal stress s The peak shear strength of the joint /b The total friction angle of the flat surface Ac The potential contact area A0 The maximum possible contact area & Murat Karakus
[email protected] 1
Faculty of Engineering, China University of Geosciences, Wuhan, China
2
School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, Australia
hmax C rc /r a 0
Z Lx Ly * * *
a; b; c Leq dSs dSxy x n, y n, z n xs, ys hs hx,
y
SABC Ss,ABC Sxy,ABC
The maximum apparent dip angle in the shear direction A ‘‘roughness’’ parameter The compressive strength of the intact material obtained from a standard uniaxial test The residual friction angle The angle between the schistosity plane and the normal to the joint The position of the horizontal plane Length in the x-direction of roughness surface sample Length in the y-direction of roughness surface sample The vectors of the tetrahedron sides The equivalent length in the shear direction The projections of shear direction The projections of horizontal plane The inner normal vector of ABC plane The unit vectors in x- and y-direction The included angle of ABC with the shear plane The included angle of ABC with the horizontal plane The area of ABC The projected area of ABC in the shear plane The projected area of ABC in the horizontal plane
1 Introduction This paper proposes a new method to account for the variation of roughness profiles in three-dimensional space in rock mass. Shear strength and hydro-mechanical
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behavior of discontinuities in rock masses are heavily influenced by the joint roughness properties (Tse and Cruden 1979; Maerz and Franklin 1990), and thus it must be defined accurately to avoid design errors. Pioneering work on this area was done by Barton who introduced the joint roughness coefficient (JRC) to define discontinuity roughness (Barton 1973). Since then, JRC has been widely used in rock engineering. It is well understood that JRC has directional dependency with which shear strength and hydro-mechanical properties vary significantly. Therefore, many researchers developed some models to estimate JRC accurately in two dimensions (e.g., Tse and Cruden 1979; Maerz and Franklin 1990; Kulatilake et al. 2006; Tatone and Grasselli 2010), which obviously lacks accounting the variation of roughness in three-dimensional (3D) surface. Some researchers, such as Herda (2006), Ge et al. (2014), have used 3D laser technology to identify full 3D profiles of discontinuities. 3D laser technology provides precise models for open discontinuity surfaces. However, it is impossible to have a complete 3D roughness profile for discontinuities within the rock mass. Therefore, 2D trace of discontinuities is usually used in calculating the JRC of discontinuities in rock mass. Recently, a new method to predict 2D JRC was proposed by Zhang et al. (2014). They introduced a new roughness index (k) using the Root Mean Square method, which considers the influence of inclination angle and amplitude of asperities and their direction on the JRC. The new roughness index, k, has been found to be reliable in determining the JRC (Zhang et al. 2014). However, the JRC values in 2D profiles may not represent the roughness accurately in all direction. Therefore, 3D JRC profiles must be developed to account for the variation of roughness profiles in three-dimensional space. In this way, the shear strength of discontinuity surfaces can be accurately calculated for any shear direction, a. This paper is structured as follows: The application of JRC in geotechnical engineering is studied in Sect. 1. Section 2 reviews the available methods used for calculating JRC values. The methodology and logical development stages of 3D JRC model are presented in Sect. 3. Lastly, 2D JRC (Zhang et al. 2014), the proposed 3D JRC and 3D JRC values from a method developed by Grasselli (2001) are calculated using FORTRAN. Subsequently, JRC and peak shear strengths by all methods and experimental results were compared. For experimental works, we collected five joint samples of limestone which are the main components of the Jiweishan Mountain rockslide. Then, we scanned the 3D surfaces of five joint samples on which direct shear tests were conducted.
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2 Two- and Three-Dimensional JRC Application in Geotechnical Engineering JRC may be estimated by visual inspection of rock joint profile and comparing with the 10-standard profiles proposed by Barton and Choubey (1977). Although this method is simple, it is considerably subjective (AlamedaHerna´ndez et al. 2014) due to the fact that comparison of joint profiles by visual inspection is influenced by human error (Herda 2006; Ge et al. 2014). Therefore, alternative methods to visual comparison have been developed by researchers to eliminate this deficiency. Myers (1962) proposed four characteristics for surface roughness identification and suggested that the Root Mean Square (Z2) was most useful. Tse and Cruden (1979) analyzed the relationship between JRC and eight parameters that have been used to characterize numerically the roughness of surfaces. The authors pointed out that the root mean square of the first derivative (Z2) and the structure function (SF) of the profile are strongly correlated with the values of the JRC. The roughness profile index (Rp) is defined as the ratio of the true length of a 2D roughness profile to its nominal length. Rougher profiles display greater ratios because the true profile length is larger due to the increased undulation and unevenness. Maerz and Franklin (1990) proposed a method for estimating JRC using a regression model. The model describes a relationship between the JRC and the roughness profile index, Rp, which is defined as the ratio of the true length of a fracture surface trace to its projected length. Recently, a number of researchers have applied the concept of a fractal dimension (D) to describe rock joint surfaces (e.g., Mandelbrot 1985; Kulatilake et al. 2006; Ge et al. 2014). Lee et al. (1990) reported that the value of the fractal dimension is directly proportional to the JRC values for the roughness profile. However, Tatone and Grasselli (2010) suggested that the inclination, h*, of the individual line segments forms the profile. They suggested this after selecting a forward or reverse direction in their analysis, and the empirical equations in the form of a power law were found to best relate the mean values of hmax =ðC þ 1Þ2D to the 2D JRC. Kulatilake et al. (2006) introduced a new parameter Dr1d 9 Kv that captures the overall roughness characteristics of natural rock joints by using fractal techniques. It has been found that the JRC is affected by the shear direction, inclination angle and asperities of the joint surface (e.g., Yang et al. 2001; Tatone and Grasselli 2012). However, there is no singular method that considers all of the factors summarized above. The evaluation methods of the JRC are based on several contact and non-contact techniques used to measure the
Author's personal copy Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface
surface topography of discontinuity surfaces in rock. The most commonly used methods are mechanical profilometry (Brown and Scholz 1985), shadow profilometry (Maerz and Franklin 1990), stylus profilometry (Kulatilake et al. 1995) and laser profilometry (Huang et al. 1992; Lanaro 2000). Stylus profilometry and laser profilometry produce very detailed profiles of the surface roughness. Recently, with the rapid improvement in 3D laser scanning technology, 3D scanners are being used to digitize the surface roughness profile and then to determine the JRC and Roughness Coefficients (Rs) (Karmen 2010). Tatone and Grasselli (2012) presented a method that uses high-resolution, 3D surface measurements to measure the directional fracture roughness and the spatial aperture distribution under zero normal stress of fractures in rock cores. The method is based on the cumulative distribution of the inclination of the line segments forming a roughness profile. Ge et al. (2014) proposed a modified 3D divider and variogram method to quantify natural rock joint roughness based on fractal dimensions. This provided possible sliding directional values (under gravitational loading) close to that reported by Ge et al. (2014).
3 Methodology for Calculating 3D JRC In order to define the 3D JRC for surface roughness, we used 2D JRC defined by Zhang et al. (2014). The 2D JRC was formulated as: JRC ¼
40 20 1 þ ebk
ð1Þ
where k is a roughness index calculated based on the Root Mean Square method. The upper, suggested values and the lower values of the JRC are obtained when the parameter b equals 30, 20 and 15, respectively. Therefore, k can be obtained by 1=3 h 0 k¼ ðZ2 Þ2=3 ð2Þ L where 1 h¼ L
Z
x¼L
j yjdx ¼ x¼0
N X jyiþ1 þ yi jðxiþ1 xi Þ i¼1
2L
ð3Þ
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 x¼L dy 0 Z2 ¼ ðmaxð0; ÞÞ2 dx L x¼0 dx ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M uX ðmaxð0; yiþ1 yi ÞÞ2 ¼t ðxiþ1 xi ÞL i¼1
ð4Þ
where h is the mean height of surface asperities in the mean line which is a horizontal line, where the mean height is the minimum, between the peak and the valley. L represents 0 the length of the 2D-profile, Z2 is the modified Root Mean Square (RMS) value, x is the x-coordinate and y is the ycoordinate, and M denotes the divided number of surface profile. The proposed method of calculating the 2D JRC produces reasonably good results. However, it still fails to define three-dimensional roughness profile. Therefore, we aim to define full three-dimensional JRC. For this, we modified Eqs. (3) and (4). In this process, the first step is to determine the position of the mean plane, which is the key component of calculating the mean asperity height in Eq. (3). The proposed method to calculate 3D JRC is given in ‘‘Appendix,’’ including methods to determine the mean plane and the theory of the modified 3D RMS.
4 Three-Dimensional JRC and Peak Shear Strength Rock joint roughness has attracted accelerating attention by many researchers and engineers over the past four decades because of its critical influence on the shear strength of rock joints. The most widely used formula for estimating the shear strengths of rock joints was proposed by Barton (1973) as: JCS s ¼ rn tan JRClog10 þ /b ð5Þ rn where s is the peak shear strength of rock joints, rn is the effective normal stress, and /b is the total friction angle of the flat surface. In his comprehensive review of the shear strength of rock joints, Barton described the important influence of joints’ surface roughness on the shear strength (Barton 1973). The JRC values of roughness profiles vary from 0 to 20, and /b and Joint Wall Compressive Strength (JCS) can be obtained accurately from shear box tests and Schmidt hammer tests, respectively. Grasselli (2001) and Tatone and Grasselli (2010, 2013) proposed a new 2D and 3D discontinuity roughness parameter to calculate the shear strength of rock joints: hmax h C Ac ¼ A0 ð6Þ hmax where Ac is the potential contact area, A0 is the maximum possible contact area, hmax is the maximum apparent dip angle in the shear direction, and C is a ‘‘roughness’’ parameter, calculated using a best-fit regression function. This is shown in Fig. 2, which is based on the 3D digitized figures (see Fig. 1), that characterizes the distribution of the apparent dip angles over the surface. C and hmax depend
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30
30
8.500
20
11.55
7.435
20
10.10 6.370
10 8.650
10
5.305
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4.240
Z
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3.175
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(e) JW03-3BR Fig. 1 3D digitized data of joint surfaces
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Y
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2.963 0.000
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0° 60° 120° 180° 240° 300°
Normalized Length, Lθ*
0.6 0.5
0° : C = 3.5054, θmax = 71.53, A0 = 0.4754
0.4
60° : C = 2.9499, θmax = 69.45, A0 = 0.6027 120° : C = 2.7313, θmax = 62.90, A0 = 0.6183
0.3
180° : C = 3.2918, θmax = 73.48, A0 = 0.5246 240° : C = 3.3926, θmax = 77.86, A0 = 0.3973
0.2
300° : C = 3.8443, θmax = 69.28, A0 = 0.381 7
0.1 0.0 0
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Angular Threshold, θ*(degrees) Fig. 2 Example of the distribution of normalized length, Lh*, as a function of the angular threshold, h*, in different shear direction (JW02-1BL)
on the specified shear direction. hmax =ð1 þ CÞ is defined to describe the roughness of the joint surface. The upper limit of the curve, where it intersects with the y-axis, is A0; the curve intersects the x-axis at hmax . Based on many lab tests, the following expression for peak shear strength was proposed by Grasselli (2001): h max rn A0 Crc s ¼ rn tanð/r Þ 1 þ e ð7Þ where s is the peak shear strength of the joint; rn is the effective normal stress; rc is the compressive strength of the intact material obtained from a standard uniaxial test; /r is the residual friction angle (measured after a standard displacement of 5 mm); and other parameters defined as above. Here, we can calculate the value of /r by the following expression suggested by Grasselli (2001): cos a 1=C /r ¼ /b þ C A1:5 ð8Þ 0 hmax 1 A0 where a is the angle between the schistosity plane and the normal to the joint. If the rock does not exhibit schistosity, a is assumed to be equal to zero. Graselli’s method considers the shear direction, which is similar to our proposed method. We implemented our new method along with Graselli’s method using Compaq Visual FORTRAN 6 to compare their performance. In order to validate our proposed model, experimental data were used. For the experimental study, rock joints from Jiweishan Mountain rockslide were taken as examples to determine the shear strength calculated by 2D and 3D JRC. On June 5, 2009, Jiweishan Mountain rockslide, a huge catastrophic rockslide fragment flow, took place under a mass up to 60 m in thickness at the crest of the Jiweishan Mountain in
Fig. 3 Limestone samples for direct shear testing
Wulong County, Chongqing, China, with a long-runout distance of 1500 m. The debris, which had a volume of over 7 9 106 m3, moved toward the valley, and disintegrated and covered an area of 0.47 km2. This catastrophic slide occurred in a sequence of coal measure rocks. The sole factor responsible for the catastrophic behavior was the apparent bedding orientation. The shear strength of the rock joint becomes the key to a stability analysis of rock mass. We collected five joint samples of limestone as shown in Fig. 3. Samples IDs are JW01-1BL, JW01-2BL, JW031BL, JW03-2BL and JW03-3BR, and their sizes are 57.0 9 49.6, 62.9 9 53.8, 53.6 9 49.7, 62.6 9 50.9 and 54.8 9 46.9, respectively (size of the blocks are in mm). We scanned the 3D surface shape of five joint samples as shown in Fig. 1 and completed direct shear tests along the y-direction (see Fig. 3). Then, we calculated the 2D and 3D JRC in different shear direction. Additionally, the shear strength of each direction was calculated by 2D and 3D JRC based on the JRC–JCS model. We found that the uniaxial compressive strength of the intact rock was 42.24 MPa, JCS is 20.815 MPa, and the basic friction angle is 34° in the laboratory tests. Table 1 gives the Z-position of the mean plane, the mean asperity height and the 3D JRC of five joint samples by the proposed method. The peak shear strengths of five samples with different normal stresses are given in Fig. 4 by our proposed method and Grasselli’s method. The comparison shows that these two methods are very close to the test data. Furthermore, the peak strength envelopes of the two methods are given in Fig. 5. Compared with Figs. 4 and 5, the shear strengths calculated by two methods are the same at low normal stress. The result shows three points. Firstly, that the relationship between peak shear strength and normal stress is
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Author's personal copy G. Zhang et al. Table 1 JRCs calculated by the proposed method for Limestone samples
Parameters
Sample no. JW02-1BL
4.95
6.66
8.64
1.68
0.90
3.06
2.62
2.36
The minimum value of JRC3D
16.40
11.41
19.42
18.22
18.95
The maximum value of JRC3D
18.04
14.65
19.81
19.39
19.74
Shear strength, τ (MPa)
2.5 2.0
τtest τsuggested τGrasselli
1.5 1.0 0.5 1.5
2.0
2.5
3.0
3.5
Normal stress, σn (MPa) Fig. 4 Comparison of shear strengths with the propose method, Grasselli’s method and direct shear box test results
28
~5 8° =4 in
σ
n
tan
φ
φ
τ= sliding on discontinuity surfaces
12
τ= σn
8
4
76+
=3.
σn
g
tren
ks
pea
τ th:
4°) 26. ( n ta
σn=JCS=20.815 MPa
16
tan (53 .4° )
Shear strength, τ (MPa)
20
2°
JW02-2BL(the proposed method) JW02-2BL(Grasselli's method) JW02-1BL(the proposed method) JW02-1BL(Grasselli's method) JW03-1BL(the proposed method) JW03-1BL(Grasselli's method) JW03-2BL(the proposed method) JW03-2BL(Grasselli's method) JW03-3BR(the proposed method) JW03-3BR(Grasselli's method)
24
4°)
26.
= : τ gth n e r l st dua resi
an( σt n
shearing in intact rock
σns=4.42 MPa 0 0
4
8
12
16
Normal stress, σn (MPa) Fig. 5 Peak shear strength at different normal stresses
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JW03-3BR
4.05
3.0
1.0
JW03-2BL
The mean asperity height (mm)
3.5
0.5
JW03-1BL
The Z-position of mean plane (mm)
4.0
0.0
JW02-2BL
20
5.82
approximately linear in Grasselli’s method, while their relationship obeys Barton-Bandis shear strength failure criterion in our proposed method. Secondly, at low normal stress, the friction angles of both methods are about 53.4 degrees. And thirdly, at high normal stress, the friction angle of Grasselli’s method is still about 48°–52°, which is far larger than the basic friction angle. In our proposed method, the true peak shear strength is calculated as 3.76 MPa with friction angle of 26.4° as shown in Fig. 5. What this demonstrates, as shown in Fig. 5, Grasselli’s method can calculate peak shear strength at low normal stress, while our proposed method can calculate peak shear strengths accurately not only at low normal stress but also in high normal stress conditions. The proposed peak strength failure criterion can be simplified as bilinear failure criteria (Patton Model) shown with two blue, dashed lines. The result of our proposed method is similar to Barton–Bandis shear strength failure criterion. More experimental data are needed to enable a complete assessment of the proposed method, especially at high normal stress. We compared JRC and shear strength of the proposed method and Grasselli’s method in Fig. 6 in which Fig. 6a lists 2D JRC and 3D JRC by the modified RMS and 3D JRC of Grasselli’s method, and Fig. 6b shows peak shear strengths of all methods. It can be seen that the 3D JRC and peak shear strength of the proposed method and of Grasselli’s method each vary smoothly with the shear direction, while the 2D JRC and peak shear strength calculated by the 2D JRC change abruptly with the shear direction. This shows that the 3D JRC and peak shear strength calculated by 3D JRC are closer to the actual roughness profiles. Additionally, although hmax /(1 ? C) and JRC both characterize the surface roughness, they are based on different theories and are not comparable in magnitude. Figure 6b shows that the peak shear strengths of two methods are very close except that the peak shear strengths of JW02-2BL and JW03-1BL deviate slightly. As a result, when we take samples of rock that are discontinuous in the field, the most likely shear direction, which would be set to the shear direction in the laboratory, should be pointed out. Then, the actual shear strength would be obtained.
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Fig. 6 Surface roughness profiles and shear strengths of various methods in different direction
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(b) Shear strength
Fig. 6 continued
5 Conclusion The joint roughness surface could be defined accurately in three dimensions. In this research, we developed a new method to predict the 3D JRC, which produces more consistent JRC values in all direction than the 2D JRC. The proposed method in this paper and Grasselli’s method both produces good fit under low normal stress conditions. The proposed method under higher normal stresses is also closer to expected behavior as shown in Fig. 5. However, the predicted directional variation of the envelopes needs to be further supported by more experimental data as the current paper used five rock joints taken from Jiweishan Mountain rockslide for
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experimental work. Therefore, the directional dependency of rock joint can be further investigated. The 3D roughness parameter, k, based on the modified RMS, considers the shear direction, the inclination angle, the amplitude of asperities and the scale effect. The relationship of k with the JRC was developed and then combined with Barton–Bandis shear strength failure criterion to estimate the peak shear strength of rock joints. Predictions from our proposed method were compared with Grasselli’s method and the results of experimental shear box tests conducted on samples taken from the Jiweishan Mountain rockslide in China. The results show that the friction angle is about 53.4° when normal stress is low, approximately
Author's personal copy Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface
4.42 MPa, and when the normal stress is greater than 4.42 MPa, the friction angle decreases and becomes stable at 26.4°, which represents the residual friction angle.
Ly
L eq O tion
Acknowledgements Financial supports from the National Natural Science Foundation of China through research Grant No. 41472265 and National Basic Research Program No. 2011CB710600 are gratefully acknowledged.
α
irec ar d e h s
Lx Fig. 9 Calculation of the equivalent length
Appendix See Figs. 7, 8 and 9.
30
(1,M+1) 20
(N+1,M+1)
(1,1)
z
10
0
-10
α
10
40
30
20
y
30
20
-200
(N+1,1) shear direction
x 50
40
10
0
(a) (b) Fig. 7 3D digital surface model (note: the positive x-direction is set to the zero angle and contra-rotates with the positive z-direction)
(a) A, B and C in
(b) A, B with C in the
(c) B, C with A in the reverse
(d) A, C with B in the reverse
the same side
reverse side
side
side
Fig. 8 Space relationship of triangle of the unit with the mean plane (notes ABC—part of roughness surface, A’B’C’—the projection of roughness surface in mean plane)
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Author's personal copy G. Zhang et al.
The Mean Plane As shown in Fig. 7a, a 3D surface can be digitized by 3D laser technology. According to the surface size and scanning precision, it can be divided into N ? 1 in x-direction and M ? 1 in y-direction. Thus, the 3D surface is discretized into N9M units and made up of (N ? 1)9(M ? 1) points, and each unit contains four points. Any unit made up of points A (i, j), B (i ? 1, j), C (i, j ? 1) and D (i ? 1, j ? 1) will have coordinates of (xi, j, yi, j, zi, j), (xi?1, j, yi?1, j, zi?1, j), (xi, j?1, yi, j?1, zi, j?1) and (xi?1, j?1, yi?1, j?1, zi?1, j?1), respectively. The unit can be thought of as two triangles of ABC and BCD, as shown in Fig. 7b. The whole 3D surface can be divided into 2(N 9 M) triangles. Their relationship with the horizontal plane A’B’C’D’ is shown in Fig. 7b, where A’, B’, C’ and D’ are the projection of A, B, C and D in the horizontal plane, respectively. Then, for the whole 3D roughness surface, the mean asperity height can be obtained by: Z Lx Z Ly Vtotal 1 h¼ ¼ jz z0 jdxdy Lx Ly Lx Ly 0 0 Nþ1 Mþ1 X Vði;jÞðiþ1;jÞði;jþ1Þ þ Vðiþ1;jÞði;jþ1Þðiþ1;jþ1Þ X ð9Þ ¼ Lx Ly i¼1 j¼1 Vði;jÞðiþ1;jÞði;jþ1Þ ¼ VABCC0 B0 A0 ; Vðiþ1;jÞði;jþ1Þðiþ1;jþ1Þ ¼ VBCDD0 C0 B0
ð10Þ
where z’ represents the position of the horizontal plane, Lx and Ly are the length in the x-direction and y-direction of roughness surface sample, respectively. The value of h varies by changing the position of the horizontal plane, which is called the mean plane, where the value of h is the minimum. In fact, according to the relative relationship between each triangle and the mean plane, the volume in Fig. 7b can be divided into four conditions as shown in Fig. 8. As shown in Fig. 8, the volume of each case should be calculated by different formulae because of the different shapes. However, no matter how complex the shape is, it always can be divided into several tetrahedrons, for which the volume can be calculated easily according to the coordinates of four vertexes. Assume there is a tetrahedron with known vertexes, such as (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4). Its volume can be calculated by 1 * * * V ¼ ða bÞ c 6 *
*
*
ð11Þ
where a, b and c are the vectors of the tetrahedron sides, and they can be calculated by
8* a ¼ ðx2 x1 ; y2 y1 ; z2 z1 Þ > > < * b ¼ ðx3 x1 ; y3 y1 ; z3 z1 Þ > > :* c ¼ ðx4 x1 ; y4 y1 ; z4 z1 Þ
ð12Þ
In Fig. 8, the first case can be divided into AA’B’C’, ABCC’ and ABB’C’, the second case is made up of AA’B’D, ABB’D, BB’OD and CC’OD, the third is made up of CC’B’D, CBB’D, BB’OD and AA’OD, and the last is made up of AA’C’D, ACC’D, CC’OD and BB’OD. Therefore, the volume of each can be calculated by Eq. (11). Additionally, it is substituted into Eqs. (10) and (9), such that the mean asperity height can be obtained. In addition, the length of the rock sample in the shear direction should be obtained in order to calculate the parameter, k, in Eq. (2). However, it is difficult to determine this shear length in reality. Instead, an equivalent value can be used, as shown in Fig. 9, where the ellipse is made up of the end-point of the equivalent length. Then, the formula for the equivalent shear length is given as: Lx Ly Leq ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLx sin aÞ2 þ ðLy cos aÞ2
ð13Þ
where Leq is the equivalent length in the shear direction, and a is the angle between the shear direction and the xcoordinate. The Modified Root Mean Square Method The Root Mean Square is the second step in calculating the roughness index. We will now transfer Eq. (4) to a new form in order to meet the three-dimensional requirement as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 dSs 2 0 Z2 ¼ maxð0; Þ dSxy Lx Ly dSxy vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u
u 1 2ðNMÞ X maxð0; Ss;i Þ 2 t ¼ ð14Þ Lx Ly i¼1 Sxy;i where dSs and dSxy are the projections of a triangle in the shear plane whose normal vector is the shear direction and the horizontal plane, respectively. They can be calculated by the coordinates of three vertexes in the triangle. Taking ABC as an example in Fig. 7b, the inner normal vector of ABC plane can be expressed by (xn, yn, zn) (here zn should be a negative), which are calculated by 8 > < xn ¼ ðyiþ1;j yi;j Þðzi;jþ1 zi;j Þ ðziþ1;j zi;j Þðyi;jþ1 yi;j Þ yn ¼ ðziþ1;j zi;j Þðxi;jþ1 xi;j Þ ðxiþ1;j xi;j Þðzi;jþ1 zi;j Þ > : zn ¼ ðxiþ1;j xi;j Þðyi;jþ1 yi;j Þ ðyiþ1;j yi;j Þðxi;jþ1 xi;j Þ ð15Þ
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Author's personal copy Estimation of Joint Roughness Coefficient from Three-Dimensional Discontinuity Surface
Assuming the shear direction is horizontal, its unit vector might be written as (xs, ys, 0), where we can calculate xs and ys as: xs ¼ cos a; ys ¼ sin a
ð16Þ
Therefore, according to the formula for calculating the included angles of two vectors, the angles of the plane ABC (as shown in Fig. 7b) with the shear direction and ABC with the horizontal plane are given by, respectively 8 xn cos a þ yn sin a > > < cos hs ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2n þ y2n þ z2n ð17Þ zn > > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cos h xy : x2n þ y2n þ z2n where hs and hxy are the included angles of ABC with the shear plane and the horizontal plane, respectively. Thus, the projected areas of ABC in the shear plane and the horizontal plane can be given by, Ss;ABC ¼ SABC cos hs ; Sxy;ABC ¼ SABC cos hxy
ð18Þ
where SABC is the area of ABC, Ss,ABC is the projected area of ABC in the shear plane, and Sxy,ABC is the projected area of ABC in the horizontal plane. Similarly, Ss,i and Sxy,i can be calculated in the same manner. It should be pointed out that the revised RMS method proposed by Zhang et al. (2014) considers the shear direction. Firstly, the plane does not affect the shear strength calculated by the method proposed in this paper if it reverses to the shear direction. Therefore, those planes are not to be considered in Eq. (14) if the angles of the inner normal vector of those planes with the shear direction are larger than 90°. Furthermore, the projected area of any ABC in the shear plane is considered if the plane faces to the shear direction as given in Eq. (18). The detailed steps for calculating the 3D JRC are as follows: 1.
2.
3.
Obtain the real joint surface profile, 3D geometry (xi, yi, zi), by scanning the joint surface shown in Fig. 7a. Then, triangulate the roughness surface. According to the shear direction, divide the roughness surface into a rectangle for which the long edge is Leq and parallel to the shear direction. The short edge is perpendicular to the shear direction as shown in Fig. 7. Then, Leq is equal to the quadrature length of the inner ellipse of the roughness surface in the shear direction given in Eq. (13). Assess every triangle to determine whether it is in reverse or in the shear direction. If the triangles are in the reverse direction, then they are to be disregarded in the analysis. Only the projected areas of the triangle that are in the shear direction and the horizontal plane are to be calculated in Eq. (18).
4.
Lastly, obtain the modified RMS, Z20 , using Eq. (14).
According to the above equations and steps, we implemented the theoretical approach using Compaq Visual FORTRAN 6 to calculate 3D JRC. Furthermore, the 2D JRCs of every roughness surface in different shear direction were calculated using the method proposed by Zhang et al. (2014).
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