A New 2D Discontinuity Roughness Parameter and Its Correlation With JRC

International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1391–1400

Contents lists available at ScienceDirect

International Journal ofRock Mechanics & Mining Sciences

1365-16

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijrmms

Technical Note

A new 2D discontinuity roughness parameter and its correlation with JRC

Bryan S.A. Tatone, Giovanni Grasselli n

Lassonde Institute, Department of Civil Engineering, University of Toronto, 35 St. George St., Toronto, Ontario, Canada M5S 1A4

a r t i c l e i n f o

Article history:

Received 16 November 2009

Received in revised form

28 May 2010

Accepted 13 June 2010Available online 23 July 2010

1. Introduction

It has long been recognized that the roughness of rockdiscontinuities, when clean and unfilled, can have a significantimpact on both the hydraulic and shear strength characteristics ofdiscontinuous rock masses. In response, several approaches toparameterizing roughness have been proposed over the years,including empirical (e.g. [1,2]), statistical (e.g. [3,4]), and fractalmethods (e.g. [5,6]).

As noted by recent publications by the authors [7,8], themajority of discontinuity roughness evaluations to date have beenbased on the analysis of 2D profiles rather than the 3D surfacetopography. It was noted, in agreement with several otherresearchers, that this approach can lead to incomplete and biasedroughness estimates. To overcome these drawbacks, a 3Devaluation methodology was developed based on the prior workof Grasselli [9–11], in which the associated roughness metric wasdefined in terms of the maximum apparent asperity inclination,y*

max and an empirical fitting parameter C as, y*max/(C+1) [7,8].

Despite the relative merits of the 3D roughness evaluationmethodology, estimates of roughness obtained using 2Dapproaches cannot be abandoned. Considering the analysis of2D profiles formed the conventional approach to roughnessestimation in rock engineering for many years, it is valuable tounderstand how 2D parameters compare to 3D parameters. Theobjectives of this paper are as follows: (1) present a 2D roughnessevaluation methodology analogous to a previously developed 3Dmethodology [8]; (2) compare pseudo-3D roughness estimatesobtained with the 2D methodology with those obtained withthe 3D methodology described in [8]; and (3) establish empi-rical relationships between the new 2D roughness parameter,y*

max/(C+1)2D, and the well-known joint roughness coefficient

09/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

016/j.ijrmms.2010.06.006

esponding author. Tel.:+1 416 978 0125; fax: +1 416 978 3674.

ail address: [email protected] (G. Grasselli).

(JRC) to enable shear strength estimation according to the Barton–Bandis shear strength criterion.

In accomplishing these objectives, several points of progressin roughness quantification will be realized. Firstly, it will bepossible to analyze 2D roughness profiles in the direction ofshearing (i.e., forwards or backwards), which is a step forwardcompared to other 2D methods. Secondly, insight will be gainedinto the suitability of using the 2D methodology to characterize3D roughness, giving those without access to 3D measuringsystems a potential alternative approach to characterize rough-ness anisotropy comparable to the 3D methodology. Lastly,by relating the new 2D parameter to JRC, new users of theparameters can quickly grasp relative differences in roughnessassociated with different values of y*

max/(C+1)2D.

2. Rationale

When shearing rock joints, only a small fraction of the totaljoint surface area is damaged [11–15]. The shape, extent anddistribution of these damaged zones are controlled by manyfactors including: the roughness of the surface, which accounts forthe size, and the shape of the asperities; the shear direction; theapplied normal stress; the total displacement; and the mechanicalproperties of the intact asperities. Nevertheless, the damagedareas are strictly related to the specific surface topography andare typically restricted to those asperity faces that have a localdip-direction opposite the shear direction [11,16] and preferen-tially develop in areas comprised of the steepest faces [11,16–19].Thus, it follows that a roughness parameter describing thetopography of the surface should be based on the distribution ofthe local apparent dip of the asperities with respect to the sheardirection to capture its influence on shear strength.

Considering the previously developed 3D methodology [8], thedistribution of the local apparent asperity dip was characterizedby evaluating the apparent inclination of each triangular facet of a

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B.S.A. Tatone, G. Grasselli / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1391–14001392

triangulated surface model with respect to a selected analysisdirection. The roughness metric for the surface was based onthe shape of the resulting distribution. Considering a 2D profilefrom a rough surface, there are only two possible directions tochoose from (i.e. forward and reverse). Thus, the degree of shearresistance offered by the roughness of the surface in a specificdirection must be approximated by evaluating the relativeproportion of steeply dipping line segments in a 2D profileoriented in the direction of interest. Based on the shape of thisdistribution, an analogous 2D roughness metric can be defined.The following section of this article presents a step-by-stepdescription of the 2D evaluation methodology.

3. 2D roughness evaluation methodology

The methodology for estimating the roughness of 2D profilescan be divided into four steps similar to the 3D methodology [8].These steps include: (1) acquisition of 2D profiles; (2) alignmentof the profiles; (3) analysis of the aligned profiles; and (4)evaluation of the roughness metric for each profile. A detaileddescription of these steps is provided in the following sub-sections.

3.1. Acquisition of 2D profiles

Similar to the 3D methodology, the first step in analyzingthe 2D roughness involves measuring the discontinuity surface.Two-dimensional profiles can either be measured directly froma discontinuity surface using a profilometer or they can beextracted from triangulated irregular network (TIN) surfacemodels acquired with 3D measurement systems, such as: laserscanners, photogrammetry systems, or structured light projectioninstruments (e.g., Fig. 1). The coordinates defining these profiles,whether collected directly or extracted from a TIN, can subse-quently be utilized for roughness analysis.

3.2. Alignment of 2D profiles

Following acquisition, the 2D profiles must be aligned toestablish a line of reference to measure the inclination of the linesegments defining the asperities. There are two approaches thatmay be used to establish this reference line. First, if profiles are

Fig. 1. Extraction of 2D profiles from a triangulated surface: (a) triangulated surface sh

Note the dashed line in (b) represents the best-fit plane through the surface rather tha

measured directly with a profilometer, a best-fit linear regressionline can be created through the points defining the profile.Afterwards, the profile can be rotated such that the best-lineis horizontal. Alternatively, if profiles are extracted from aTIN surface, the surface can be aligned by setting a best-fit planethrough the surface to be horizontal. Profiles can then beextracted and analyzed without any further alignment. If thelatter approach is employed, the coordinate axes of the TINmodel should be rotated such that the x-axis is oriented along thelength of the profile being extracted. In doing so, the coordinatesdefining the profiles can be expressed by x and z coordinate pairs(i.e. y¼0 along the profile), which simplifies the analysis of theprofile.

3.3. Analysis of 2D profiles

To begin analyzing a 2D profile, an analysis direction must bespecified. Unlike a 3D surface where any direction between 01 and3601 can be considered, there are only two possible analysisdirections for a 2D profile: forward and reverse. After selecting theforward or reverse analysis direction, the inclination, y*, of theindividual line segments forming the profiles can be evaluated.

Following the same approach as the 3D methodology, butwhere profile length replaces surface area, it is possible todistinguish the fraction of the total profile length that is moresteeply inclined than progressively greater threshold values of y*.This fractional length is referred to as the normalized length, Ly*,in that it is defined by the length of the profile inclined moresteeply than y* divided by the total length of the profile, Lt.

Fig. 2 further illustrates the characterization of a 2D profile(Section 1 from Fig. 1) based on the inclination of the individualline segments. As shown in the figure, when threshold values of y*

equal to 01, 51, 101, 201, and 301 are considered, the correspondingvalues of Ly* are 0.490, 0.328, 0.225, 0.071, and 0.022, respectively.

3.4. Calculation of 2D roughness

Given the cumulative distribution of the normalized length,Ly*, as a function increasing threshold values of y*, it is possibleto estimate the 2D equivalent of the 3D roughness parametery*

max/(C+1), termed y*max/(C+1)2D. To do so, Eq. (1) is fit to the

cumulative distribution with length terms replacing the

owing location of sections to be extracted and (b) profile view of Sections 1 and 2.

n the best-fit lines through the individual sections.

Fig. 2. The use of the angular threshold concept to characterize a 2D profile

(Section 1 from Fig. 1) in the forward analysis direction (left to right). The various

colours are indicative of the line segments that are steeper than the given

threshold value of y*. The normalized length, Ly*, corresponding to each threshold

value is also given.

B.S.A. Tatone, G. Grasselli / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1391–1400 1393

analogous areal terms of the 3D methodology

Ly* ¼ L0y*

max�y*

y*max

!C

ð1Þ

where L0 is the normalized length of the profile corresponding toan angular threshold of 01 in the chosen analysis direction (i.e. thelength of the profile defined by an apparent dip greater than 01divided by the total profile length); y*

max is the maximuminclination of the profile in the chosen analysis direction; and C

is a dimensionless fitting parameter, calculated via a non-linearleast-squares regression analysis, that characterizes the shape ofthe cumulative distribution [11].

Typical cumulative distributions of Ly* in forward and reversedirections of a 2D profile (Section 1 from Fig. 1) are illustrated inFig. 3a and b, respectively. Also shown are the best-fit linesdefined by Eq. (1) along with the values of C, y*

max, and resultingvalues of y*

max/(C+1)2D.

4. Comparison of results from 2D and 3D analyses

Considering Sections 1 and 2 shown in Fig. 1, the forward andreverse analysis directions correspond to the positive and

negative y-direction and positive and negative x-direction,respectively. Since the positive x- and y-directions correspond toanalysis directions of 01 and 901 in the 3D evaluation methodol-ogy, it is possible to plot 2D and 3D roughness parameters on thesame polar plot for comparison. If several additional profiles inmultiple orientations are analyzed, these too can be plotted on thesame polar plot to obtain a pseudo-3D roughness estimate for thediscontinuity surface. This section of the paper compares pseudo-3D roughness parameters obtained using the 2D methodologywith those obtained using the 3D methodology proposed in [8].This comparison is accomplished via the analysis of a 200 mm2

tensile fracture in limestone. Additional pseudo-3D and 3Droughness values for other discontinuity surfaces can be foundin [7].

4.1. Roughness magnitude

To compare the values of y*max/(C+1) obtained with the 3D

methodology with the corresponding values of y*max/(C+1)2D, 6

groups of 11 2D profiles oriented in directions between 01 and3601 at increments of 301 were extracted from a digitized surface,as shown in Fig. 4a. The 2D roughness values for each of theseprofiles are plotted in Fig. 4b. As shown by the figure, the 2Dvalues bracket the corresponding 3D values in all analysisdirections while the mean 2D values are in approximateagreement with the 3D values. The deviation between the 2Dand 3D roughness values varies depending on the analysisdirection. The mean 2D roughness value and corresponding 3Dvalue in each analysis direction vary from a minimum of 1% to amaximum of 11%.

4.2. Roughness anisotropy

The shapes of the roughness polar plots are indicative ofroughness anisotropy. Roughness values that are approximatelythe same in all directions (i.e. isotropic) produce a nearly circularpolar plot while, roughness values that display a distinctdifference with direction can result in elliptical or sinusoidal-shaped plots. A simple quantitative description of the anisotropyin surface roughness can be obtained by considering the ratiobetween the maximum and minimum roughness values on thepolar plot [8].

For the limestone fracture depicted in Fig. 4, the polar plots ofthe 3D and pseudo-3D roughness display an elliptical shape.Considering the 3D roughness values, the major and minor axes ofthe ellipse are oriented in the 175–3551 and 85–2651 directions,respectively, and the maximum and minimum 3D roughnessvalues of 12.85 and 11.06 produce an anisotropy value of 1.16.Considering the pseudo-3D roughness, as defined by the mean 2Droughness in each analysis direction, the major and minor axes ofthe ellipse are oriented in the 150–3301 and 60–2401 directions,respectively, and the maximum and minimum values of 13.28 and10.67 produce an anisotropy value of 1.24.

In general, the orientation and degree of roughness anisotropydefined by the 3D and pseudo-3D roughness values were inreasonable agreement. Despite the limited area of the fracturesurface that was sampled by the 66 2D profiles and the limitedanalysis orientations considered, the major and minor axes of thepolar plots varied by only 251 and the anisotropy values varied byless than 7%. Thus, reasonable estimates of directional anisotropylike that characterized by the 3D methodology can be obtainedby considering the average 2D roughness of multiple profiles invarying orientations. With the addition of more profiles in moreorientations, the discrepancies between the 3D and pseudo-3Droughness could potentially decrease.

Fig. 3. Example of the distribution of normalized length, Ly*, as a function of the angular threshold, y*, for a 2D profile (Section 1 from Fig. 1): (a) analysis in the forward

direction and (b) analysis in the reverse direction.

Fig. 4. (a) Location of 2D profiles extracted from a 3D TIN surface. (b) Polar plot containing 3D roughness values obtained from the triangulated surface and corresponding

2D roughness values obtained from the 2D profiles.


4.3. Limitations of 2D analyses

Although the preceding sub-sections of this paper have shownthat roughness values based on 2D profiles can be used to obtainreasonable approximations of the 3D roughness, the limitations ofusing 2D profiles over 3D surfaces must be recognized. It is impor-tant to appreciate that when using 2D profiles, only topographyencountered by the profiles is considered. Hence, some importantgeometric features of the surface may be neglected if they are nottransected by one of the profiles. The exclusion of such featurescould lead to a misrepresentation of roughness, anisotropy and,consequently, a misrepresentation of relative shear resistance.

In addition, considering the variability of the 2D values in eachanalysis direction, it can be seen that roughness estimates basedon 2D profiles of the same orientation and in close proximity canboth overestimate and underestimate the 3D roughness value.Therefore, it is suggested that several profiles in each direction becollected and analyzed to establish upper bound, lower bound andmean estimates of the 2D roughness. In doing so, one can reducethe risk of obtaining misleading estimates of surface roughness.Nevertheless, caution must always be exercised when using anyroughness parameter based on 2D profiles; even if several profilesare considered.

5. Correlation with JRC

5.1. Background

The joint roughness coefficient (JRC) in conjunction with theBarton–Bandis shear criterion is the most widely used descriptionof roughness and discontinuity shear strength, respectively.Reliable estimates of the JRC for a rock discontinuity can bedetermined through back-calculation via the results of a tilt/pulltest or direct shear test. However, in many cases such testing maybe limited by time and budgetary constraints. Therefore, tofacilitate estimation of a JRC value without back-calculation,Barton and Choubey [2] published a set of ten standard roughnessprofiles corresponding to different ranges of JRC values. Theseten standard profiles were taken as the most representative of136 specimens sheared in the laboratory [2]. A description ofthese samples and the corresponding back-calculated JRC values isprovided in Table 1.

Shortly after the initial publication of the 10 standard profilesmany researchers and practitioners realized the subjective natureof visually comparing joint surfaces to the standard profiles [20].To overcome the subjective nature of using the standard profiles,several researchers have attempted to establish correlations

Table 1Description and back-calculated JRC value of the rock joints that define the ten standard JRC profiles [2].

Sample no. Rock type Description of joint JRC back-calculated

1 Slate Smooth, planar: cleavage joints, iron stained 0.4

2 Aplite Smooth, planar: tectonic joints, unweathered 2.8

3 Gneiss Undulating, planar: foliation joints, unweathered 5.8

4 Granite Rough, planar: tectonic joints, slightly weathered 6.7

5 Granite Rough, planar: tectonic joints, slightly weathered 9.5

6 Hornfels Rough, undulating: bedding joints, calcite coatings 10.8

7 Aplite Rough, undulating: tectonic joints, slightly weathered 12.8

8 Aplite Rough, undulating; relief joints, partly oxidized 14.5

9 Hornfels Rough, irregular: bedding joints, calcite coatings 16.7

10 Soapstone Rough, irregular: artificial tension fractures, fresh surfaces 18.7


between the standard JRC values and other objective measures ofroughness. Generally, this task was accomplished by digitizing thestandard profiles, assessing the roughness of the profiles with anobjective parameter, and attempting to establish an empiricalrelation between the standard JRC value and the objectiveroughness parameter.

Over the years, empirical equations relating the statisticalparameter, Z2 [21–24]; the roughness coefficient, RP [23–25]; andthe fractal dimension, D [24,26–31] to JRC have been developed toprovide a method of objectively quantify JRC values. The mostcommonly cited equations are those of Tse and Cruden [21] andMaerz et al. [25], which consider the relation of Z2 and RP to JRC,respectively:

JRC ¼ 32:2þ32:47logðZ2Þ ð2Þ

JRC ¼ cðRP�1Þ, where c¼ 400�411 ð3Þ

In establishing the above relationships, the standard JRC

profiles were digitized with a sampling interval of approximately0.5 mm. As noted by Yu and Vayssade [23], the values of Z2 and RP

are sensitive to the sampling interval used to digitize the profiles.Thus, when using equations such as (2) and (3) to estimate JRC,it is imperative that the sampling interval used to obtain 2Dprofiles is consistent with that used to initially develop theequations. The work of Hsiung et al. [32] further illustrates theimportance of sampling interval when using empirical relations toestimate JRC, albeit indirectly. In their work, estimated JRC valuesfor the same profiles are shown to vary by up to 100% when usingempirical equations originally developed using 2D profilesacquired with different sampling intervals.

In addition, it is important to note that the ten standardprofiles were originally obtained from laboratory specimens usinga profile comb similar the one illustrated in Fig. 5a. As illustratedin Fig. 5b, a profile comb is only capable of obtaining measure-ments at a fixed horizontal interval over the continuous discon-tinuity surface, typically between 0.5 and 1 mm. As a result, somefeatures of the surface with a base length less than this intervalwill be neglected. At the same time some step-like features maybe added upon tracing the profile comb to paper, as illustratedin Fig. 5c.

In an attempt to obtain improved empirical relationships forJRC, some researchers decreased the sampling interval used todigitize the standard JRC profiles to values less than 0.5 mm.However, because the instrument initially used to create theprofiles was only capable of obtaining measurement points every0.5–1.0 mm, smaller sampling intervals will not yield any addi-tional information. In fact, decreasing the sampling interval mayresult in the inclusion of some of the small-scale step-like featuresintroduced by the profile comb into the calculation of objectiveroughness parameters.

5.2. Digitization of standard JRC profiles

To digitize the ten standard JRC profiles of Barton and Choubey[2] in this study, a copy of the original printed publication wasscanned in black and white with a 1200 DPI flatbed scanner. Theimage was saved in TIFF format and imported into AutoCAD 2006for digitization. In AutoCAD, the image was scaled to real-worlddimensions using the 10 cm scale bar in the original figure. Next,a series of vertical lines spaced 0.5 mm apart were constructedacross the length of the profiles and polylines were used to tracethe profiles with the intermediate points falling on the intersec-tion of the vertical lines with the profile. Once each profile wastraced, the coordinates defining the polylines were exported toASCII files using a LISP function.

Following digitization of the profiles, it was noted that theoriginal JRC profiles were not aligned such that the average plane(best-fit straight line) was horizontal. Instead, the best-fit linethrough all but the JRC 4–6 profile had a non-zero slope. To realignthe profiles, the slopes of the best-fit lines were used to calculatedegrees of rotation required to make them horizontal. Withrequired angular rotations between 11 counter-clockwise to 21clockwise, differences between many of the original and realignedprofiles before and after rotation are nearly imperceptible (Fig. 6).Nevertheless, the aligned profiles were selected to establish arelation with y*

max/(C+1)2D as the evaluation methodologyrequires a best-fit line through the profile be established toserve as reference for measuring asperity inclinations. For theconvenience of further use, listings of the x, y coordinatesrepresenting both the original and realigned profiles areavailable in [7].

5.3. Verification of digitization

To verify the digitization of the standard profiles, the rough-ness parameters Z2 and RP for the profiles were calculated andcompared to previously published values. Bearing in mind theintrinsic sampling limitations of a profile comb, only studiesutilizing sampling intervals of 0.5 and 1.0 mm were considered.The Z2 values obtained in the current study were compared tothose given by Tse and Cruden [21], Yu and Vayssade [23], andYang et al. [22], while the RP values were compared to those givenby Maerz et al. [25] and Yu and Vayssade [23]. Note thatcomparisons of these values are presented graphically herein,while tabulated values are available in [7].

For sampling intervals of 0.5 and 1.0 mm, the current andpreviously published values of Z2 and RP for each standard JRC

profile are plotted in Fig. 7a and c and Fig. 8a and c, respectively.As indicated by the plots, the data from the current study is inagreement with the previously published values for bothsampling intervals. Using the new values, new empirical

Fig. 5. The use of a profile comb to obtain 2D profiles of a rough rock joint: (a) comb applied to a surface such that the pins conform to the rock surface; (b) zoomed-in view

of pins contacting rock surface illustrating the tool’s sampling interval and how it is unable to capture surface features smaller than this interval; and (c) recorded profile

demonstrating how step-like features may be added to the profile when tracing the profile comb.

Fig. 6. The ten standard JRC profiles of Barton and Choubey [2] re-digitized with a horizontal point spacing of 0.5 mm. The profiles as printed in the original publication are

shown by the dotted lines, while the profiles realigned such that a best-fit linear regression line through each profile is horizontal are shown by the solid lines. Note that the

angles of rotation required to realign the profiles are shown in the right-most column of the figure (clockwise +ve).



equations were derived to estimate JRC from Z2 as

JRC ¼ 51:85ðZ2Þ0:60�10:37 ð0:5mm sampling intervalÞ ð4Þ

JRC ¼ 55:03ðZ2Þ0:74�6:10 ð1:0mm sampling intervalÞ ð5Þ

and from RP as

JRC ¼ 3:36� 10�2þ

1:27� 10�3

lnðRPÞ

" #�1

ð0:5mm sampling intervalÞ

ð6Þ

JRC ¼ 3:38� 10�2þ

1:07� 10�3

lnðRPÞ

" #�1

ð1:0 mm sampling intervalÞ:

ð7Þ

A graphical comparison of these new equations with previousrelations are shown in Fig. 7b and d and in Fig. 8b and d. For bothroughness parameters and sampling intervals, all the empiricalrelations are in close agreement. The agreement between the RP andZ2 values and the corresponding empirical JRC relationships indicatethat the digitization procedure adopted in the current studyaccurately captured the geometry of the ten standard JRC profiles.

5.4. New empirical relation: JRC versus y*max/(C+1)2D

With the Z2 and RP values for the digitized profiles inagreement with previously published values, a relationship

Fig. 7. JRC versus Z2 as measured from the ten standard JRC profiles: (a) Z2 values for the

empirical equations for JRC based on a 0.5 mm sampling interval; (c) Z2 values for the st

of empirical equations for JRC based on a 1.0 mm sampling interval.

between JRC and y*max/(C+1)2D could be investigated. Tables 2

and 3 summarize the values of y*max/(C+1)2D for the standard JRC

profiles digitized with a sampling interval of 0.5 and 1.0 mm,respectively. The tables include the results of analyzing theprofiles in the forward and reverse directions according to themethodology described in Section 3. The results of forward andreverse analyses are not always equal indicating the presence ofanisotropy with respect to shear direction. These observations arein agreement with previous research (e.g., [33]), where directional2D roughness parameters for 2D profiles were found to vary in theforward and backward directions.

The values of y*max/(C+1)2D in the forward and backward

directions alongside the mean of both values were plottedagainst their corresponding JRC values in Fig. 9. Empiricalequations in the form of a power law were found to best relatethe mean values of y*

max/(C+1)2D to JRC. The resulting equations toestimate JRC for sampling intervals of 0.5 and 1.0 mm can beexpressed as

JRC ¼ 3:95 y*max=½Cþ1�2D

� �0:7�7:98 ðfor 0:5mm sampling intervalÞ

ð8Þ

JRC ¼ 2:40 y*max=½Cþ1�2D

� �0:85�4:42 ðfor 1:0mm sampling intervalÞ

ð9Þ

These above equations in relation to the measured values areshown by the solid lines in Fig. 9.

standard JRC profiles digitized with a 0.5 mm sampling interval; (b) comparison of

andard JRC profiles digitized with a 1.0 mm sampling interval; and (d) comparison

Fig. 8. JRC versus RP as measured from the ten standard JRC profiles: (a) RP values for the standard JRC profiles digitized with a 0.5 mm sampling interval; (b) comparison of

empirical equations for JRC based on a 0.5 mm sampling interval; (c) RP values for the standard JRC profiles digitized with a 1.0 mm sampling interval; (d) comparison of

empirical equations for JRC based on a 1.0 mm sampling interval.

Table 2

Summary of y*max/(C+1)2D values for the ten standard JRC profiles digitized with a

sampling interval of 0.5 mm and realigned such that the best-fit line through the

profiles is horizontal.

JRC profile Actual JRC value

(from Table 1)y*

max /(C+1)2D (0.5 mm sampling interval)

Forward directiona Reverse directionb Average

0–2 0.4 2.69 2.98 2.83

2–4 2.8 5.20 4.14 4.67

4–6 5.8 5.43 5.12 5.27

6–8 6.7 8.27 7.66 7.97

8–10 9.5 10.36 6.17 8.26

10–12 10.8 9.37 9.31 9.34

12–14 12.8 10.03 9.48 9.75

14–16 14.5 10.63 12.96 11.80

16–18 16.7 13.48 13.04 13.26

18–20 18.7 17.03 14.44 15.74

a Analysis direction is from left to right with respect to the standard JRC

profiles.b Analysis direction is from right to left with respect to the standard JRC

profiles.

Table 3

Summary of y*max/(C+1)2D values for the ten standard JRC profiles digitized with a

sampling interval of 1.0 mm and realigned such that the best-fit line through the

profiles is horizontal.

JRC

profile

Actual JRC

value (from

Table 1)

y*max/(C+1)2D (1.0 mm sampling interval)

Forward

directiona

Reverse

directionb

Average

0–2 0.4 2.44 2.02 2.23

2–4 2.8 4.02 3.41 3.71

4–6 5.8 5.25 5.31 5.28

6–8 6.7 8.51 5.88 7.20

8–10 9.5 8.70 6.14 7.42

10–12 10.8 8.84 9.35 9.10

12–14 12.8 8.67 9.57 9.12

14–16 14.5 10.21 13.32 11.77

16–18 16.7 12.22 13.22 12.72

18–20 18.7 14.94 13.81 14.37

a Analysis direction is from left to right with respect to the standard JRC

profiles.b Analysis direction is from right to left with respect to the standard JRC

profiles.


6. Summary and conclusions

This paper has developed a roughness evaluation methodologyfor 2D roughness profiles. The methodology was adapted from the3D roughness methodology previously presented in [8]. However,instead of basing the roughness metric on the cumulative

distribution of the apparent dip of the individual triangles of aTIN surface, the 2D roughness metric, termed y*

max/(C+1)2D, isbased on the cumulative distribution of the inclination of the linesegments forming a profile. Comparison of the 2D roughness

Fig. 9. JRC versus y*max/(C+1)2D as measured from the ten standard JRC profiles: (a) for profiles digitized with a sampling interval of 0.5 mm and (b) for profiles digitized

with a sampling interval of 1.0 mm.


values of several profiles with 3D values for the same surfaceshowed that the 2D values bracketed the 3D values, while themean 2D roughness value in a specified orientation was found tobe in closer agreement with the 3D values. Based on this result, itwas recommended that an average roughness values from severalprofiles in the same orientation be taken as the representativeroughness value to reduce the likelihood of obtaining misleadingroughness estimates with the 2D methodology.

In addition, y*max/(C+1)2D was correlated with the well-known

joint roughness coefficient (JRC). To do so, the ten standardprofiles of Barton and Choubey [2] were digitized with samplingintervals of 0.5 and 1.0 mm and analyzed according to the 2Devaluation methodology. By plotting the JRC value for each profileversus the corresponding value of y*

max/(C+1)2D, empiricalequations for JRC were derived. These equations serve as a usefuladdition to previously published empirical equations developedto objectively quantify JRC. Moreover, these relations allowusers of the new 2D roughness parameter to quickly grasp therelative differences in roughness represented by different valuesof y*

max/(C+1)2D.

Acknowledgements

This work has been supported by the Natural Science andEngineering Research Council of Canada in the form of DiscoveryGrant No. 341275 and RTI Grant no. 345516 held by G. Grasselliand an Alexander Graham Bell Canada Graduate Scholarship heldby B.S.A. Tatone. The authors would also like two anonymousreviewers for their comments and suggestions which helpedimprove this paper.

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Documents

A New 2D Discontinuity Roughness Parameter and Its Correlation With JRC