Statistical Estimate of Uniaxial Compressive Strength of Rock Based on Shore Hardness

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. Re e ences [1] T. Hólmgei sdó i , P.R. Thomas, Use o he D-762 sho e ha dness scle oscope o es ing small ock olumes, In e na ional Jou nal o Rock Mechanics and Mining Sciences 35 (1998) 85-92. h p://dx.doi.o g/10.1016/S0148-9062(97)00317-3. [2] E. Yaúa , Y. E do÷an, Es ima ion o ock physicomechanical p ope ies using ha dness me hods, Enginee ing Geology 71 (2004) 281-288. h p://dx.doi.o g/10.1016/S0013-7952(03)00141-8. [3] F.I. Shalabi, E. Co ding, O. Al-Ha amleh, Es ima ion o ock enginee ing p ope ies using ha dness es s, Enginee ing Geology 90 (2007) 138-147. h p://dx.doi.o g/10.1016/j.enggeo.2006.12.006. [4] C. Pohl, De e mina ion o cha ac e is ic soil alues by s a is ical me hods, ISGSR 2011: p oceedings o he 3 d In e na ional Symposium on Geo echnical Sa e y and Risk (2011) 427-434. [5] J. Likeš, J. Machek, P obabili y calcula ions (in Czech), STNL, P ague, 1981. [6] R.B. D'Agos ino, M.A. S ephens, Goodness-o - i echniques, Ma cel Dekke , New Yo k, 1986. [7] M. Meloun, M. Hill, J. Mili ký, K. Kupka, Analysis o La ge and Small Samples o Biochemical and Clinical Da a, Clin Chem Lab Med 39 (2001) 53-61. h ps://doi.o g/10.1515/CCLM.2001.013. [8] P.S. Ho n, A.J. Pesce, B.E. Copeland, A obus app oach o e e ence in e al es ima ion and e alua ion, Clin Chem 44 (1998) 622-631. [9] P.S. Ho n, Some easy T-s a is ics, J. Am. S a is . Assoc 78 (1983) 930-936. [10] M. VoĜecho ský, L. Miþa, J. Boš ík, Assessmen o he load-bea ing capaci y o a shallow ounda ion - Pa II. Ve i ica ion o design eliabili y using ully p obabilis ic me hod (in Czech), S a ební obzo 2 (2012) 40-45. [11] D. Dee e, R. Mille , Enginee ing classi ica ion and index p ope ies o in ac ock, Uni e si y o Illinois, U bana, 1966. [12] M. Kizil, Rock Tes s - Sho e Scle oscope, The Uni e si y o Queensland, h p://www.minme .uq.edu.au/~mkizil/5E364_Soil_Rock_Lec u e/sdld013.h m, 2016 (accessed 2016-09-12). [13] G.B. Baeche , J.T. Ch isi an, Reliabili y and s a is ics in geo echnical enginee ing, Wiley, Chiches e , 2003. [14] D. No ák, R. Rusina, M. VoĜecho ský, FReET e sion 1.5 - Pa 2 FReET M / A Use Manual, in: FReET P og am Documen a ion: Re ision 10/2012, Ce enka Consul ing, B no, 2012, pp. 23. [15] D. No ák, M. VoĜecho ský, R. Rusina, FReET e sion 1.5 - Pa 1 FReET Theo y Manual, in: FReET P og am Documen a ion: Re ision 10/2012, Ce enka Consul ing, B no, 2012, pp. 52.