IOP Con e ence Se ies: Ma e ials Science and Enginee ing
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E alua ion o Rock Join Coe icien s
To ci e his a icle: Ondej Audy and Tomáš Ficke 2017 IOP Con . Se .: Ma e . Sci. Eng. 245 032007
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Rela ed con en
Rock Join Coe icien s De i ed om he
Th ee-Dimensional Fou ie Relie s
Ondej Audy and Tomáš Ficke
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NOVAE AS DISTANCE INDICATORS.
S. an den Be gh and C. J. P i che
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Moni o ing and e alua ion o ood alle gen
labeling egula ion: a selec ion and anking
model o compliance indica o s and
espec i e me ics
L F R C iollo, M F L Almeida and R F Calili
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032007 doi:10.1088/1757-899X/245/3/032007
E alua ion o Rock Join Coe icien s
Ondřej Audy 1, Tomáš Ficke 1
1 B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ve eří 95,
CZ-602 00 B no, Czech Republic
[email p o ec ed]
Abs ac . A compu e me hod o e alua ion o ock join coe icien s is desc ibed and se e al
applica ions a e p esen ed. The me hod is based on wo absolu e nume ical indica o s ha a e
o med by means o he Fou ie eplicas o ock join p o iles. The i s indica o quan i ies he
e ical dep h o p o iles and he second indica o classi ies wa y cha ac e o p o iles. The
absolu e indica o s ha e eplaced he o me ly used ela i e indica o s ha showed some
a i icial beha io in some cases. This con ibu ion is ocused on p ac ical compu a ions es ing
he unc ionali y o he newly in oduced indica o s.
1. In oduc ion
The concep o join ock coe icien s (JRC) was in oduced by Ba on and Choubey [1, 2] o y yea s
ago. Ye , i has been used in geo echnical p ac ice up o p esen days. Soon a e in oducing his
concep , many au ho s ied o exp ess he coe icien s JRC
Rin a ious analy ical o ms.
Thse and C uden [3] in es iga ed eigh di e en pa ame e s as po en ial candida es o JRC.
These eigh pa ame e s include wo classical pa ame e s a
R and q
R aken om he a senal o
oughness me ology o me allic su aces, and oo mean squa e alues 2
Z, 3
Z compu ed om he
i s and second de i a i es o he measu ed p o iles, espec i ely. The eigh pa ame e s also include
he pa ame e 4
Z ha ep esen s he pe cen age excess o he dis ance measu ed along he p o ile
whe e he slope is posi i e o e he dis ance whe e he slope is nega i e; wo s a is ical pa ame e s
called mean squa e alues (MSV) and he au oco ela ion unc ion (ACF); and, inally, he s uc u e
unc ion (SF). Thse and C uden [3] compu ed all eigh pa ame e s om he measu ed wo-dimensional
(2D) p o iles )(xy wi hin he leng h in e al (0, L) acco ding o he ollowing o mulae


L
adxy
L
R
0
||
1 , 

L
qdxy
L
R
0
2
1 ( oo mean squa e) (1)







Ldx
dx
dy
L
Z
0
2
2
1, 








Ldx
dx
yd
L
Z
0
2
2
2
3
1,


  nega i eiposi i ei xx
L
Z)()(
1
4 (2)


Ldxy
L
MSV
0
2
1, 
Ldxxxyxy
L
ACF
0
)()(
1, 
LdxxxyxySF
0
2
])()([ (3)
2
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WMCAUS IOP Publishing
IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032007 doi:10.1088/1757-899X/245/3/032007
Thse and C uden [3] ound ha only wo o he eigh pa ame e s, namely 2
Zand SF , a e s ongly
co ela ed wi h Ba on’s join oughness coe icien JRC, and sugges ed he co esponding eg ession
ela ions
)log(47.322.32 2
ZJRC  , )log(58.1628.37 SFRJRC


(4)
Mae z e al. [4] measu ed la ge a eas o join ock su aces using an in e es ing echnique called
shadow p o ilome y and, as a desc ip o o he join oughness coe icien , p oposed he so-called
oughness p o ile index p
R ela ed o he so-called mic o-a e age angle i, de ined as an a e age o
inclina ion angles j
I be ween adjacen poin s along he sampling line








 
p
n
jjR
I
n
i1
a ccos||
1
1
(5)
Mae z e al. [4] also sugges ed a eg ession pa e n )( pJRC RR in he ollowing o m
)1(411


p
RJRC (6)
Hong e al. [5], [6] no iced ha he con en ional me hods o digi ized su ace p o iles usually
unde es ima e he su ace oughness pa ame e s due o aliasing and he unde ec ed dead zones o
su ace ex u e.
Lee e al. [7] p oposed he ac al dimension as a ool o an objec i e quan i ica ion o oughness
p o iles o ock discon inui ies. They compu ed dimension D om wo-dimensional p o iles by he
ya d-s ick me hod, and sugges ed a eg ession equa ion as ollows:
2
015.0
1
9304.16
015.0
1
7844.3787804.0 












 DD
JRC (7)
Kula ilake e al. [8] used a spec al unc ion )( S
D
s K S 25
)( 
 (8)
whe e is he spa ial equency o he wo-dimensional p o ile and s
K is a cons an . They classi ied
he oughness o join su aces by he h ee pa ame e s ( s
KID ,, ); he symbol
I
ep esen ed he
inclina ion angle o he aspe i ies.
Al hough many o he sugges ed eg ession pa e ns men ioned abo e p o ide good esul s, he
Ba on isual assessmen o join oughness coe icien s emains he mos equen ly used me hod in
geo echnical p ac ice. The isual compa a i e me hod p oposed by Ba on [1] employs he measu ed
oughness coe icien JRC assigned o each s anda d da abase 2D p o ile. The measu ed JRC alues
ep esen one g ea ad an age o his me hod, since measu ed da a a e usually close o eali y han
hose gene a ed by heo e ical models. Howe e , isual compa ison o su aces is a he subjec i e
me hod. Fo his eason, i would be use ul o de elop a ully compu e ized objec i e p ocedu e ha
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WMCAUS IOP Publishing
IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032007 doi:10.1088/1757-899X/245/3/032007
would eplace he subjec i e isual compa ison o he in es iga ed ock join p o iles wi h Ba on's
s anda d p o iles. De eloping such a compu e ized p ocedu e is a goal o he p esen con ibu ion.
2. Compu e me hod
The compu e me hod o assessing he ock join p o iles equi es he ock join p o iles o be
digi ally scanned in he o m o a disc e e unc ion ),( ji yx . This disc e e unc ion can be i ed by
he Fou ie pa ial sum ),( jiN yxF as ollows
)sinsinsincos
cossincoscos(),(),(
1
0,
q
yn
p
xk
d
q
yn
p
xk
c
q
yn
p
xk
b
q
yn
p
xk
ayxFyx
j
i
kn
j
i
kn
j
i
kn
N
nk
j
i
knknjiNji










 

 (9)
whe e
)},(,),({ qqyppx






 (10)
dydx
q
yn
p
xk
yx
pq
akn


coscos),(
1
 (11)
dydx
q
yn
p
xk
yx
pq
bkn


cossin),(
1
 (12)
dydx
q
yn
p
xk
yx
pq
ckn


sincos),(
1
 (13)
dydx
q
yn
p
xk
yx
pq
dkn


sinsin),(
1
 (14)














041
0,00,021
)0,0(1
nk
nko nk
nk
kn

(15)
The symbol N de ines he uppe bound )1(

N o he subsc ip s n, k, and es ic s he numbe o
e ms in he Fou ie pa ial sum (9).
To cha ac e ize wa y cha ac e o p o iles, he Fou ie ma ix ),( nkMN is in oduced along wi h wo
indica o s )(NS and )(NH
2222 )()()()(),( knknknknknknknknN dcbankM 

(16)
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WMCAUS IOP Publishing
IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032007 doi:10.1088/1757-899X/245/3/032007






1
0
)(
1
0
|),(),(|)(
N
nN
o
N
N
k
nkMnkMNS (simila i y indica o ) (17)
|)()(|)( )( NRNRNH o
(heigh indica o ) (18)





K
i
L
jji
low
Nji
yxFyx
LK
NR
11
2)(
)],(),([
1
)(
(ho izon al p o ile wid h) (19)
whe e LK  is a pixel esolu ion o he used digi al images c ea ed du ing scanning p ocedu e. Rock
join s usually ha e a ce ain wa y base le el
),(
)( ji
low
N
yxF
which is no a pa o hei inhe en
su ace s uc u e and hus i should be emo ed om he scanned p o ile, i.e.
),(),(),(
)()( ji
co
ji
low
Nji
yx yxFyx 
. So ha he Fou ie coe icien s (11) - (15) as well as he
ma ix (16) a e compu ed om he co ec ed p o ile
),(
)( ji
co
yx
.
Bo h he absolu e indica o s )(NS and )(NH we e discussed in ou sepa a e con e ence
con ibu ion [9] and hus he es s conce ning hei capabili y o classi ying dynamical con o mi y o
p o iles can be ound he e.
3. Resul s and discussions
Eigh specimens o ock su aces ha e been ga he ed. Two o hem (Su ace 1 and Su ace 2) ha e
been chosen as es ed su aces whe eas he emaining se en ha e se ed as da abase su aces. In he
i s case, when Su ace 1 was es ed, he da abase su aces consis ed o su aces nos. 2 - 8. In he
second case, when Su ace 2 was es ed, he da abase su aces consis ed o su aces nos. 1, 3 - 8. The
esul ed alues o bo h he indica o s S and H ha e been plo ed g aphically in igu e 1. The bold
a ows in his igu e ma ks he o al minima o he g aphs. The minima indica e couples o he mos
simila p o iles (in es iga ed su ace e sus da abase su ace). Fo example, he minimum o S
indica o associa ed wi h Su ace 1 is loca ed a he da abase no. 4 which means ha Su ace 1 and
da abase su ace 4 ep esen he wo mos simila su aces among all he possible couples o su aces
con aining Su ace 1.
Figu e 1. Resul ed alues o he simila i y and heigh indica o s ela ed o selec ed wo p o iles
(Su ace 1, Su ace 2) and da abase p o iles (1 - 8)

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WMCAUS IOP Publishing
IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032007 doi:10.1088/1757-899X/245/3/032007
As seen om igu e 1, bo h he indica o s S and H de e mines he same couples o he mos simila
su aces, namely Su ace 1 e sus da abase su ace no. 4 and Su ace 2 e sus da abase su ace no. 8.
By expec ing isually all possible couples o su aces, we came o he same conclusions. This ac
migh suppo he unc ionali y o bo h he indica o s. Howe e , he iden ical esul s o hese indica o s
is no a qui e na u al ma e . Occasionally, di e en esul s may appea . This is unde s andable since
su ace p ope ies ela ed o p o ile heigh s and p o ile shapes a e no igh ly co ela ed and hus hei
a ious combina ions may occu . Fo example, wo p o iles may ha e compa able heigh s bu qui e
di e en wa y shapes. In such a case he indica o s H and S will ine i ably de e mine di e en
couples o he mos simila su aces. The ques ion is how such a si ua ion should be sol ed. The
answe is easy. Since he compa ison is done be ween an in es iga ed specimen and he s anda d
da abase p o iles, whose JRC a e known, he ele an couple will be ha associa ed wi h he smalle
da abase JRC. Such a solu ion will always be a he side o highe sa e y.
4. Conclusions
The compu e ized compa ison o ock join p o iles wi h s anda d da abase p o ile is an objec i e
p ocedu e based on nume ical indica o s. In his con ibu ion, wo indica o s o wa y shape and p o ile
heigh ha e been in oduced and subjec ed o p elimina y es s o e i y hei unc ionali y. These
p elimina y es s ha e indica ed ha employmen o he wo indica o s could be p ospec i e bu a
u he es s a e necessa y o a inal decision.
Acknowledgmen
Suppo o his p ojec was p o ided by he G an Agency o he Czech Republic (p ojec no. 13-
03403S). The Fou ie compu a ions we e pe o med by he so wa e w i en by Assoc. P o . D.
Ma išek, PhD.
Re e ences
[1] N. Ba on, Re iew o a new shea -s eng h c i e ion o ock join s. Enginee ing Geology, ol.
7, pp. 287-332, 1973.
[2] N. Ba on and V. Choubey, "The shea s eng h o ock join s in heo y and p ac ice", Rock
Mechanics, ol. 10, pp. 1–65, 1971.
[3] R. Thse and D. M. C uden, "Es ima ing join oughness coe icien s", In . J. Rock Mech. Min.
Sci. & Geomech. Abs ., ol. 16, pp. 303–307, 1979.
[4] N. H. Mae z, J. A. F anklin and C. P. Benne , "Join oughness measu emen using shadow
p o ilome y", In . J. Rock Mech. Min. Sci. & Geomech. Abs ., ol. 27, pp. 329–343, 1990.
[5] E. S. Hong, I. M. Lee and J. S. Lee, "Measu emen o ock join oughness by 3D scanne ",
Geo ech. Tes ing J., ol. 29, pp. 1–8, 2006.
[6] E. S. Hong, I. M. Lee and J. S. Lee, "Unde es ima ion o oughness in ough ock join s", In . J.
Nume . Anal. Me h. Geomech., ol. 32, pp. 1385–1403, 2008.
[7] Y. H. Lee, J. R. Ca , D. J. Ba and C. J. Haas, "The ac al dimension as a measu e o he
oughness o ock discon inui y p o iles", In . J. Rock Mech. Min. Sci. & Geomech. Abs ., ol.
27, pp. 453–464, 1990.
[8] P. H. S W. Kula ilake, G. Shou, T. H. Huang and R. M. Mo gan, "New peak shea s eng h
c i e ia o aniso opic ock join s", In . J. Rock Mech. Min. Sci. & Geomech. Abs ., ol. 32, pp.
673–697, 1995.
[9] O. Audy and T. Ficke , "Rock join coe icien s de i ed om he h ee-dimensional Fou ie
elie s", Wo ld Mul idisciplina y Ci il Enginee ing-A chi ec u e-U ban Planning Symposium-
WMCAUS, 12 -16 June, 2017, P ague.