Scale Effect and Anisotropic Analysis of Rock Joint Roughness Coefficient Neutrosophic Interval Statistical Numbers Based on Neutrosophic Statistics

Journal of Soft Computing in Civil Engineering 2-4 (2018) 62-71
How to cite this article: Jiang W, Ye J, Cui W. Scale effect and anisotropic analysis of rock joint roughness coefficient
neutrosophic interval statistical numbers based on neutrosophic statistics. J Soft Comput Civ Eng 2018;2(4):62–71.
https://doi.org/10.22115/scce.2018.143079.1086.
2588-2872/ © 2018 The Authors. Published by Pouyan Press.
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Scale Effect and Anisotropic Analysis of Rock Joint Roughness
Coefficient Neutrosophic Interval Statistical Numbers Based on
Neutrosophic Statistics
W. Jiang1
, J. Ye1,2*
, W. Cui2
1. School of Civil Engineering, Shaoxing University, Shaoxing, Zhejiang 312000, China
2. Department of Electrical Engineering and Automation, Shaoxing University, Shaoxing, Zhejiang 312000, China
Corresponding author: yejun@usx.edu.cn
https://doi.org/10.22115/SCCE.2018.143079.1086
ARTICLE INFO ABSTRACT
Article history:
Received: 03 August 2018
Revised: 22 September 2018
Accepted: 26 September 2018
In rock mechanics, mechanical properties of rock masses in
nature imply complexity and diversity. The shear strength of
rock mass is a key factor for affecting the stability of the
rock mass. Then, the joint roughness coefficient (JRC) of
rock indicates an important parameter in the shear strength
and stability analysis of rock mass. Since the nature of the
rock mass is indeterminate and incomplete to some extent,
we cannot always express rock JRC by a certain/exact
number. Therefore, this paper introduces neutrosophic
interval statistical numbers (NISNs) based on the concepts of
neutrosophic numbers and neutrosophic interval probability
to express JRC data of the rock mass in the indeterminate
setting. Then we present the calculational method of the
neutrosophic average value and standard deviation of NISNs
based on neutrosophic statistics. Next, by an actual case, the
neutrosophic average value and standard deviation of the
rock JRC NISNs are used to analyze the scale effect and
anisotropy of the rock body corresponding to different
sample lengths and measuring directions. Lastly, the analysis
method of the scale effect and anisotropy for JRC NISNs
shows its effectiveness and rationality in the actual case
study.
Keywords:
Joint roughness coefficient
(JRC);
Neutrosophic interval
probability;
Neutrosophic interval statistical
number;
Neutrosophic statistics;
Scale effect;
Anisotropy.

W. Jiang et al./ Journal of Soft Computing in Civil Engineering 2-4 (2018) 62-71 63
1. Introduction
In 1973, Barton [1] first put forward the concept of joint roughness coefficient (JRC) and
indicated its important influence on the shear strength of rocks. It is used to express the empirical
coefficient of rough surface undulation to indicate the shear strength of the rock mass structural
plane [2]. However, mechanical parameters in nature, such as rock JRC, usually contain
imprecise, incomplete, and uncertain information, which is difficultly expressed by certain/exact
values. Thus, it is obvious that the classical statistical methods cannot be fit for the expression
and analysis of the uncertain JRC data. So Ye et al. [3] first introduced the neutrosophic
functions of JRC and shear strength for analyzing the scale effect and anisotropy of JRC and the
shear strength of rock mass based on neutrosophic/interval functions [4]. Since the neutrosophic
number (NN) introduced in [5–7] is very suitable for expressing the hybrid information of both
certain and uncertain measuring values, Ye et al. [8] presented the expression and analysis
method of JRC based on NN functions. Next, Chen et al. [9] utilized the NNs of JRC to analyze
the scale effect and anisotropy of the uncertain JRC data using neutrosophic statistics [7]. Chen
et al. [10] further proposed the concepts of neutrosophic interval probability (NIP) and
neutrosophic interval statistical numbers (NISNs) based on the concepts of neutrosophic
statistics and NNs, and then applied them to the expressions of uncertain JRC data (JRC NISNs).
Unfortunately, there is no study on analyzing scale effect and anisotropy of JRC NISNs by using
the neutrosophic statistical analysis method of JRC NISNs in existing literature [3–10]. To solve
this gap, this study proposes the calculational method of the neutrosophic average value (NAV)
and neutrosophic standard deviation (NSD) of NISNs based on neutrosophic statistics for the
first time, and then applies them to analyze the scale effect and anisotropy of the rock mass based
on the measured data of rock mass in Changshan County, Zhejiang Province, China as an actual
case. However, the analysis method of the scale effect and anisotropy for JRC NISNs can
provide an effective and reasonable way under uncertain environments.
In order to do work, Section 2 reviews some basic concepts of NN, NIP, and NISN. In Section 3,
we present the calculational method of NAV and NSD for NISNs based on neutrosophic
statistics. In Section 4, the indeterminate JRC data are represented by NISNs based on an actual
case, and then the scale effect and anisotropy of the JRC NISNs are analyzed by using NAVs and
NSDs of JRC NISNs. Lastly, conclusions and next work are indicated in Section 5.
2. Basic concepts of NNs and NISNs
2.1. Neutrosophic numbers
The JRC property of rock surface shape is usually uncertain and incomplete in the measuring and
data processing process due to a lack of known/exact information in the indeterminate setting.
We cannot always express it by a certain/exact value, which limits the possibility of the exact
expression and analysis to some extent. Then NN [5–7] can express the problem of partial
certainty and/or partial uncertainty because it can be simply expressed as y = p+qI (p and q are
any real numbers), where p is the certain part and qI is the uncertain part with its indeterminacy
I. Obviously, NN easily represents the certain and/or uncertain information in indeterminate

64 W. Jiang et al./ Journal of Soft Computing in Civil Engineering 2-4 (2018) 62-71
setting. For instance, y = 3+2I for I  [0,1] or I  [2,3] indicates y  [3,5] or y  [7,9], which
implies a changeable interval number depending on I. Hence, NN show its convenient and
flexible advantage in the expression of certain and/or uncertain information [8–10].
2.2. Neutrosophic interval probability
Chen et al. [10] first proposed the concept of NIP to express the NIP of uncertain JRC data in the
indeterminate setting.
Set a = [al
, au
] as the interval range of measuring data for a group of samples. Chen et al. [10]
defined a NIP N and expressed it as N = <[al
, au
], (Tp, Ip, Fp)>, where Tp is its truth probability in
the certain interval, Ip is its indeterminate probability in the uncertain interval; and Fp is its falsity
probability in the almost impossible/failure interval, for Tp + Ip + Fp = 1.
For instance, suppose that there is a group of n samples regarding the same size. Then we
calculate the average value m and standard deviation  of the JRC data obtained from the n
samples. Thus, the interval value a = [al
, au
] is obtained when al
is the minimum value (the lower
bound) of the measuring data and au
is the maximum value (the upper bound) of the measuring
data. Then, we can define [m, m+] as the confident/truth interval, [m3, m)  (m+,
m+3] as the uncertain intervals, and [𝑎𝑙
, 𝑚 − 3σ) ∪ (𝑚 + 3σ, 𝑎𝑢
] as the remaining/incredible
intervals. Hence, we can calculate the truth probability Tp = nT/n for the frequency nT in [m,
m+], the uncertain probability Ip = nI/n for the frequency nI in [m3, m)  (m+, m+3],
and the falsity probability Fp = nF/n for the frequency nF in [al
, m3)  (m+3, au
].
2.3. Neutrosophic interval statistical numbers
Chen et al. [10] also presented the concept of NISN and used it for the expression of JRC with
uncertain information corresponding to the NN and confidence degree.
Chen et al. [10] defined the following confidence degree:
𝑐 = 𝑇𝑝/√𝑇𝑝
2 + 𝐼𝑝
2 + 𝐹
𝑝
2, (1)
and then presented a NISN, which is expressed as
𝑆 = 𝑚 + (1  c)I for I  [Il
, Iu
], (2)
where m and  are the average value and standard deviation of a group of JRC data, respectively.
Firstly, we consider that the indeterminate JRC value is represented by NISN based on an actual
example below.
Example 1. In an actual example, we take a group of 290 measured data (n = 290) of the sample
length L = 100cm and measuring direction 0o
, and then express the indeterminate JRC value by
NISN. Thus, Table 1 indicates the statistical values of the obtained JRC data.
Table 1
Statistical data of the obtained JRC data for the actual example.
n 𝑚 σ [𝑎𝑙
, 𝑎𝑢
] 𝑛𝑇 𝑛𝐼 𝑛𝐹
290 13.20 4.35 [4.63, 25.82] 205 70 15

Hence, the indeterminate JRC value is expressed by NISN below.
From Table 1, we can calculate the truth, indeterminacy, and falsity probabilities in the interval
[4.63, 25.82] as follows:
𝑃𝑇 = 205/290 = 0.71, 𝑃𝐼 = 70/290 = 0.24, 𝑃𝐹 = 15/290 = 0.05.
Then, the NIP of JRC is expressed as N = <[4.63, 25.82], (0.71, 0.24, 0.05)>.
By Eq. (1), the confidence degree is
c = 𝑇𝑝/√𝑇𝑝
2 + 𝐼𝑝
2 + 𝐹
𝑝
2 =
0.71
√0.712+0.242+0.052
= 0.9065.
By Eq. (2) the NISN of JRC is expressed as
𝑆 = 𝑚 + (1  c)I = 9.8725+0.09354.35I = 9.8725+0.4067I.
Suppose the indeterminacy I is specified as I  [1, 1]. Then, there is S = [9.4658, 10.2792].
3. Neutrosophic statistics of NISNs
In this section, we propose the neutrosophic statistical method to calculate the NAV and NSD of
NISNs.
For a group of Si = mi + (1  ci)iI for I  [Il
, Iu
] and i = 1, 2, …, n, we set bi = (1  ci)i, then
there is Si = mi + biI for I  [Il
, Iu
]. Thus the NAV of Si is presented as the following formula:
𝑆𝑚 = 𝐴𝑚 + 𝐵𝑚𝐼 for I  [Il
, Iu
], (3)
where 𝐴𝑚 = 1
𝑛
∑ 𝑚𝑖
𝑛
𝑖=1 and 𝐵𝑚 = 1
𝑛
∑ 𝑏𝑖
𝑛
𝑖=1 for the average value mi and the indeterminacy
coefficient bi = (1  ci)i.
Then, the NSD of NISNs Si is proposed as
 = √1
𝑛
∑ (𝑆𝑚 − 𝑆𝑖)2
𝑛
𝑖=1 , (4)
where 𝑆𝑚 − 𝑆𝑖 is calculated by
𝑆𝑚 − 𝑆𝑖 = 𝐴𝑚 − 𝑚𝑖 + (𝐵𝑚 − 𝑏𝑖)𝐼 for I  [Il
, Iu
], (5)
and then there is the following result:
(𝑆𝑚 − 𝑆𝑖)2
= {𝑚𝑖𝑛[(𝐴𝑚 + 𝐵𝑚𝐼𝑙
)(𝑚𝑖 + 𝑏𝑖𝐼𝑙), (𝐴𝑚 + 𝐵𝑚𝐼𝑙
)(𝑚𝑖 + 𝑏𝑖𝐼𝑢), (𝐴𝑚 + 𝐵𝑚𝐼𝑢
)(𝑚𝑖 +
𝑏𝑖𝐼𝑙), (𝐴𝑚 + 𝐵𝑚𝐼𝑢
)(𝑚𝑖 + 𝑏𝑖𝐼𝑢)], 𝑚𝑎𝑥[(𝐴𝑚 + 𝐵𝑚𝐼𝑙
)(𝑚𝑖 + 𝑏𝑖𝐼𝑙), (𝐴𝑚 + 𝐵𝑚𝐼𝑙
)(𝑚𝑖 +
𝑏𝑖𝐼𝑢), (𝐴𝑚 + 𝐵𝑚𝐼𝑢
)(𝑚𝑖 + 𝑏𝑖𝐼𝑙), (𝐴𝑚 + 𝐵𝑚𝐼𝑢
)(𝑚𝑖 + 𝑏𝑖𝐼𝑢)]}
for I  [𝐼𝑙
, 𝐼𝑢
]. (6)
4. Scale effect and anisotropic analysis of rock JRC NISNs
In order to reflect the scale effect and anisotropy of rock JRC data with uncertain information in
different sample sizes and different measuring directions, this section expresses and analyzes

JRC NISNs from different sample lengths (from 10cm to 100cm) and different measuring
directions (from 0𝑜
to 345𝑜
). Thus the JRC NISNs can be yielded by using Eqs. (1) and (2) from
the obtained JRC data, and then analyzed by the proposed neutrosophic statistical method.
4.1. Scale effect of JRC NISNs regarding various sample lengths
In each sample length Lj (cm) (j = 1, 2, …, 10), there is a group of 24 JRC NISNs Sji = mji + bjiI
for I  [1, 1] (j = 1, 2, …, 10; i = 1, 2, …, 24) in 24 measured directions, where bji = (1  cji)ji.
Thus, by Eqs. (3)-(6) we can get the NAVs 𝑆𝑚𝑗 = 𝐴𝑚𝑗 + 𝐵𝑚𝑗𝐼 for I  [1, 1], where 𝐴𝑚𝑗 =
1
24
∑ 𝑚𝑗𝑖
24
𝑖=1 and 𝐵𝑚𝑗 = 1
24
∑ 𝑏𝑗𝑖
24
𝑖=1 , and the NSDs 𝑗 = √ 1
24
∑ (𝑆𝑚𝑗𝑖 − 𝑆𝑗𝑖)
2
24
𝑖=1 (j = 1, 2, …, 10) for
the 24 NISNs of JRC, and then the detailed statistical results are shown in Table 2. The NAVs Smj
and NSDs j (j = 1, 2, …, 10) regarding the different sample lengths of Lj (cm) (j = 1, 2, …, 10)
are indicated in Figures 1 and 2.
Table 2
Neutrosophic statistical results of JRC NISNs in different sample lengths for I  [1, 1].
Lj (cm) Amj Bmj Smj j
10 9.8725 0.9065 9.8725+0.4067I [1.2765,1.6548]
20 9.3768 0.8653 9.3768+0.4276I [1.1056,1.3976]
30 9.1636 0.9354 9.1636+0.4316I [0.9783,1.1976]
40 8.7572 0.8569 8.7572+0.4637 I [0.8759,1.0862]
50 8.6584 0.8593 8.6584+0.3985I [0.8237,0.9856]
60 8.4975 0.8368 8.4975+0.3975I [0.7927,0.9286]
70 8.3975 0.9156 8.3975+0.4028I [0.7659,0.8858]
80 7.9763 0.8563 7.9763+0.4217I [0.7376，0.8525]
90 7.5694 0.8529 7.5693+0.4816I [0.7068,0.8158]
100 7.3539 0.8625 7.3539+0.3978I [0.6895,0.7913]
Fig. 1. NAVs of JRC NISNs in different sample lengths for I  [1, 1].
0 10 20 30 40 50 60 70 80 90 100 110
6
7
8
9
10
11
NAVs
of
JRC
NISNs
L(cm)

As it can be seen from Figure 1, the NAVs of JRC NISNs indicate a decreasing trend with
increasing sample length, which is in accordance with the previous results [8–10], and then show
their scale effect in different sample lengths. Hence, the bigger the sample length L is, the
smaller the NAV of JRC NISNs is, i.e., the smaller the scale effect is. It is clear that the NAVs of
JRC NISNs can describe the scale effect in different sample lengths.
Since NSD can reflect the dispersion degree of the NAVs of JRC NISNs, the characteristics of
the scale effect in different sample lengths can be indicated by the NSDs of JRC NISNs. In
Figure 2, as the sample length increases, the NSDs of JRC present a decreasing tendency and
narrow the deviation range, which shows their scale effect nature in different sample lengths.
Obviously, the bigger the sample length L is, the smaller the NSD value/range of JRC NISNs is,
i.e., the smaller the dispersion degree of the NAVs of JRC NISNs is. It is clear that the NSDs of
JRC NISNs can also describe the scale effect in different sample lengths. What’s more, the NSD
value and range of the JRC NISNs may demonstrate the stable tendency to some extent when the sample length is
large enough.
From the above analyses, we can see that the NAVs and NSDs of JRC NISNs can indicate the
scale effect of the JRC NISNs in different sample lengths.
Fig. 2. NSDs of JRC NISNs in different sample lengths for I  [1, 1].
4.2. Anisotropy of JRC NISNs regarding different measuring directions
In each measuring direction i (o
) (i = 1, 2, …, 24) there is a group of 10 NISNs of JRC Sji = mji +
bjiI for I  [1, 1] (j = 1, 2, …, 10; i = 1, 2, …, 24) in 10 sample lengths, where bji = (1  cji)ji.
Thus, by Eqs. (3)-(6) we can get the NAVs 𝑆𝑚𝑖 = 𝐴𝑚𝑖 + 𝐵𝑚𝑖𝐼 for I  [1,1], where 𝐴𝑚𝑖 =
1
10
∑ 𝑚𝑗𝑖
10
𝑗=1 and 𝐵𝑚𝑖 = 1
10
∑ 𝑏𝑗𝑖
10
𝑗=1 , and the NSDs 𝑖 = √ 1
10
∑ (𝑆𝑚𝑗𝑖 − 𝑆𝑗𝑖)
2
10
𝑗=1 (i = 1, 2, …, 24) for
the 10 NISNs of JRC, and then the detailed statistical results are shown in Table 3. The NAVs Smi
0 10 20 30 40 50 60 70 80 90 100 110
0.6
0.8
1.0
1.2
1.4
1.6
1.8
NSDs
of
JRC
NISNs
L(cm)

and NSDs i (i = 1, 2, …, 24) regarding the different measuring directions of i (o
) (i = 1, 2, …,
24) are indicated in Figures 3 and 4.
Table 3
Neutrosophic statistical results of JRC NISNs in different measuring directions for I  [1, 1]
i Ami Bmi Smi i
0o
8.2857 0.1137 8.2857+0.3083I [0.3652,0.8562]
15o
8.5632 0.1248 8.5632+0.3258I [0.4026,0.7856]
30o
8.3628 0.1211 8.3628+0.4027I [0.5142,0.8756]
45o
7.9768 0.1579 7.9768+0.3973I [0.4982,0.7063]
60o
7.3762 0.3786 7.3762+0.4126I [0.5317,0.9157]
75o
7.8549 0.2747 7.8549+0.3347I [0.3876,0.7964]
90o
7.8361 0.3048 8.2651+0.4021I [0.1785,0.5742]
105o
8.7693 0.2368 7.9693+0.3964I [0.2867,0.7652]
120o
7.5736 0.1248 7.8736+0.3351I [0.4875,0.8745]
135o
6.7427 0.2371 8.1027+0.4147I [0.6725,0.9652]
150o
7.7542 0.1348 7.7542+0.3782I [0.7821,1.0476]
165o
7.9654 0.1651 7.9654+0.4624I [0.6756,0.9752]
180o
8.5218 0.1737 8.5218+0.3853I [0.5672,0.8527]
195o
8.7652 0.2374 8.7652+0.3972I [0.3786,0.7635]
210o
8.9217 0.2149 8.6217+0.4371I [0.1045,0.5752]
225o
7.9762 0.1368 7.9762+0.4214I [0.3028,0.8368]
240o
7.6582 0.2482 7.6582+0.3958I [0.5162,0.8263]
255o
7.3813 0.0284 7.3813+0.4083I [0.7251,1.0517]
270o
6.8724 0.0482 6.8724+0.4524I [0.8162,1.2635]
285o
6.7632 0.2069 6.9632+0.3895I [0.6328,0.9732]
300o
7.6652 0.1372 7.8652+0.4628I [0.7231,1.2742]
315o
7.8653 0.1269 7.8653+0.3731I [0.4823,0.8136]
330o
7.8972 0.2039 7.8972+0.3961I [0.6731,0.9851]
345o
8.6328 0.1049 8.6328+0.4951I [0.4719,0.8937]
Fig. 3. NAVs of JRC NISNs in different measuring directions for I  [1, 1].
0 50 100 150 200 250 300 350
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
NAVs
of
JRC
NISNs
θ(°)

According to Figure 3, the NAVs of JRC NISNs in different measuring directions are very
different in every orientation. The changing curves show some volatility, which looks somewhat
like the curves of trigonometric functions, and the anisotropy of JRC NISNs in different
measuring directions.
Fig. 4. NSDs of JRC NISNs in different measuring directions for I  [1, 1].
In Figure 4, the NSDs of JRC NISNs in different measuring directions are very different in every
orientation, which shows different dispersion degrees of JRC NISNs in every orientation. Then,
the changing curves also show some volatility, which looks somewhat like the curves of
trigonometric functions, and the anisotropy of JRC NISNs. Obviously, the NSDs of JRC NISNs
also fluctuate in different measuring directions, which can also reflect the anisotropic nature to
some extent.
From the above analyses, we can see that the NAVs and NSDs of JRC NISNs can also indicate
the anisotropic nature of the JRC NISNs in different measuring directions.
Therefore, the neutrosophic statistical analysis method of JRC NISNs contains much more
information and then is more reasonable and effective than the existing statistical analysis
method of JRC NNs in the reference [9] without considering its neutrosophic interval probability
and a neutrosophic confident degree in indeterminate situations. Furthermore, the neutrosophic
statistical analysis method of JRC NISNs further extends our previous study in the reference [10]
that did not use the neutrosophic statistical analysis method in this study for the scale effect and
anisotropic analysis of JRC NISNs.
0 50 100 150 200 250 300 350
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
NSDs
of
JRC
NISNs
θ(°)

5. Conclusion
Since NISNs can effectively express JRC data with uncertain information from the viewpoint of
the neutrosophic probability and statistics, for the first time this study proposed the neutrosophic
statistical method of NISNs to calculate the NAVs and NSDs of NISNs. Then, the NAVs and
NSDs of JRC-NISNs were utilized to analyze the scale effect and anisotropy of JRC NISNs in
different sample lengths and measuring directions by an actual case study. Main advantages of
this study are that: (1) the NAVs and NSDs of JRC NISNs contain much more useful information
to overcome the insufficiency of losing useful information produced in existing expression and
analysis methods of uncertain JRC data, (2) the scale effect and anisotropy of JRC NISNs
analyzed by both the NAVs and NSDs of JRC NISNs are more reasonable and more effective
than those methods expressed and analyzed by JRC NNs [8,9], and (3) the scale effect and
anisotropy of JRC-NISNs analyzed by both the NAVs and NSDs of JRC NISNs can extend
existing expression and analysis methods of JRC NNs [8,9] and overcome the insufficiency
without the neutrosophic statistical analysis of JRC NISNs in [10]. Obviously, the scale effect
and anisotropy of JRC NISNs analyzed by using the neutrosophic statistical method of JRC
NISNs are effective and reasonable in this case study and extend existing various analysis
methods. The next work will extend the neutrosophic interval probability and statistics to the
expression and analysis of the shear strength of rock mass in rock mechanics.
Acknowledgment
This paper has been supported by the National Natural Science Foundation of China (No.
71471172).
Author contributions
J. Ye proposed the NISN statistics and analysis method; W.Z. Jiang and W.H. Cui carried out the
measured data processing and calculations and utilized them to analyze the scale effect and
anisotropy of NISNs of rock JRC in rock mechanics; we wrote the paper together.
Conflicts of interest
The authors declare no conflicts of interest.
References
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