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ORIGINAL PAPER
Determination of Joint Roughness Coefficients Using RoughnessParameters
Hyun-Sic Jang • Seong-Seung Kang •
Bo-An Jang
Received: 8 August 2013 / Accepted: 16 December 2013
� Springer-Verlag Wien 2014
Abstract This study used precisely digitized standard
roughness profiles to determine roughness parameters such
as statistical and 2D discontinuity roughness, and fractal
dimensions. Our methods were based on the relationship
between the joint roughness coefficient (JRC) values and
roughness parameters calculated using power law equa-
tions. Statistical and 2D roughness parameters, and fractal
dimensions correlated well with JRC values, and had cor-
relation coefficients of over 0.96. However, all of these
relationships have a 4th profile (JRC 6–8) that deviates by
more than ±5 % from the JRC values given in the standard
roughness profiles. This indicates that this profile is sta-
tistically different than the others. We suggest that fractal
dimensions should be measured within the entire range of
the divider, instead of merely measuring values within a
suitable range. Normalized intercept values also correlated
with the JRC values, similarly to the fractal dimension
values discussed above. The root mean square first deriv-
ative values, roughness profile indexes, 2D roughness
parameter, and fractal dimension values decreased as the
sampling interval increased. However, the structure func-
tion values increased very rapidly with increasing sampling
intervals. This indicates that the roughness parameters are
not independent of the sampling interval, and that the
different relationships between the JRC values and these
roughness parameters are dependent on the sampling
interval.
Keywords Roughness parameters � Joint roughness
coefficient � Standard roughness profiles � Digitization �Power law equation
List of Symbols
a, b, c Regression coefficients of power law equation
ACF Auto-correlation function
Ah� Potential contact areas
an Normalized intercept yielded by dividing the
intercept (log a) by the nominal length of the
profile
C Dimensionless parameter
CLA Centerline average values
D Fractal dimension
iave Average roughness angles
JRC Joint roughness coefficient
Lh� Normalized length
log a Intercept of the Log L(r)-Log r plot
L(r) Total length of the profile
MSV Mean square roughness height
P Roughness parameter (Z2, SF, Rp-1,
h�max= C þ 1ð Þ2D and D-1)
r Divider value
RMS Root mean square roughness height values
Rp Roughness profile indexes
SDi Standard deviation of roughness angle
SF Structure function
SI Sampling interval
Z1 Mean square first derivative
H.-S. Jang � B.-A. Jang (&)
Department of Geophysics, Kangwon National University,
1 Kangwondaehak-gil, Chuncheon, Gangwon-do 200-701,
Republic of Korea
e-mail: [email protected]
H.-S. Jang
e-mail: [email protected]
S.-S. Kang
Department of Energy Resources Engineering, Chosun
University, 309 Pilmun-daero Dong-gu, Gwangju 501-759,
Republic of Korea
e-mail: [email protected]
123
Rock Mech Rock Eng
DOI 10.1007/s00603-013-0535-z
Z2 Root mean square first derivative values
Z3 Root mean square second derivative
Z4 Percentage excess of distance
h�cr Threshold apparent inclinations
h�max Maximum apparent inclination
1 Introduction
It is important to accurately determine joint roughness,
because it is a critical value used for calculating the
mechanical properties of rock joints. In addition, the max-
imum shear strength of a joint is dependent on its roughness.
Joint roughness has generally been determined using joint
roughness coefficient (JRC) values, as introduced by Barton
(1973). The JRC was later refined by defining 10 standard
roughness profiles in additional work by Barton and
Choubey (1977). The International Society of Rock
Mechanics uses this methodology as the standard method
for determining joint roughness (ISRM 1978), and it has
been widely used in geotechnical and rock engineering.
However, this method uses visual comparisons to estimate
JRC values, meaning that results may vary because of the
subjectivity and experience level of the investigator
(Hsiung et al. 1993; Wakabayashi and Fukushige 1995).
Recent developments have meant that the roughness of a
joint can be precisely measured using laser profilometer or
digital measurement systems, which have the potential to
produce more accurate JRC evaluations. A number of sta-
tistical parameters have been used to determine joint
roughness, such as the root mean square roughness height
values (RMS), root mean square first derivative values (Z2),
average roughness angles (iave), structure function (SF)
values, and roughness profile indexes (Rp). More recent
research has focused on fractal analysis (Turk et al. 1987;
Lee et al. 1990; Wakabayashi and Fukushige 1995; Kulat-
ilake et al. 1995; Jang et al. 2006).
These statistical parameters correlate relatively well
with JRC values and are easily determined, although a
number of different relationships between these parameters
and JRC values have been proposed (Tse and Cruden 1979;
Maerz et al. 1990; Yu and Vayssade 1991; Tatone and
Grasselli 2010). These parameters are also highly depen-
dent on the sampling interval (Miller et al. 1990; Yu and
Vayssade 1991; Chun and Kim 2001). Fractal dimension
values have also been reported to correlate well with JRC
values, although it is difficult to derive distinctive fractal
dimensions for roughness profiles with self-affine charac-
teristics (Carpinteri and Chiaia 1995; Kulatilake et al.
1997). Grasselli and Egger (2003) proposed a 3D discon-
tinuity roughness parameter that was defined by estimating
the contact area of a shearing joint surface. Tatone and
Grasselli (2010) introduced a similar concept to both 3D
surface topography and 2D profiles. They proposed a 2D
discontinuity roughness parameter to be used during JRC
determination.
This study precisely digitized 10 standard roughness
profiles. They were then used to examine the issues and
efficacies of various methods for calculating JRC values,
including Z2, SF, and Rp values, and the 2D discontinuity
roughness parameter of Tatone and Grasselli (2010). In
addition, we then used a divider method for fractal analysis,
and investigated the effect of the sampling interval on JRC
values using standard roughness profiles digitized at four
different sampling intervals (0.1, 0.5, 1.0, and 2.0 mm).
2 Digitization of Standard Roughness Profiles
Barton and Choubey (1977) used 136 individual shear tests
on rock joint specimens to define 10 standard roughness
profiles. They allocated a range of JRC values for each
profile (Table 1). These standard roughness profiles are
widely used to visually estimate joint roughness values.
Previous research has used a standard roughness profile
digitized at 0.5 mm intervals (a result of using a profile
comb with a 1-mm distance between teeth) to measure
roughness profiles (Fig. 1).
In this study, we have digitized the standard roughness
profiles, which we then used to determine JRC values. We
scanned these profiles using a 1,200 dot per inch (dpi)
resolution, and then converted the resulting images into
bitmap image files. Next, the images were digitized at 0.1-
mm intervals using Origin software (Fig. 2), with an
additional three profiles digitized at sampling intervals of
0.5, 1.0, and 2.0 mm.
The majority of these profiles have horizontal lengths of
*99.0 mm; the 7th profile (JRC 12–14) is the shortest
(96.0 mm) and the 8th profile (JRC 14–16) is the longest
(101.0 mm). Barton (1982) reported that the amplitude of
asperities is closely related to JRC values, and that the
amplitudes of asperities measured from standard roughness
profiles tend to increase as JRC values increase. However,
in our study the amplitudes of the 3rd (JRC 4–6), 9th (JRC
16–18), and 10th (JRC 18–20) profiles are smaller than
those of the immediately preceding profiles (Fig. 3).
3 Determination of JRC Values Using Statistical
Parameters
Various statistical parameters have been used to examine
joint roughness; for example, centerline average values,
mean square roughness height, RMS, mean square first
derivative, Z2, root mean square second derivative,
H.-S. Jang et al.
123
percentage excess of distance, average roughness angle i,
standard deviation of roughness angle i (SDi), SF, auto-
correlation function, and Rp values (Wu and Ali 1978;
Krahn and Morgenstern 1979; Tse and Cruden 1979;
Reeves 1985; Maerz et al. 1990; Yu and Vayssade 1991;
Yang et al. 2001; Kim and Lee 2009; Tatone and Grasselli
2010). Tse and Cruden (1979) evaluated these statistical
parameters and reported that the Z2 and SF values corre-
lated well with JRC values. A correlation analysis by Yu
and Vayssade (1991) determined that JRC values corre-
lated well with the Z2, SF, SDi, and Rp values. In addition,
Chun and Kim (2001) reported that the joint asperity slope
estimates provided by statistical parameters (such as the
average roughness angle or Z2 values) correlated better
with JRC values than other parameters. Furthermore,
Miller et al. (1990) suggested that Z2 values correlated well
with JRC values. However, it should be noted that Z2
values alone are not sufficient for evaluating joint rough-
ness, as these values vary with the sampling interval (Yu
and Vayssade 1991; Chun and Kim 2001).
In this paper, we present the Z2, SF, and Rp values for
standard roughness profiles that were digitized at a 0.5-mm
sampling interval. We have evaluated the correlation
between these values and the JRC values, and compared
the results with previous research. Each of these parameters
is calculated using a different statistical approach. The Z2
values are related to the roughness slope, SF values are
related to the degree of change in roughness height, and the
Table 1 Ten standard
roughness profiles suggested by
Barton and Choubey (1977)
The numbers within parentheses
are the exact JRC values
calculated using back-analysis
Profile No.
Rock type Typical roughness profiles JRC range
1 Slate 0–2 (0.4)
2 Aplite 2–4 (2.8)
3 Gneiss
(muscovite)4–6 (5.8)
4 Granite 6–8 (6.7)
5 Granite 8–10 (9.5)
6 Hornfels(nodular)
10–12 (10.8)
7 Aplite 12–14 (12.8)
8 Aplite 14–16 (14.5)
9 Hornfels(nodular)
16–18 (16.7)
10 Soapstone 18–20 (18.7)
SCALE
Fig. 1 Diagram illustrating the
use of a profile comb to obtain
2D profiles of a rough rock joint
(from Tatone and Grasselli
2010)
Determination of Joint Roughness Coefficients
123
Rp values are related to the actual length of the profile. The
profiles shown in Fig. 4 have Z2, SF, and Rp values that
were determined usingZ2 ¼
1
L
Z x¼L
x¼0
dy
dx
� �2
dx
" #1=2
¼ 1
L
Xn�1
i¼1
yiþ1 � yið Þ2
xiþ1 � xi
" #1=2
;
ð1Þ
SF ¼ 1
L
Z x¼L
x¼0
f ðxþ dxÞ � f ðxÞ½ �2dx
¼ 1
L
Xn�1
i¼1
yiþ1 � yið Þ2 xiþ1 � xið Þ; ð2Þ
Rp ¼
Pn�1
i¼1
xiþ1 � xið Þ2þ yiþ1 � yið Þ2h i1=2
L: ð3Þ
Z2 is the most widely used parameter in roughness
analysis. Tse and Cruden (1979) proposed a relationship
between Z2 and JRC [Eq. (4)] based on digitizing a stan-
dard profile at a 0.5-mm sampling interval. In addition, Yu
and Vayssade (1991) reported a linear relationship between
Z2 and JRC, as shown in Eq. (5). Tatone and Grasselli
(2010) reported a power law relationship between these
two variables [Eq. (6)].
JRC ¼ 32:2 + 32:47 log Z2; Tse and Cruden 1979ð Þ: ð4ÞJRC ¼ 61:79 Z2 � 3:47; Yu and Vayssade 1991ð Þ: ð5Þ
JRC ¼ 51:85 Z2ð Þ0:60�10:37; Tatone and Grasselli 2010ð Þ:ð6Þ
The Z2 values calculated from each of the profiles dig-
itized in this study are shown in Fig. 5a. The JRC values
increase as Z2 increases, and the best fit line between these
two variables is provided by a power law equation such as
that shown in Eq. (7). Although the Z2 values for the 4th,
9th, and 10th profiles deviate slightly from the JRC values,
the overall correlation is strong (coefficient of determina-
tion, R2 = 0.972).
JRC ¼ 51:16 Z2ð Þ0:531� 11:44: ð7Þ
Fig. 4 Diagram used to define
the statistical parameters for a
joint profile. Here, yi is the
height of a joint profile at xi, and
Dx is the distance between xi?1
and xi. L is the horizontal length
of a joint profile (from
Kulatilake et al. 1995)
Fig. 2 Digitization of standard roughness profiles
Fig. 3 Amplitudes of asperity for standard roughness profiles
H.-S. Jang et al.
123
We calculated the JRC values using Eqs. (4–7) and
determined the Z2 values using the standard roughness
profiles that we digitized in this study. The calculated JRC
values and the reported standard roughness profiles (Barton
and Choubey 1977) are plotted together in Fig. 5b. If they
are identical, the points will lie along the diagonal 1:1 line
shown in this figure. The dotted lines indicate values that
lie within ±5 % of this 1:1 line. All of the JRC values
determined using Eq. (7), and except the 4th value are
distributed within the ±5 % range. This indicates that this
approach can effectively estimate JRC values. A total of
seven JRC values calculated using Eq. (5) are outside of
this ±5 % range, although there are no significant outliers
within this dataset. The JRC values calculated by Eq. (6)
are similar to those determined using Eq. (7), although the
scatter is somewhat larger in the former. Overall, the two
relationships are very similar. In addition, the JRC values
for the 1st to 3rd profiles calculated using Eq. (4) are rel-
atively low, indicating that this relationship is not suitable
for the calculation of JRC values for smooth profiles.
The SF values also correlate well with JRC values. Tse
and Cruden (1979) reported a logarithmic relationship
between the two [Eq. (8)], and Yu and Vayssade (1991)
proposed a square root equation that defines the relation-
ship between SF and JRC [Eq. (9)].
JRC ¼ 37:28þ 16:58 log SF; Tse and Cruden 1979ð Þ:ð8Þ
JRC ¼ 121:13ffiffiffiffiffiffiSFp
� 3:28; Yu and Vayssade 1991ð Þ:ð9Þ
Figure 6a shows the SF values calculated for the stan-
dard roughness profiles digitized during this study, and
demonstrates that they positively correlate with the JRC
values in the regressed power law relationship of Eq. (10).
The 4th, 9th, and 10th profiles have values that slightly
deviate from this relationship, similarly to the Z2 parame-
ter, although the relationship has a very good overall cor-
relation of R2 = 0.972.
JRC ¼ 73:95 SFð Þ0:266
� 11:38: ð10Þ
Figure 6b shows the JRC values calculated using Eqs.
(8–10) compared with values for the standard profiles
shown in Table 1. All of the JRC values calculated using
Eq. (10) are within ±5 % of the values in Table 1, except
the 4th profile. This is similar to the JRC values calculated
by Yu and Vayssade (1991), which also have relatively
small errors. However, the values calculated by Tse and
Cruden (1979) deviated significantly from all of the JRC
values listed in Table 1.
The Rp parameter is a simple analysis tool that has been
widely used in roughness analyses of various materials. Rp
values start at 1, although the majority of researchers use
an Rp-1 value for convenience. Maerz et al. (1990) sug-
gested that the correlation between Rp-1 and JRC can be
modeled using the linear relationship given in Eq. (11),
whereas Yu and Vayssade (1991) proposed a square root
relationship [Eq. (12)] that is similar to the relationship
they proposed for SF. Tatone and Grasselli (2010) derived
Eq. (13) based on the correlation between the Rp and JRC
values.
JRC ¼ 411:1 Rp � 1� �
; Maerz et al: 1990ð Þ: ð11Þ
JRC ¼ 92:07ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRp � 1
p� 3:28; Yu and Vayssade 1991ð Þ:
ð12Þ
Fig. 5 Relationships between JRC and Z2 as calculated from the standard profiles used in this study (a), and comparison of JRC values
calculated using different relationships (b)
Determination of Joint Roughness Coefficients
123
JRC ¼ 3:36� 10�2 þ 1:24� 10�3
ln Rp
� �" #�1
;
Tatone and Grasselli 2010ð Þ:ð13Þ
Figure 7a shows the Rp - 1 values calculated in this
study; these values have a similar distribution to the Z2 and
SF values, and have a good fit to the power law equation
shown in Eq. (14) (R2 = 0.973).
JRC ¼ 65:9 Rp � 1� �0:302� 9:65: ð14Þ
Figure 7b shows the JRC values calculated in this study
[Eq. (14)], in previous studies [Eqs. (11–13)], and the
values given in Table 1. The JRC values derived using
Eq. (14) are within ±5 % of the actual values, except for
the values calculated for the 4th profile. This is the same
for the Z2 and SF parameters. These results are similar to
those using Eq. (12) (Yu and Vayssade 1991) and Eq. (13)
(Tatone and Grasselli 2010), whereas the relationship in
Eq. (11) (Maerz et al. 1990) contains significant deviations
for the 3rd, 5th, 6th, and 10th profiles.
Fig. 6 Relationship between the JRC and SF values calculated using the standard profiles in this study (a), and comparison of JRC values
calculated using different relationships (b)
Fig. 7 Relationship between JRC and Rp - 1 calculated using the standard profiles used this study (a), and comparisons of values calculated by
various relationships (b)
H.-S. Jang et al.
123
Various types of equations, including logarithmic, lin-
ear, square root, and power law, were used in previous
studies that examined the relationships between statistical
parameters and JRC values. Of these equations, the power
law equation shown in Eq. (15) can represent all of the
relationships between statistical parameters and JRC
values.
JRC ¼ a P½ �bþc; ð15Þ
where P is a roughness parameter (Z2, SF, Rp - 1, etc.),
and a, b, and c are regression coefficients. The JRC values
calculated by the relationships presented by Tse and Cru-
den (1979) and Maerz et al. (1990) differ significantly from
the values calculated during this study (Figs. 6b, 7, 8b),
although these deviations may indicate digitization prob-
lems. Their research was undertaken more than 20 years
ago, when high-precision digitization may not have been
possible. However, the JRC values calculated using the
relationship of Yu and Vayssade (1991) are similar to those
calculated during this study, even though their research was
also performed more than 20 years ago. The JRC values
calculated by the relationship outlined in Tatone and
Grasselli (2010) are almost the same as those presented
here, indicating accurate digitization in both studies.
4 Determination of JRC Using 2D Roughness
Parameters
Grasselli and Egger (2003) investigated areas of contact
during shearing along joints and proposed a 3D disconti-
nuity roughness parameter. They assumed that not all areas
of a joint are in contact during shearing, but only areas with
positive apparent inclinations along the shearing direction.
This assumption allows for the calculation of potential
contact areas Ah� with respect to the threshold apparent
inclinations h�cr from 0� to 90�, along the direction of
shearing. The authors also suggested that a 3D roughness
parameter could be determined using the relationship
between potential contact areas and an apparent inclination
threshold.
Tatone and Grasselli (2010) introduced a 2D roughness
parameter (h�max= C þ 1ð Þ2D) using the same concept as the
3D roughness parameter, where a maximum apparent
inclination h�max and a dimensionless parameter C can be
calculated from the relationship between the normalized
length (Lh� ) and the inclination threshold. The normalized
length is the total length of a portion that has a steeper
inclination than the inclination threshold h�, divided by the
total profile length. Tatone and Grasselli (2010) examined
the average h�max= C þ 1ð Þ2D values of standard roughness
profiles measured from forward and reverse directions at a
sampling interval of 0.5 mm. This analysis led to the fol-
lowing power law relationship between these values and
the JRC values.
JRC ¼ 3:95 h�max= C þ 1ð Þ2D
� �0:7�7:98;
Tatone and Grasselli 2010ð Þ:ð16Þ
We calculated the average h�max= C þ 1ð Þ2D values from
the standard roughness profiles digitized at a 0.5-mm
sampling interval. We analyzed the relationship between
these values and the JRC values, yielding an accurate
power law equation [Eq. (17)] with an R2 value of 0.978
(Fig. 8a).
Fig. 8 Relationships between JRC and h�max= C þ 1ð Þ2D calculated using the standard profiles in this study (a), and comparing these results with
values calculated by Tatone and Grasselli (2010) (b)
Determination of Joint Roughness Coefficients
123
JRC ¼ 5:30 h�max= C þ 1ð Þ2D
� �0:605� 9:49: ð17Þ
The JRC values calculated using Eqs. (16), (17) are very
similar (Fig. 8b). However, the JRC value for the 4th
profile calculated using Eq. (17) falls outside the ±5 %
error range, as do the JRC values calculated for the 4th and
10th profiles using Eq. (16).
The JRC values calculated in this study using the 2D
roughness parameters are very similar to values calculated
in previous studies, indicating that the digitization of
standard profiles in both this and previous studies is
accurate and precise.
5 Determination of JRC Values Using Fractal Analysis
The term fractal is etymologically related to the Latin
‘fractum’, which means ‘broken’ (Mandelbrot 1983).
Fractal shapes have statistically similar morphologies that
appear at various magnification levels on fractured surfaces
in a self-similar system. Fractal analysis quantifies this
complexity as a fractal dimension (D), and enables the
mathematical expression of complex or irregular shapes in
natural objects. Divider, box-counting, variogram, and
power spectral analysis methods are commonly used in
fractal analysis (Cox and Wang 1993; Carpinteri and
Chiaia 1995; Kulatilake et al. 1995; Seidel and Haberfield
1995; Chun and Kim 2001). The divider method is the most
popular method in fractal analysis. It measures a profile
length (L) in a number of sections, determined using a
divider value (r). The relationship between L, r, and the
fractal dimension D shown in Eqs. (18), (19) is derived by
taking the logarithmic values of both sides of the equation.
L ¼ arð1�DÞ ð18Þlog L ¼ log aþ ð1� DÞ log r ð19Þ
The total length of the profile, L(r), is a function of the
divider value (r). It negatively correlates with divider
values, resulting in a straight line with a negative gradient
on a log–log diagram. This line has a gradient of 1-D as
expressed in Eq. (19), and an intercept at log a (Fig. 9).
A number of researchers have conduced fractal analyses
of roughness profiles using the divider method. They have
developed various equations [Eqs. (20)–(24)] that attempt
to explain the correlations between fractal dimensions or
normalized intercepts and JRC values (Turk et al. 1987;
Carr and Warriner 1989; Lee et al. 1990; Wakabayashi and
Fukushige 1995; Jang et al. 2006).
JRC ¼ �1133:6þ 1141:6D; Turk et al: 1987ð Þ; ð20Þ
JRC ¼ �1022:55 + 1023:92D; Carr and Warriner 1989ð Þ;ð21Þ
JRC ¼ �0:878þ 37:784D� 1
0:0015
� �
� 16:93D� 1
0:0015
� �2
; Lee et al: 1990ð Þ; ð22Þ
JRC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D� 1
4:413� 10�5
r; Wakabayashi and Fukushige 1995ð Þ;
ð23Þ
JRC ¼ �256:22
1þ eðan�0:9892Þ=0:00462þ 21:42; Jang et al: 2006ð Þ;
ð24Þ
where an is the normalized intercept calculated by dividing
the intercept by the length of the profile.
Kulatilake et al. (1997) suggested a new suitable range
concept and argued that fractal dimensions should be
estimated within this suitable range, although other
researchers have used the entire range of dividers. This
concept is illustrated in Fig. 10, where feature sizes are
indicated by the lengths of lines between two digitized
points. If a divider span is considerably shorter than the
size of these features (Divider span 1), this span will trace
the profile without defining any peaks or valleys. This
results in an L(r) that is almost identical for all divider
spans, and flattens the slope of the resulting log L(r)-log
r plot (Fig. 10). If the divider span is much larger than the
feature size (Divider span 2), the length should be close to
the horizontal length of the profile. This is primarily
because the divider span will bridge peaks or valleys within
the profile and flatten the slope of the log L(r)-log r plot.
This indicates that the correct slope of the log L(r)-log
r plot, and therefore the correct D value, can be obtained by
fitting a regression line to the non-flattening portion of this
Fig. 9 Diagram showing variations in log L(r) vs. log r values (as
determined using the divider method), where r is the divider span and
L(r) is the total length of the profile measured by r
H.-S. Jang et al.
123
plot. This non-flattening portion is termed the suitable
range.
We studied the relationships between the JRC and
D values, and between the JRC and an values, using two
different regression ranges (Fig. 11). We calculated the
fractal dimensions by regression using the whole range of
dividers that correlate well with JRC values (R2 = 0.990),
although the fractal dimensions measured using the suit-
able range of dividers had a lower correlation
(R2 = 0.852). The fractal dimensions calculated for the 4th
to 8th profiles (JRC 6–16) using this suitable range also
significantly deviate from the best fit curve (Fig. 11a). This
indicates that fractal dimensions should be calculated by
regression using the whole range of dividers, in contrast to
the arguments of Kulatilake et al. (1997). Here, we present
a new power law equation for the relationship between
fractal dimensions and JRC values [Eq. (25)]:
JRC ¼ 103:37ðD� 1Þ0:300 � 8:54
(over the whole range):ð25Þ
Normalized intercepts measured using both ranges cor-
relate well with JRC values, although the values deter-
mined using the suitable range correlate better than those
Fig. 10 Suitable range of r-values for the estimation of fractal dimensions using the divider method (Kulatilake et al. 1997)
Fig. 11 Relationships between a JRC and fractal dimension D values, and b JRC and normalized intercept an values over two different ranges
Determination of Joint Roughness Coefficients
123
calculated using the whole range (Fig. 11b). We also pro-
pose a new power law equation for the correlation between
normalized intercepts and JRC values,
JRC ¼ 138:71ðan � 1Þ0:393 � 5:15
(in the suitable range):ð26Þ
JRC values calculated using the equations outlined here
and from previous studies, and the values of the standard
profiles of Barton and Choubey (1977) are shown in Fig. 12.
If the values calculated using the two different methods lie
on the 1:1 diagonal line, then the results are identical. Points
lying above the line indicate that the standard profile JRC
values are larger than the calculated values, and vice versa. It
should be noted that the JRC values calculated by the
equations determined in previous studies are lower than the
values of the standard profiles. In Fig. 12, the dashed lines
represent a ±5 % deviation from the correct JRC values.
Almost all of the JRC values calculated in this study and
those calculated by Jang et al. (2006) are within this range of
uncertainty. However, the majority of JRC values calculated
in other studies fall outside this error range, possibly
reflecting errors introduced during digitization. This result is
the same as for the statistical parameters.
6 Influence of Sampling Interval on Variations in JRC
Values
The profile of a joint must be digitized when determining
JRC values using statistical parameters, 2D roughness
parameters, or fractal dimensions. The JRC values deter-
mined during analysis may vary with the sampling interval
(Miller et al. 1990; Yu and Vayssade 1991; Chun and Kim
2001). Here, we investigate the effect of the sampling
interval on JRC values using standard roughness profiles
digitized at four different sampling intervals (0.1, 0.5, 1.0,
and 2.0 mm). We calculated the statistical parameters, 2D
roughness parameters, and fractal dimensions for these
profiles.
Figure 13 shows the roughness parameters of the stan-
dard profiles with respect to the sampling interval. We
derived the relationships between the JRC values and
roughness parameters using a power law equation. The Z2,
Rp-1, 2D roughness parameter (h�max= C þ 1ð Þ2D), and D
values decrease as the sampling interval increases,
although the differences in the calculated values are small.
In addition, the Z2, Rp-1, and h�max= C þ 1ð Þ2D values for
the 4th profile (JRC = 6–8) have a different pattern com-
pared with the other profiles, when sampled at an interval
\1.0 mm. However, the D values for the 8th profile
increase as the sampling interval increases. The relation-
ships between JRC values and the Z2, Rp-1,
h�max= C þ 1ð Þ2D, and D values at each sampling interval are
shown as power law equations in Table 2. The correlation
coefficients for each of these equations increase with the
sampling interval, yielding R2 values of 0.99 for Z2, Rp-1
and h�max= C þ 1ð Þ2D at a sampling interval of 2.0 mm.
However, the R2 value of the fractal dimension is highest at
a sampling interval of 0.5 mm.
The SF values increase very rapidly with increasing
sampling interval, and they are nearly 0 at a sampling
interval of 0.1 mm. This indicates that small errors during
profile digitization result in large differences in SF values,
and therefore JRC estimates. The R2 values also increase
with the sampling interval, reaching 0.99 at a sampling
interval of 2.0 mm. The best fit curve equations for these
relationships are given in Table 2.
The R2 values for Z2 and SF are identical for all sam-
pling intervals. This is primarily because Z2 and SF have
similar characteristics. That is, the Z2 values are calculated
by squaring the height difference between two adjacent
points divided by the horizontal length between these
points [Eq. (1)], and the SF values are calculated by mul-
tiplying the squared height differences between two adja-
cent points by the horizontal length between these points
[Eq. (2)].
All of these results clearly indicate that the JRC values
are dependent on the sampling interval. The Z2, Rp-1,
h�max= C þ 1ð Þ2D, and D values vary similarly with the
changing sampling interval, whereas the SF values vary
more significantly. This indicates that the correct rela-
tionships must be used to estimate JRC values.
Fig. 12 Comparison of JRC values calculated from the equations
suggested by previous studies
H.-S. Jang et al.
123
Table 3 provides some examples of inaccurate JRC
estimations that can arise when the wrong relationships are
used. These examples use the standard roughness profiles
that were digitized at a sampling interval of 0.1 mm before
Z2 parameters were calculated. We estimated the JRC
values using relationships with sampling intervals of 0.1,
0.5, 1.0, and 2.0 mm (Table 3). The JRC values calculated
at a sampling interval of 2.0 mm are the largest, and the
differences between JRC values calculated at sampling
intervals of 2.0 and 0.1 mm range from 2.67 to 4.72.
7 Conclusions
In this study, we have presented precisely digitized stan-
dard roughness profiles. We used these profiles to calculate
Fig. 13 Relationships between
JRC values and roughness
parameters at four different
sampling intervals. The filled
diamond, checked circle,
checked triangle and square
symbols represent sampling
intervals of 0.1, 0.5, 1.0, and
2.0 mm, respectively
Determination of Joint Roughness Coefficients
123
roughness parameters such as statistical parameters, the 2D
discontinuity roughness parameter of Tatone and Grasselli
(2010), and fractal dimensions. We calculated the rela-
tionships between the JRC values and these roughness
parameters using power law equations.
The statistical parameter values (e.g., Z2, SF, and Rp)
measured from the standard roughness profiles digitized in
this study correlate well with JRC values. However, all the
relationships have deviations larger than ±5 % for the 4th
profile (JRC 6–8), indicating that its statistical properties
are slightly different from the others. The JRC values
calculated using the relationships outlined by Tse and
Cruden (1979) and Maerz et al. (1990) have the largest
errors. These relationships were outlined more than
20 years ago and these errors may be the result of inac-
curate digitization. In comparison, the JRC values calcu-
lated using the relationships outlined by Yu and Vayssade
(1991) and Tatone and Grasselli (2010) fall within the
±5 % error range. It should be noted that the JRC values
calculated using the relationship suggested by Yu and
Vayssade (1991) are accurate, even though this study used
standard roughness profiles that were digitized more than
20 years ago.
The relationship between the JRC values and 2D
roughness parameters (h�max= C þ 1ð Þ2D) outlined in this
study and by Tatone and Grasselli (2010) are almost
Table 2 Relationships between the JRC values and roughness parameters, determined at different sampling intervals
P Z2 SF Rp-1
Sampling interval (mm) Sampling interval (mm) Sampling interval (mm)
0.1 0.5 1.0 2.0 0.1 0.5 1.0 2.0 0.1 0.5 1.0 2.0
a 54.57 51.16 53.15 54.14 135.11 73.95 53.15 34.49 64.37 65.90 73.64 72.85
b 0.394 0.531 0.692 0.650 0.197 0.266 0.346 0.325 0.248 0.302 0.377 0.350
c -19.13 -11.44 -6.32 -6.40 -19.15 -11.38 -6.31 -6.40 -14.35 -9.65 -5.52 -5.69
R2 0.962 0.972 0.986 0.990 0.962 0.972 0.986 0.990 0.964 0.973 0.987 0.990
P h�max= C þ 1ð Þ2D D-1
Sampling interval (mm) Sampling interval (mm)
0.1 0.5 1.0 2.0 0.1 0.5 1.0 2.0
a 6.82 5.30 3.00 2.78 103.37 107.76 106.74 96.29 Empirical equation JRC ¼ a P½ �bþc
b 0.538 0.605 0.768 0.813 0.300 0.319 0.316 0.276
c -12.13 -9.49 -4.83 -3.98 -8.54 -6.99 -6.63 -8.13
R2 0.971 0.978 0.990 0.992 0.990 0.991 0.985 0.970
Table 3 JRC values calculated using the roughness parameter relationships outlined in Table 2, at sampling intervals of 0.1, 0.5, 1.0, and
2.0 mm
Profile no. Exact JRC JRC calculated by Z2
Eq. for SI = 0.1 mm Eq. for SI = 0.5 mm Eq. for SI = 1.0 mm Eq. for SI = 2.0 mm
1 0.4 0.14 1.13 2.22 3.31
2 2.8 3.24 3.93 4.78 6.02
3 5.8 5.06 5.65 6.42 7.73
4 6.7 9.33 9.83 10.62 12.08
5 9.5 9.79 10.30 11.11 12.58
6 10.8 9.68 10.18 10.99 12.46
7 12.8 12.87 13.47 14.50 16.03
8 14.5 13.58 14.22 15.32 16.85
9 16.7 15.45 16.22 17.54 19.09
10 18.7 19.55 20.73 22.73 24.27
The Z2 values are measured from the standard roughness profiles digitized at a 0.1 mm sampling interval, and the exact JRC values are as in
Table 1
SI sampling interval
H.-S. Jang et al.
123
identical. The JRC values calculated using both relation-
ships are almost identical to the JRC values given in the
standard roughness profiles, indicating that the digitization
in this study is both accurate and precise. The JRC values
for the 4th profile significantly deviate from the values
given in the standard roughness profiles, an identical result
to that calculated using statistical parameters.
The JRC values correlate well with the D values measured
by the divider method. The fractal dimensions measured
using the whole range of dividers correlate better with JRC
values than those measured using only the suitable range of
dividers. This result is in contrast to the findings of Kulatilake
et al. (1997). In addition to the D values, the an values also
correlate well with JRC values, yielding high correlation
coefficients regardless of the range of values used during
regression. The relationships suggested by other researchers
yield much lower JRC values than those reported for the
standard roughness profiles.
The Z2, Rp, 2D roughness parameter, and D values
decrease as the sampling interval increases, but the SF
values increase very rapidly as the sampling interval
increases. This indicates that the roughness parameters are
not independent of the sampling interval, and the differing
relationships between JRC values and roughness parame-
ters may be a consequence of the sampling interval used.
This suggests that the JRC value estimates may be not
accurate if the wrong relationship is used. The differences
between estimates at differing sampling intervals are
minimized when using fractal dimensions, and are maxi-
mized when using SF values.
Acknowledgments This research was supported by the Basic Sci-
ence Research Program of the National Research Foundation of Korea
(NRF), funded by the Ministry of Education, Science and Technology
(2011-0007281).
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