Suggested New Statistical Parameter for Estimating Joint Roughness Coefficient considering the Shear Direction

The 10 standard roughness joint proﬁles provided a visual comparison to get the joint roughness coeﬃcient (JRC) of rock joint surface, but the accuracy of this method is inﬂuenced by human factors. Therefore, many researchers try to evaluate the roughness morphology of joint surface through the statistical parameter method. However, JRC obtained from most of the existing statistical parameters did not reﬂect the directional property of joint surface. Considering the 10 standard proﬁles as models of diﬀerent roughness joints, we proposed a new idea for the accurate estimation of JRC. Based on the concept of area diﬀerence, the average of positive area diﬀerence ( Sa ) and sum of positive area diﬀerence ( Ss ) were ﬁrst proposed to reﬂect the roughness of joint surfaces on the basis of directional property, and their ﬁtting relationship with JRC was also investigated. The result showed that the Sa and Ss calculated by shearing from right to left (FRTL) and JRC backcalculated from right to left (FRTL) came to a satisfying power law. The correlation between JRC and Sa was better than that of Ss . The deviation between the predicted value calculated by Sa and the true value was smaller than that obtained from the existing statistical parameters. Therefore, Sa was recommended as a new statistical parameter to predict the JRC value of joint proﬁle. As the sampling interval increased from 0.5 to 4 mm, the correlation between Sa and JRC gradually decreased, and the accuracy of the prediction results also declined. Compared with the single JRC values for joint proﬁles mentioned in the literature, the forward and reverse JRC were obtained. Based on the laboratory direct shear test of the natural joint surface, the JRC values of two joint surfaces in four shear directions were backcalculated by the JRC-JCS model. Based on 3D scanning and point cloud data processing technology, JRC of joint surface in diﬀerent directions were obtained by Sa method, and they are very close to those obtained by JRC-JCS model. It is conﬁrmed that Sa could accurately estimate the joint roughness coeﬃcient and reﬂect its anisotropy.


Introduction
e mechanical behavior of rock mass is much more complex than the complete rock because of its intricate joints and structural planes. e safety and stability of rock mass are mainly controlled by shear deformation along the joints, which is particularly prominent in slope and tunnel engineering [1][2][3][4]. For unfilled rock joint, the irregular morphology of the joint surface is one important factor to control the shear strength of joints. erefore, how to describe it quickly and accurately is of crucial importance [5][6][7][8]. Barton [9] proposed the JRC-JCS model widely used in engineering through the direct shear test of 136 natural joint specimens and put forward the joint roughness coefficient (JRC) to quantitatively describe the irregular morphology of the joint for the first time. rough some simple experiments, we can obtain the value of peak shear strength, actual normal stress, basic friction angle, and compressive strength of joint wall (JCS) defined in JRC-JCS model, and then we can get the JRC value through back calculation. However, since we usually hope that the JRC value of joint can be obtained before the direct shear test and used to predict the shear strength of joints, this method was usually used to verify the accuracy of roughness evaluation method.
In order to predict the joint roughness, Barton and Choubey [10] proposed 10 standard roughness profiles based on the direct shear tests of 136 natural rock specimens. e JRC value of the standard profile is obtained by back calculation from JRC-JCS model with a range of 0 to 20, as shown in Figure 1.
Compared with the surface morphology of natural joints with the standard profiles, the most similar one is selected to obtain the range of joint roughness coefficient. is process is called the empirical comparison method.
e International Society for Rock Mechanics [11] recommend these 10 profiles as the standard method for evaluating the joint roughness coefficient, because the empirical comparison method had given the way to determine JRC value for the first time.
e aforementioned empirical comparison method is very convenient and fast. However, the process is greatly affected by experience of engineer, which leads to the floating of the joint roughness coefficient value [13,14]. erefore, in order to obtain the roughness coefficient of joints more accurately, statistical parameters such as the straight edge method [15], modified straight edge method [16], fractal dimension (D) [17,18], root mean square roughness index (RMS) and mean square value roughness index (MSV) [19], root mean square of the first deviation of profiles (Z 2 ) [19][20][21][22][23][24][25][26][27], structure function (SF) [19][20][21][22][23], standard deviation of the angle (SDi) [21,22], roughness profile index (R P ) [21][22][23][24]28], maximum inclination (θ * max ) [23,24], mean positive angle (θ p+ ) [29], modified root mean square (Z′' 2 ) [30], and support vector regression (SVR) model [31] had been presented. ese aforementioned parameters can be calculated from geometric coordinates of joint profile. In Figure 2 [12], x i and y i are the abscissa and ordinate of joint profile, △x is the distance between the abscissa of adjacent coordinate point and becomes a fixed value at the same sampling interval, and L is the horizontal length of joint profile. e aforementioned statistical parameters have a good correlation with JRC; however, the regression formula between them needs to be applied at a specific sampling interval.
At present, many researchers believed that the rock joint surface has anisotropy, which led to the joint surface sheared in different directions showing different strength and deformation characteristics. In addition, the shear direction of rock joints periodically reversed under cyclic shear conditions. e shear strength values are usually not equal, although the joint morphology does not change much before the last shearing process [32][33][34][35]. In the JRC-JCS model proposed by Barton [10], JCS, normal stress, and basic friction angle are all parameters without direction property. erefore, only the JRC have different values when the shear direction is changed.
us, it can be concluded that the contribution of the joint surface morphology to the joint roughness in different directions is different and it should be reconsidered on the basis of direction property. e surface morphology of natural joints is random, and it is impossible to have the same feature in different directions.
e default shear direction of the 10 standard profiles proposed by Barton was from left to right (FLTR), which mean that the JRC values in Figure 1 represent the roughness under condition of FLTR. JRC values of the 10 standard profiles under the condition of shearing from right to left (FRTL) are unknown. It is not the same as them obtained under the condition of FLTR. e aforementioned statistical parameters can predict the JRC value of joint profile, but most of them can reflect the direction property of roughness. e predicted shear strengths of all joints at the shear condition of FLTR and FRTL are the same when using a statistical parameter such as Z 2 to evaluate JRC and shear strength. Although some scholars [23,24,[29][30][31] had proposed directional parameters, the correlation between some of them and JRC is not very good or has not been verified. Obviously, the research on statistical parameters considering shear direction is not rich enough. It is very meaningful to come up with a statistical parameter that can consider the direction property of joint morphology and accurately predict the JRC value.

Preparation for Research
e data acquisition of joint morphology is the primary task of roughness evaluation. e morphology acquisition and direct shear test of natural joint surface are troublesome, and the JRC obtained are usually used to evaluate the threedimensional roughness of joint surface. In addition, it is difficult to find joint surfaces with different roughness. So, the 10 standard roughness profiles proposed by Barton [9] were usually used as the basis of evaluation method of joint roughness, as shown in Table 1. e main reason is that their JRC values have been determined, and these values are gradient. Before doing the research, we need to get the discrete data of the joint profiles and validate the reliability of the acquired data to ensure our following research work is accurate.

Digitization of Ten Standard Roughness Profiles.
e original data of the 10 standard roughness profiles were measured by a profile comb ( Figure 3). e sampling interval of the obtained data was 0.5 mm, while the spacing between the comb teeth was 1 mm. It can be seen that the measured data had a slight error from the actual morphology of joint surface. Digitization process of joint Profile was equivalent to reacquisition of the original data. So, it is not reliable to obtain discrete data of the standard profiles with sampling interval less than 0.5 mm because this has exceeded the original accuracy of the joint profile. So, the sampling interval of digitization process was set to 0.5 mm in this section.
We first saved these profiles shown in Figure 1 as images in JPG format by screenshot software and imported these images into GETDATA software. Before the acquisition process, we first need to remove useless pixels from the joint image and set the axes for it. e average horizontal length of 10 standard roughness joint profiles was about 99 mm. According to the processing idea of Zheng and Qi [25] and Huan et al. [35], the horizontal length of the 10 joint profiles   was assumed to be 100 mm, and the JRC values of them were assumed to be unchanged. e geometric coordinates of the profile could be acquired by autocapture mode, and the point capture mode was used to fill the vacant part and modify the singularity point. e data point was at the center of the joint profile by default as shown in Figure 4. e 10 joint profiles in images were digitized at sampling intervals of 0.5, 1.0, 2.0, and 4.0 mm, respectively, for a subsequent study of the effects of sampling intervals. Figure 5 showed the modified 10 standard roughness profiles and the coordinate of starting point was set to (0, 0).

Verification of Profile Data.
As mentioned earlier, various statistical parameters had been proposed to describe the morphology of joint surface. In this paper, we first recommended Z 2 and R P as representatives to verify the accuracy of the obtained joint data. eir definitions are as follows: where N is the number of discrete points obtained from joint profile and M represents the number of positive slopes. e values of Z 2 , and R P were calculated for the standard roughness profiles at 0.5 mm sampling interval ( Figure 6). Except for the singularity obtained from the fourth profile, the values of Z 2 , and R P tended to increase with the increasing of JRC values. e best fitting curves of statistical parameters and JRC were established using the following: where P represents the statistical parameters and a, b, and c represents the regression coefficient. Under the condition of 0.5 mm sampling interval, the correlation coefficients of Z 2 and R P with JRC calculated in our paper were larger than those of Jang et al. [23] and Zheng and Qi [25] (Table 1). is has further confirmed that the data of the Barton standard profiles digitized in our paper were highly accurate. In order to further discuss the rationality of some statistical parameters considering shear direction in evaluating joint roughness, we selected mean positive angle (θ + p ) proposed by Belem et al. [29] and Z 2 ′ proposed by Zhang et al. [30] to verify their correlation with JRC. eir definitions are as follows: Under the condition of 0.5 mm sampling interval, the relationship curves between the aforementioned two statistical parameters and JRC were shown in Figure 7. It could be seen that the correlation between these two statistical parameters and JRC was not very good, and R 2 was 0.9313 and 0.9271, respectively. erefore, it was necessary to find new statistical parameters to estimate the joint roughness considering the shear direction.

Derivation of New Statistical Parameters
In order to fill the gaps in the existing studies, two new statistical parameters were proposed in this paper to characterize the roughness of joint profile based on local area. In the definition of DA, the local roughness and shear direction were considered simultaneously.
As shown in Figure    Advances in Civil Engineering was defined as the local area, and the difference between adjacent areas was defined as DA. ey were calculated using the following formula: As can be seen from Figure 9, when the joint is smooth, the value of DA is close to zero. When the joint profile is inclined upwards, the value of DA is positive.
When the joint profile is inclined downwards, the value of DA is negative. e distribution of DA calculated from Barton's standard roughness profiles had been obtained, as shown in Figure 10.
e abscissa was the numerical range of DA, and the ordinate was the frequency of occurrence. It could be seen that as the joint roughness increased, the numerical range of DA increased obviously. For example, the range of DA obtained from the first profile (JRC � 0-2) was from −0.05 mm 2 to 0.06 mm 2 , the range of DA obtained from the fifth profile (JRC � 8-10) was from −0.12 mm 2 to 0.12 mm 2 , and the range of DA obtained from the last one (JRC � 18-20) was from −0.25 mm 2 to 0.3 mm 2 . It could be seen that when  Advances in Civil Engineering the roughness of the joint profile changed, the distribution characteristics of DA were very different. In addition, the distribution characteristics of positive DA and negative DA calculated from each profile were also quite different, which could be used to describe the orientation of the morphology. Under the shear condition of FLTR and FRTL, only the positive DA is counted. e average of positive DA (Sa) and sum of positive DA (Ss) are defined to accurately describe the roughness of joint profile considering the directional property, as shown in (5) and (6). ere is a certain relationship between Sa and Ss calculated from the same profile as shown in (7):   Advances in Civil Engineering where m is the number of positive DA. e values of m calculated from different profiles were different. e values of Sa and Ss calculated from standard roughness profiles have been obtained under the shear condition of FLTR and FRTL, respectively, as shown in Table 2. e sampling interval for the data of joint profiles is 0.5 mm.
It could be seen that the values of Sa and Ss obtained under the shear condition of FLTR tend to increase as backcalculated JRC values increase. However, the values of Sa and Ss calculated from the shear condition of FLTR and backcalculated JRC did not maintain a linear increase. e values of Sa calculated from the shear condition of FRTL had singularities in the 4th, 9th, and 10th profiles, and the values of Ss calculated from the shear condition of FRTL had singularities in the 4th and 9th profile. e fitting function of Sa and Ss and the backcalculated JRC value were established by the power law function (Equation (3)) as shown in Figure 11.
It could be seen that Sa and Ss obtained under the shear condition of FLTR had a strong correlation with backcalculated JRC (R 2 � 0.9908, R 2 � 0.9811). However, the correlation between Sa and Ss obtained under the shear condition of FRTL with JRC was not good (R 2 � 0.9409, e correlation between the statistical parameters Sa and Ss under the shear condition of FRTL with backcalculated JRC had a better correlation. ey had the same directional property as the backcalculated JRC. Sa was better correlated with JRC than Ss; therefore Sa was considered to be able to describe the joint profile roughness well, considering the directional property. Equation (8) was recommended for estimating the joint JRC at 0.5 mm sampling interval. e JRC value can only describe the joint roughness corresponding to the shear direction considered in the calculation of statistical parameters.
In order to verify the accuracy of the statistical parameters in estimating the JRC, the deviation value between JRC pre and JRC true was proposed as shown in (9). JRC pre was the predicted JRC value obtained from the regression formula, and JRC true represented the actual JRC value of the joint profile. e results of deviation calculated from Z 2 , R p , and Sa are shown in Table 3: It could be seen that the maximum deviation of Z 2 and R P in evaluating the joint JRC was 1.738 and 1.737. e deviations of these two statistical parameters in evaluating the JRC value of the 3th, 4th, and 5th profile were both greater than one. In contrast, the maximum deviation of Sa was 0.865, and no deviation was greater than 1. However, the JRC pre value obtained from the first profile was less than zero, which indicated that the Sa cannot well evaluate the roughness of smooth joints. e JRC value of joint profile was supposed to be greater than 0, so the Sa calculated at 0.5 mm sampling interval should be greater than 0.01489 mm 2 . e mean deviation was proposed to describe the accuracy of the statistical parameters in evaluating the JRC of joint profile. It could be seen that the mean deviation obtained by Sa was smaller than Z 2 and R P , which further showed that Sa could well estimate the roughness of joint profile.
In order to better understand the directional properties of joint roughness, the forward JRC was defined to describe the roughness obtained under the shear condition of FLTR, and the reverse JRC was defined to describe the roughness obtained under the shear condition of FRTL. Under the shear condition of FRTL, the values of Sa corresponding to standard roughness profiles were calculated by (8). e reverse JRC of each profile could be obtained by substituting the data of Sa into (8) as shown in Table 4. It could be seen that the values of the forward JRC and reverse JRC of each profile were very different. e forward JRC values were greater than the reverse JRC except for the 4th, 6th, and 8th profile. e reverse JRC value of the first profile was negative, which further illustrated that Sa could not predict the roughness of smooth joint. e numerical difference between forward JRC and reverse JRC values calculated from the Sa well verified the directional property of joint roughness, which could not be achieved by statistical parameters, such as Z 2 and R P . Under the shear condition of FRTL, the ordinal numbers of the standard roughness profiles had changed according to the value of the reverse JRC. Table 4 showed the reordering of standard roughness profiles, which had been adjusted in the 4th, 5th, 8th, and 10th profile.

Effect of Sampling Interval
As mentioned earlier, the values of existing statistical parameters and their correlation with the roughness are affected by the sampling interval of the joint profile. High precision data can better describe the true morphological features of the joint profile, but it is not easy to obtain. In this paper, Barton's 10 standard roughness profiles were digitized at 0.5, 1, 2, and 4 mm sampling interval, respectively.
e Sa was calculated under the shear condition of FLTR and regressed with value of the forward JRC. e results were shown in Table 5 and Figure 12.
As can be seen from Table 5, the value of Sa increased significantly with the increase of sampling interval, which indicated that Sa has a great sensitivity to the sampling interval. A small error in the calculation will result in a large deviation in the value of Sa, which will lead to a deviation of the JRC pre value. It is necessary to ensure that the Sa value and corresponding regression formula are under the same sampling interval condition in order to obtain an accurate JRC pre value.
As can be seen from Figure 12, as the sampling interval gradually increased from 0.5 to 4 mm, the correlation between Sa and forward JRC decreases. When the sampling interval is 0.5 and 1 mm, the value of Sa increased with the increase of the joint number. However, when the sampling Advances in Civil Engineering 9 interval was 2 mm, the Sa calculated from the 3rd and 9th profiles was less than the value calculated from the preceding profile. When the sampling interval was 4 mm, the Sa values calculated from the 9th and 10th profiles were less than the value calculated from the preceding profile. e deviation between JRC pre and JRC true was obtained for each sampling interval using (9), as shown in Table 6. e deviation was always less than one at the 0.5 mm sampling interval. When the sampling interval was 1 mm, the deviation of the 3rd profile was greater than one. When the sampling interval was 2 mm, deviation greater than one occurred five times and the deviation of the 9th profile was greater than two. When the sampling interval was 4 mm, deviation greater than one occurred six times. What is more,  deviation greater than two occurred four times, and the deviation of the 10th profile was greater than three. As the sampling interval increases from 0.5 mm to 4 mm, the calculated mean deviation increased from 0.47 to 1.505, which indicated that as the sampling interval increased, the accuracy of JRC pre calculated by the regression formula decreased.
As the sampling interval increased, the number of points obtained from the joint profile decreased significantly. For example, the number of discrete points was 201 at a spacing interval of 0.5 mm, while the number of discrete points was only 26 at a spacing interval of 4 mm. It could be considered that the profile data obtained under the condition of large sampling interval ignored many morphologies, and the calculated statistical parameters could not describe the roughness of joint profile very well. erefore, we recommended using the regression formula with a sampling interval of 0.5 mm to accurately predict the JRC of joint profile.

Verification of Using Sa to Evaluate JRC
In order to verify the advantage of Sa in evaluating the roughness of joint profile, the roughness of the joint profile given in the literatures [20,30,[36][37][38][39] and the natural joint surface were separately evaluated.

JRC Estimation of the Joint Profiles Given in the Literature
Studies. We first performed the corresponding digitization process on the joint profile given by literatures [20,30,[36][37][38][39] at 0.5 mm sampling interval. In many cases, natural joints are longer than the 10 standard profiles. Considering the influence of size effect on JRC value, Barton [40] proposed the following:   where JRC n is the joint roughness coefficient value of joint for study, L n is the length of natural joints, and L 0 represents the laboratory scale and the length of joint is close to 100 mm. e Sa and corresponding JRC values calculated under the shear condition of FLTR and FRTL are shown in Table 7. e JRC values of joint 1, 3,6,7,8,18,19,20, and 21 were corrected by (10). e results showed that the difference between the forward and reverse JRC of the joint profile was sometimes small but at other times large. In addition, it can be seen that the joint JRC can exceed the upper limit of 20 set by Barton, which is consistent with research findings of Du et al. [16]. is showed that the JRC calculated by the statistical parameter Sa not only can well reflect the rough morphology but also can distinguish the differences of roughness in different shear directions.

JRC Estimation of the Natural Joint Surface.
e two joint specimens (numbers J-1 and J-2) used to estimate the roughness were collected from Taibai Mountain, Xi'an City, Shaanxi Province, China. e rock type is granite, and the plane size is about 100 mm × 100 mm. e joint surfaces were produced by artificial load damage at the quarry. e 3D point cloud data of joint surfaces were obtained by the 3D Scanner Go! SCAN 20, and then they were imported into Geomagic Studio software. e joint surface data was separated through data processing, and the joint surface mold was made by 3D printing technology. e rough joint samples used in direct shear test were poured with cement mortar, and the smooth joint samples and uniaxial specimens were made additionally. e mass ratio of water, cement, and standard sand is 1 : 2 : 3. en, the direct shear tests were carried out in four shear directions, and the normal stress was 2, 4, 6, and 4 MPa separately. Figure 13 showed the production process of joint samples with J-1 joint as an example.
In this section, the roughness of joint surface in four different directions was represented by averaging the roughness of the joint profile. In each shear direction, 201 joint profiles were extracted with an interval of 0.5 mm by Geomagic Studio software. e JRC of each joint profile was obtained by calculating its Sa value at 0.5 mm sampling interval and substituting it into (8).  [10], and its value was the average value of the calculation results under four normal stresses: Table 8 showed the comparison between the JRC results of two joint surfaces predicted by Sa and backcalculated by the JRC-JCS model. It could be seen that the JRC prediction results obtained by Sa were very close to those obtained from the laboratory tests. e minimum absolute error was 5%, and the maximum absolute error was 25.78%. In addition, the roughness of joints J-1 and J-2 had obvious anisotropy.
e aforementioned results showed that the new statistical  parameter Sa proposed in this paper can well reflect the joint surface roughness considering the shear direction.

Conclusion
As a parameter to quantitatively evaluate the roughness, JRC is very important for predicting the shear strength of rock joints and evaluating the stability of rock mass. At present, most statistical parameters used for estimating the JRC values of joint profile ignored the directional differences of rough morphology. erefore, this paper proposed a new statistical parameter Sa for solving this problem. First, the 10 standard roughness joint profiles proposed by Barton were digitized using data processing software as a research basis. en, the calculation results of the existing parameters, such as Z 2 and R P , were similar to those of the previous ones, which indicated that the data process can be considered correct. By calculating the existing statistical  parameters (θ + p and Z 2 ′ ) considering the shear direction, it was found that the correlation between them and JRC was not very good. e concept of area difference (DA) was proposed to describe the roughness of the joint surface, and only the case of a positive area difference was considered to reflect the roughness in a certain direction. e results showed that Sa and Ss obtained under the shear condition of FLTR satisfied strong relevance with JRC backcalculated under the shear condition of FLTR. In addition, the correlation between Sa and Ss obtained under the shear condition of FRTL and JRC backcalculated under the shear condition of FLTR was poor. e correlation coefficient between Sa and JRC was higher. So, the Sa was recommended as a new statistical parameter to predict the JRC of the joint profile. e reverse JRC of the standard profiles were obtained by substituting Sa shear condition obtained under the shear condition of FRTL into (8). e results showed that the JRC of each profile varied greatly in different directions, which indicated that Sa can well reflect the directionality of roughness. As the sampling interval increased, the correlation coefficient of the fitting formula between Sa and JRC decreased rapidly, and the deviations between JRC pre and JRC true gradually increased. ose indicated that the sampling interval of joint profile should be as small as possible or cannot accurately reflect the roughness morphology. e roughness of joint profile obtained from the literature was reestimated in this paper. e forward and reverse JRC results showed that the roughness of the joint profile in different directions was independent and varied greatly. What is more, the point cloud data of two natural rock joint surfaces were obtained by 3D scanning technology, and the corresponding physical tests were carried out based on joint specimens made of mortar. e JRC of joint surface in different directions were predicted using the Sa, and the result was very close to that of JRC-JCS model. It could be confirmed that the new statistical parameter Sa proposed in this paper can well describe the joint roughness considering the shear direction.
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.