P ocedia Enginee ing 191 ( 2017 ) 248 – 255
1877-7058 © 2017 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Pee - e iew unde esponsibili y o he o ganizing commi ee o EUROCK 2017
doi: 10.1016/j.p oeng.2017.05.178
ScienceDi ec
A ailable online a www.sciencedi ec .com
Symposium o he In e na ional Socie y o Rock Mechanics
S a is ical Es ima e o Uniaxial Comp essi e S eng h o Rock
Based on Sho e Ha dness
Ma in Zá acký*, Jan Š e aĖák, Vladisla Ho ák, Lumí Miþa
Depa men o Geo echnics, Facul y o Ci il Enginee ing, B no Uni e si y o Technology, Ve e i 95, 602 00 B no, Czech Republic
Abs ac
The pape p esen s he use o ad anced s ochas ic simula ion echniques o es ima ing he s eng h beha io o ock ma e ials.
The Sho e Rebound ha dness was measu ed on i y ock specimens coming om ele en di e en geological locali ies in Czech
Republic. The d y uni weigh o e e y es ed ock ma e ial was de e mined also. Uniaxial comp essi e s eng h o ock was
e alua ed hen by conduc ing he comp ession es on e e y specimen. Empi ical dis ibu ion o Sho e ha dness and d y uni
weigh a iables ob ained om labo a o y es s was app oxima ed by he bes i ed heo e ical p obabili y dis ibu ion.
The s ochas ic simula ion using La in Hype cube Sampling was conduc ed based on hose dis ibu ions. Two di e en equa ions
used o es ima ing he comp essi e s eng h o ock on he basis o Sho e ha dness in p ac ice was used as model unc ions.
Compa ison and s a is ical e alua ion o uniaxial comp essi e s eng h o ock de e mined by comp ession es and hose
ob ained as a esul o s ochas ic simula ion is discussed. The desc ip ion o p obabili y dis ibu ion o uniaxial s eng h
is ob ained as a esul o in oduced analysis, which can be used as inpu o ully p obabilis ic design models o ock ma e ials.
© 2017 The Au ho s. Published by Else ie L d.
Pee - e iew unde esponsibili y o he o ganizing commi ee o EUROCK 2017.
Keywo ds:Rock mechanics; Sho e Scle oscope; Rebound ha dness; Uniaxial comp essi e s eng h o ock; S ochas ic simula ion echniques;
La in Hype cube Sampling; Tes o goodness o i ; P obabili y dis ibu ion
1. In oduc ion
Se e al s udies using Sho e ha dness ha e been done o es ima e s eng h pa ame e s o in ac ock. Fo example:
[1], [2] o [3]. Ex ension o e e enced knowledge and p ac ical expe ience in using o Sho e ha dness pa ame e
om egion o Czech Republic is p esen ed in his pape . The au ho s collec ed ela i ely wide ange o ock ypes,
* Co esponding au ho . Tel.: +420-541-147-232.
E-mail add ess:[email p o ec ed]
© 2017 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Pee - e iew unde esponsibili y o he o ganizing commi ee o EUROCK 2017
249
Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
hence he esul s o examina ion o cu en ly used co ela ions a e gene al alid. Sho e ha dness o ock specimens
was measu ed i s ly. Uniaxial comp essi e s eng h was es ed on he same specimens so ob ained esul s can be
co ela ed and eliably compa ed. Two basic o mulas o es ima ion o UCS om Sho e ha dness we e chosen o
de ailed analysis. Resul s o measu emen s we e hen used as inpu s o s ochas ic simula ion me hod and sensi i i y
analysis o e alua e he possibili y o es ima ion o ock ma e ial s eng h in his way.
2. Desc ip ion o es ed ocks and labo a o y es s
2.1. Loca ions o ock sampling
Samples o he ocks come om 11 di e en loca ions a ound Czech Republic, see Table 1. Places a e
concen a ed in No he n Bohemia, in su oundings o P ague and in Mo a ia. Thus, selec ed ocks a e co e ing
conside able pa o he coun y and hey consis o wide a ie y o di e en ock ypes. The da a con ained in his
pape gi e in e es ing oppo uni y o analyse possible co ela ions o cha ac e is ics among a ious ock ypes.
Table 1. Desc ip ion o loca ions and ock ypes.
No. Name o locali y Rock ype O igin
1 Dolní Kounice G anodio i e igneous
2 Ús i nad Labem T achy e igneous
3 Velké Opa o ice Sandy ma li e sedimen a y
4 H ob Pa agneiss me amo phic
5 ýe o y schody Limes one sedimen a y
6 Dolní
Žleb Sands one sedimen a y
7 Vlas Čjo ice O hogneiss me amo phic
8 Hanušo ice Amphiboli e me amo phic
9 Vilémo Phylli e / Qua zi e me amo phic
10 V ané nad Vl a ou Tu i e sedimen a y
11 Š Čcho ice Shale sedimen a y
2.2. Labo a o y es s
The e we e 5 samples o ock es ed coming om each locali y. Excep ions a e locali y no. 4 – H ob whe e only
wo samples we e possible o p epa e and locali y no. 7 – Vlas Čjo ice wi h ou ex ac ed samples. The es
specimens we e p epa ed om d ill co es he e o e hey had cylind ical shape wi h diame e 44 mm and heigh
app ox. 75 mm. The densi y was iden i ied acco ding p ecise measu emen o dimensions and weigh o each
sample.
Fu he he e was es ed scle oscopic ha dness wi h appa a us Sho e– ype D (manu ac u e : The Sho e Ins umen
& M g. Co. N. Y.). The e we e always 10 ebounds eco ded on down and up base. I is 20 ebounds o each
specimen o ally. Ob ained alues we e s a is ically p ocessed a e wa ds and he uniaxial comp essi e s eng h was
calcula ed acco ding co ela ions (11) and (12). Finally, he uniaxial comp essi e s eng h was es ed di ec ly
in hyd aulic p ess Ad an es 9 (manu ac u e : Con ols) in echnological cen e AdMaS. The speed o loading was
se o app ox. 0.3 MPa/s un il ailu e. Mois u e o samples du ing he es was equal o labo a o y en i onmen .
250 Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
3. S a is ical analysis o measu ed alues o es ed cha ac e is ics o ock
3.1. Theo e ical p obabili y dis ibu ion o inpu pa ame e s
In he case o ock p ope ies, a no mal dis ibu ion o en al eady shows an adequa e compliance. Fo ock
pa ame e s, which show ypically a la ge sca e ings he logno mal dis ibu ion is p e e able acco ding [4].
The no mal dis ibu ion (o Gaussian) o d y uni weigh o ock was p esupposed i s . The no mal dis ibu ion
N (ȝ, ı2) is a con inuous p obabili y dis ibu ion desc ibed by pa ame e s ȝ and ı2. I is de ined by he p obabili y
densi y unc ion (PDF) [5]:

2
2
2
() 1
2
x
x
e
P
V
VS (1)
The pa ame e ȝ is he mean o he dis ibu ion and he pa ame e ı2 is i s a iance. The sample a iance
X
was
used as an es ima o o he mean ȝ. The Sho e ha dness (wi h highe CoV) was app oxima ed by Logno mal
dis ibu ion ln N(ȝ,ı2), which is a con inuous p obabili y dis ibu ion o andom a iable, whose loga i hm
is no mally dis ibu ed. In o he wo ds, i he andom a iable X is logno mally dis ibu ed, hen Y=log(X).
Dis ibu ion is desc ibed by pa ame e s Ȝ and ȗ. I is de ined by he p obabili y densi y unc ion (PDF).


2
2
ln
2
2
1
;; 2
0;0;0
x
x
x e
x
x
O
]
O] ]S
 O ]!
(2)
3.2. Tes o goodness o i
The Ande son–Da ling es is a s a is ical es o whe he a gi en sample o da a is d awn om a gi en
p obabili y dis ibu ion. The Ande son-Da ling (AD) es was used o decide i a da a in samples o d y uni weigh s
and Sho e ha dness comes om a popula ion wi h p esupposed dis ibu ions. The null hypo hesis “The da a ollow
he p esupposed dis ibu ion” is es ed.
The Ande son-Da ling s a is ic was gi en by he o mula:
ܣܦ ൌ െ݊ െ ଵ
௡σሺ
ʹ݅ െ ͳሻכൣ݈݊ܨሺܺ௜ሻ൅݈݊൫ͳെܨ
ሺܺ௡ି௜ାଵሻ൯൧
௡
௜ୀଵ (3)
whe e n is sample size, F(X) is cumula i e dis ibu ion unc ion o he es ed dis ibu ion and i is he i h sample,
when he da a is so ed in ascending o de . The adjus ed alue o s a is ic is gi en by:
2
0,75 2,25
*1AD AD nn
§·


¨¸
©¹
. (4)
P- alue o adjus ed AD s a is ic was possible o de e mine and use o conclude, i he es was signi ican in case
o his s udy. The p- alue is he p obabili y o ge ing a mo e ex eme esul i he null hypo hesis is ue.
I he p- alue was highe han he signi icance le el Į= 0.05, he null hypo hesis was accep ed. The e a e di e en
equa ions o calcula ion p- alue depending on he alue o AD* [6]. AD es o he Logno mal dis ibu ion was
implemen ed by ans o ming he da a using a loga i hm and using abo e es o no mali y. Desc ip ion
o p obabili y dis ibu ions based on abo e desc ibed analysis is summa ized in wo ables below:
251
Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
Table 2. D y uni weigh o ock ȖሾȀ͵ሿ – No mal p obabili y dis ibu ions.
No. Name o locali y Rock ype Mean [kg/m3] S d COV
1 Dolní Kounice G anodio i e 2618 12 0.005
2 Ús i nad Labem T achy e 2423 24 0.010
3 Velké Opa o ice Sandy ma li e 2152 74 0.045
5 ýe o y schody Limes one 2669 4 0.001
7 Vlas Čjo ice O hogneiss 2579 16 0.006
8 Hanušo ice Amphiboli e 2869 69 0.024
10 V ané nad Vl a ou Tu i e 2626 13 0.057
11 Š Čcho ice Shale 2690 16 0.006
Table 3. Sho e ha dness Sh [-] – Logno mal p obabili y dis ibu ions.
No. Name o locali y Rock ype Mean [-] S d COV
1 Dolní Kounice G anodio i e 68 7 0.097
2 Ús i nad Labem T achy e 67 6 0.087
3 Velké Opa o ice Sandy ma li e 37 5 0.139
5 ýe o y schody Limes one 46 11 0.109
7 Vlas Čjo ice O hogneiss 67 8 0.119
8 Hanušo ice Amphiboli e 59 10 0.172
10 V ané nad Vl a ou Tu i e 74 7 0.095
11 Š Čcho ice Shale 65 5 0.081
3.3. Analysis o small samples
The analysis o small samples is no eliable and esul s con ains unusually high a e o unce ain y [7]. Fo
numbe o samples n 4  20 a p ocedu e based on o de s a is ics was in oduced by Ho n [8, 9]. This app oach
is based on a dep h which co esponds o he sample qua iles. The pi o dep h is exp essed by
2
2/)1n(
in H
(5)
O
2
1
2
)1n(
in H

(6)
acco ding o which H is an in ege . The Lowe pi o is
)H(xxL (7)
And he uppe pi o is
)H1n(xxU (8)
252 Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
The es ima e o he pa ame e o loca ion is hen exp essed by he pi o hal sum
2UL
Lxx
P
(9)
The esul s a e summa ized in Table 4.
Table 4. Ul ima e comp essi e s eng h o ock in uniaxial comp ession ıc [MPa] – alues om uniaxial comp ession es .
No. Name o locali y Rock ype Pi o hal sum [MPa]
1 Dolní Kounice G anodio i e 66
2 Ús i nad Labem T achy e 68
3 Velké Opa o ice Sandy ma li e 52
5 ýe o y schody Limes one 45
7 Vlas Čjo ice O hogneiss 71
8 Hanušo ice Amphiboli e 49
10 V ané nad Vl a ou Tu i e 72
11 Š Čcho ice Shale 28
3.4. T ans o ma ion o andom a iables using simula ion me hods
The si ua ion a ises in he case o de e mining he ul ima e comp essi e s eng h o ock in uniaxial comp ession
whe e i is necessa y o ind he p obabili y dis ibu ion o he andom a iable Y (ıc), which is a unc ion
o he ec o X o andom a iables (Ȗ, Sh), whose p obabili y dis ibu ions a e known:


YhX (10)
whe e h is a eal unc ion o wo eal a iables de ined on he ield o alues o a andom ec o X. I can be said,
ha he andom a iable Y is ans o ma ion o andom ec o X [10]. The ans o ma ion model unc ion was
conside ed al e na i ely by wo equa ions. Equa ion (11) uses bo h pa ame e s o Sho e ha dness Sh and uni
weigh Ȗ [11]:

895,6.10 62,3Sh..00066,0
1
J
V
(11)
And equa ion (12) [12], in whe e he ul ima e comp essi e s eng h depends only on he Sho e ha dness Sh
)12Sh.(54,3
1
V
(12)
Simula ion me hods such as Mon e Ca lo o a iance educ ion echniques such as LHS me hod can be
ad an ageously used o accele a e and acili a e he ans o ma ion and o calcula ion o s a is ical momen s o PDF
assigned o andom a iables Y in his speci ic example ocused on de e mina ion o ul ima e comp essi e uniaxial
s eng h [13]. A andom ec o X (wi h desc ip ion o p obabili y dis ibu ion and momen s cha ac e is ics o i s
componen s calcula ed in he p e ious pa ag aphs) se es as he andom inpu o u he p ocessing, which consis s
o he ollowing s eps:
xDe ining he ans o ma ion unc ion


YhX ;
xCalcula ing ealiza ion o unc ion


YhX o all gene a ed ealiza ions o ec o X;

253
Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
xAssigning he mos sui able p obabili y dis ibu ion o he se o da a consis s o he ealiza ions o he unc ion


YhX and calcula ing i s s a is ical cen al momen s.
La in Hype cube Sampling echnique LHS–mean wi h 10e5 simula ions was used o gene a ing ealiza ions
o wo-dimensional ec o X. The pa ame e s space was desc ibed by p obabili y dis ibu ions summa ized
in he pa ag aph 2.1. The ad an age o his me hod consis s in he ac , ha i equi es less numbe o simula ions
while conse ing signi icance es ima es o s a is ical pa ame e s usually [14]. So wa e ool F ee was used o
pe o ming he desc ibed analysis [15].
4. Resul s
Table 5. Resul s – es ima e o uniaxial comp essi e s eng h o ock calcula ed acco ding equa ion (11).
No. Name o locali y Rock ype Mean [MPa] S d COV
1 Dolní Kounice G anodio i e 156 26 0.164
2 Ús i nad Labem T achy e 134 18 0.132
3 Velké Opa o ice Sandy ma li e 61 7 0.110
5 ýe o y schody Limes one 91 11 0.121
7 Vlas Čjo ice O hogneiss 152 28 0.182
8 Hanušo ice Amphiboli e 150 43 0.285
10 V ané nad Vl a ou Tu i e 185 33 0.180
11 Š Čcho ice Shale 153 21 0.138
Table 6. Resul s – es ima e o uniaxial comp essi e s eng h o ock calcula ed acco ding equa ion (12).
No. Name o locali y Rock ype Mean [MPa] S d COV
1 Dolní Kounice G anodio i e 197 23 0.118
2 Ús i nad Labem T achy e 193 21 0.106
3 Velké Opa o ice Sandy ma li e 90 18 0.206
5 ýe o y schody Limes one 119 18 0.148
7 Vlas Čjo ice O hogneiss 196 28 0.144
8 Hanušo ice Amphiboli e 168 36 0.216
10 V ané nad Vl a ou Tu i e 220 25 0.113
11 Š Čcho ice Shale 188 19 0.100
The esul o abo e p esen ed analysis is he se o p obabili y dis ibu ions desc ibed by hei momen pa ame e s
summa ized in Table 5 and Table 6. The design pa ame e s (e. g. lo e 5 % quan ile, he uppe 95 % quan ile,
he mean e c. acco ding o he ac ual design si ua ion) can be de e mined om hese dis ibu ions ia common
s a is ical me hods. Da a in Table 5 and Table 6 we e app oxima ed by he Logno mal (3 pa ) p obabili y
dis ibu ion.
Addi ionally, he ela i e e ec o each basic a iable on he esponse o model unc ion was measu ed using
he pa ial co ela ion coe icien be ween each basic inpu a iable and he esponse a iable (uniaxial s eng h).
The nonpa ame ic ank-o de s a is ical co ela ion was exp essed by he Spea man co ela ion coe icien . A high
posi i e co ela ion coe icien , in ange < 0.9; 1.0 >, was obse ed o a iable Sh o bo h model al e na i es –
equa ion (11) and equa ion (12). Opposi e, he esponse o model equa ion (11) seemed no o be so sensi i e
on a iable Ȗ, which co ela ion coe icien s was close o ze o in case o e e y es ed ock ypes.
254 Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
Fig. 1. Sensi i i y o Equa ion (11) esponse on Sho e ha dness Sh (le ) and on D y uni weigh o ock Ȗ ( igh ) o ock No.1 G anodio i e Dolní
Kounice. Adap ed om F ee g aphical esul s.
Figu e 1 (le pic u e) shows in ca esian coo dina es he s ong posi i e sensi i i y o esponse o equa ion (11)
on he Sho e ha dness Sh. The igh pic u e illus a es he small sensi i i y o he same model unc ion on he d y
uni weigh o ock Ȗ. Bo h pic u es a e ela ed o he ock samples No. 1 – G anodio i e om Dolní Kounice.
Fig. 2. Cha compa ing esul s o Mean UCS [MPa] ob ained om a ious me hods.
5. Discussion
S ochas ic simula ion echniques we e employed o make an es ima e o he uniaxial comp essi e s eng h
o a ie y o ock ypes o igina ing om Czech Republic a ea. Two empi ical equa ions o mula ing ela ionship
be ween sho e ha dness and uniaxial comp essi e s eng h ha a e commonly used in p ac ice we e examined.
Values o he uniaxial s eng h esul ing om hose ela ionships we e compa ed o he alues o uniaxial s eng h
measu ed di ec ly on ock samples es ed in hyd aulic p ess. The e can be ound some conclusions coming om
compa ison (see Fig. 1) o di ec ly and indi ec ly assessed uniaxial s eng h o ock ma e ial:
xBo h commonly used equa ions o e es ima e he uniaxial s eng h o ock. The ela ionship (12) acco ding
o [12] o e es ima es he s eng h mo e han he equa ion (11) p esen ed in [11].
xBigge di e ence can be seen be ween he di ec ly and indi ec ly de e mined s eng h in case o ock ma e ials
wi h he highe he e ogenei y (e. g. G anodi i e loc. 1) and o ho opy (e. g. Shales loc. 11).
xAs he dynamic - elas ic Sho e scle oscope ha dness es p ocedu e was o iginally de eloped o es ing o s eel
in me allu gy, he alues o s eng h o ela i ely homogeneous ma e ials (e. g. limes one loc. 5 o sandy
ma li e loc. 3) de e mined by his echnique a e appa en ly in good acco dance wi h hose measu ed di ec ly.
255
Ma in Zá acký e al. / P ocedia Enginee ing 191 ( 2017 ) 248 – 255
Based on esul s o conduc ed measu emen and calcula ions i can be ecommended o use he sho e scle oscope
o es ima ing he uniaxial s eng h o ock wi h highes cau ion and wi h aking he in o ma ion abou
o igin o he es ed ock in o accoun . The co ela ions be ween sho e ha dness and uniaxial comp essi e s eng h
should be con inuously e ised and imp o ed o achie e hei usabili y o p ac ice. Resul s o analysis
summa ized in his pape can also be used o his pu pose. Seconda y ou come o comple ed wo k is he se o ully
desc ibed p obabili y dis ibu ions o d y uni weigh o ock ha can be e en ually used in ully p obabilis ic design
me hods.
Acknowledgmen
This esea ch was inancially suppo ed by he esea ch g an No. FAST-S-15-2743 and wi hin sus ainabili y
esea ch p ojec o Minis y o Indus y and T ade o he Czech Republic No. FR-TI4/329.
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