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

Procedia Engineering 191 ( 2017 ) 248 – 255 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017 doi: 10.1016/j.proeng.2017.05.178 ScienceDirect Available online at www.sciencedirect.com Symposium of the International Society for Rock Mechanics Statistical Estimate of Uniaxial Compressive Strength of Rock Based on Shore Hardness Martin Závacký*, Jan ŠtefaĖák, Vladislav Horák, Lumír Miþa Department of Geotechnics, Faculty of Civil Engineering, Brno University of Technology, Veveri 95, 602 00 Brno, Czech Republic Abstract The paper presents the use of advanced stochastic simulation techniques for estimating the strength behavior of rock materials. The Shore Rebound hardness was measured on fifty rock specimens coming from eleven different geological localities in Czech Republic. The dry unit weight of every tested rock material was determined also. Uniaxial compressive strength of rock was evaluated then by conducting the compression test on every specimen. Empirical distribution of Shore hardness and dry unit weight variables obtained from laboratory tests was approximated by the best fitted theoretical probability distribution. The stochastic simulation using Latin Hypercube Sampling was conducted based on those distributions. Two different equations used for estimating the compressive strength of rock on the basis of Shore hardness in practice was used as model functions. Comparison and statistical evaluation of uniaxial compressive strength of rock determined by compression test and those obtained as a result of stochastic simulation is discussed. The description of probability distribution of uniaxial strength is obtained as a result of introduced analysis, which can be used as input for fully probabilistic design models of rock materials. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EUROCK 2017. Keywords:Rock mechanics; Shore Scleroscope; Rebound hardness; Uniaxial compressive strength of rock; Stochastic simulation techniques; Latin Hypercube Sampling; Test of goodness of fit; Probability distribution 1. Introduction Several studies using Shore hardness have been done to estimate strength parameters of intact rock. For example: [1], [2] or [3]. Extension of referenced knowledge and practical experience in using of Shore hardness parameter from region of Czech Republic is presented in this paper. The authors collected relatively wide range of rock types, * Corresponding author. Tel.: +420-541-147-232. E-mail address:[email protected] © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017 249 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 hence the results of examination of currently used correlations are general valid. Shore hardness of rock specimens was measured firstly. Uniaxial compressive strength was tested on the same specimens so obtained results can be correlated and reliably compared. Two basic formulas for estimation of UCS from Shore hardness were chosen for detailed analysis. Results of measurements were then used as inputs for stochastic simulation method and sensitivity analysis to evaluate the possibility of estimation of rock material strength in this way. 2. Description of tested rocks and laboratory tests 2.1. Locations of rock sampling Samples of the rocks come from 11 different locations around Czech Republic, see Table 1. Places are concentrated in Northern Bohemia, in surroundings of Prague and in Moravia. Thus, selected rocks are covering considerable part of the country and they consist of wide variety of different rock types. The data contained in this paper give interesting opportunity to analyse possible correlations of characteristics among various rock types. Table 1. Description of locations and rock types. No. Name of locality Rock type Origin 1 Dolní Kounice Granodiorite igneous 2 Ústi nad Labem Trachyte igneous 3 Velké Opatovice Sandy marlite sedimentary 4 Hrob Paragneiss metamorphic 5 ýertovy schody Limestone sedimentary 6 Dolní Žleb Sandstone sedimentary 7 VlastČjovice Orthogneiss metamorphic 8 Hanušovice Amphibolite metamorphic 9 Vilémov Phyllite / Quartzite metamorphic 10 Vrané nad Vltavou Tuffite sedimentary 11 ŠtČchovice Shale sedimentary 2.2. Laboratory tests There were 5 samples of rock tested coming from each locality. Exceptions are locality no. 4 – Hrob where only two samples were possible to prepare and locality no. 7 – VlastČjovice with four extracted samples. The test specimens were prepared from drill cores therefore they had cylindrical shape with diameter 44 mm and height approx. 75 mm. The density was identified according precise measurement of dimensions and weight for each sample. Further there was tested scleroscopic hardness with apparatus Shore–type D (manufacturer: The Shore Instrument & Mfg. Co. N. Y.). There were always 10 rebounds recorded on down and up base. It is 20 rebounds for each specimen totally. Obtained values were statistically processed afterwards and the uniaxial compressive strength was calculated according correlations (11) and (12). Finally, the uniaxial compressive strength was tested directly in hydraulic press Advantest 9 (manufacturer: Controls) in technological centre AdMaS. The speed of loading was set to approx. 0.3 MPa/s until failure. Moisture of samples during the test was equal to laboratory environment. 250 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 3. Statistical analysis of measured values of tested characteristics of rock 3.1. Theoretical probability distribution of input parameters In the case of rock properties, a normal distribution often already shows an adequate compliance. For rock parameters, which show typically a large scatterings the lognormal distribution is preferable according [4]. The normal distribution (or Gaussian) of dry unit weight of rock was presupposed first. The normal distribution N (ȝ, ı2) is a continuous probability distribution described by parameters ȝ and ı2. It is defined by the probability density function (PDF) [5]:  2 2 2 () 1 2 x x fe P V VS (1) The parameter ȝ is the mean of the distribution and the parameter ı2 is its variance. The sample variance X was used as an estimator of the mean ȝ. The Shore hardness (with higher CoV) was approximated by Lognormal distribution ln N(ȝ,ı2), which is a continuous probability distribution of random variable, whose logarithm is normally distributed. In other words, if the random variable X is lognormally distributed, then Y=log(X). Distribution is described by parameters Ȝ and ȗ. It is defined by the probability density function (PDF).   2 2 ln 2 2 1 ;; 2 0;0;0 x x fx e x x O ] O] ]S fOf]! (2) 3.2. Test of goodness of fit The Anderson–Darling test is a statistical test of whether a given sample of data is drawn from a given probability distribution. The Anderson-Darling (AD) test was used to decide if a data in samples of dry unit weights and Shore hardness comes from a population with presupposed distributions. The null hypothesis “The data follow the presupposed distribution” is tested. The Anderson-Darling statistic was given by the formula: ܣܦ ൌ െ݊ െ ଵ ௡σሺ ʹ݅ െ ͳሻכൣ݈݊ܨሺܺ௜ሻ൅݈݊൫ͳെܨ ሺܺ௡ି௜ାଵሻ൯൧ ௡ ௜ୀଵ (3) where n is sample size, F(X) is cumulative distribution function for the tested distribution and i is the ith sample, when the data is sorted in ascending order. The adjusted value of statistic is given by: 2 0,75 2,25 *1AD AD nn §·   ¨¸ ©¹ . (4) P-value for adjusted AD statistic was possible to determine and use to conclude, if the test was significant in case of this study. The p-value is the probability of getting a more extreme result if the null hypothesis is true. If the p-value was higher than the significance level Į= 0.05, the null hypothesis was accepted. There are different equations for calculation p-value depending on the value of AD* [6]. AD test for the Lognormal distribution was implemented by transforming the data using a logarithm and using above test for normality. Description of probability distributions based on above described analysis is summarized in two tables below: 251 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 Table 2. Dry unit weight of rock ȖሾȀ͵ሿ – Normal probability distributions. No. Name of locality Rock type Mean [kg/m3] Std COV 1 Dolní Kounice Granodiorite 2618 12 0.005 2 Ústi nad Labem Trachyte 2423 24 0.010 3 Velké Opatovice Sandy marlite 2152 74 0.045 5 ýertovy schody Limestone 2669 4 0.001 7 VlastČjovice Orthogneiss 2579 16 0.006 8 Hanušovice Amphibolite 2869 69 0.024 10 Vrané nad Vltavou Tuffite 2626 13 0.057 11 ŠtČchovice Shale 2690 16 0.006 Table 3. Shore hardness Sh [-] – Lognormal probability distributions. No. Name of locality Rock type Mean [-] Std COV 1 Dolní Kounice Granodiorite 68 7 0.097 2 Ústi nad Labem Trachyte 67 6 0.087 3 Velké Opatovice Sandy marlite 37 5 0.139 5 ýertovy schody Limestone 46 11 0.109 7 VlastČjovice Orthogneiss 67 8 0.119 8 Hanušovice Amphibolite 59 10 0.172 10 Vrané nad Vltavou Tuffite 74 7 0.095 11 ŠtČchovice Shale 65 5 0.081 3.3. Analysis of small samples The analysis of small samples is not reliable and results contains unusually high rate of uncertainty [7]. For number of samples n 4  20 a procedure based on order statistics was introduced by Horn [8, 9]. This approach is based on a depth which corresponds to the sample quartiles. The pivot depth is expressed by 2 2/)1n( intH (5) Or 2 1 2 )1n( intH  (6) according to which H is an integer. The Lower pivot is )H(xxL (7) And the upper pivot is )H1n(xxU (8) 252 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 The estimate of the parameter of location is then expressed by the pivot halfsum 2UL Lxx P (9) The results are summarized in Table 4. Table 4. Ultimate compressive strength for rock in uniaxial compression ıc [MPa] – values from uniaxial compression test. No. Name of locality Rock type Pivot halfsum [MPa] 1 Dolní Kounice Granodiorite 66 2 Ústi nad Labem Trachyte 68 3 Velké Opatovice Sandy marlite 52 5 ýertovy schody Limestone 45 7 VlastČjovice Orthogneiss 71 8 Hanušovice Amphibolite 49 10 Vrané nad Vltavou Tuffite 72 11 ŠtČchovice Shale 28 3.4. Transformation of random variables using simulation methods The situation arises in the case of determining the ultimate compressive strength of rock in uniaxial compression where it is necessary to find the probability distribution of the random variable Y (ıc), which is a function of the vector X of random variables (Ȗ, Sh), whose probability distributions are known:   YhX (10) where h is a real function of two real variables defined on the field of values of a random vector X. It can be said, that the random variable Y is transformation of random vector X [10]. The transformation model function was considered alternatively by two equations. Equation (11) uses both parameters of Shore hardness Sh and unit weight Ȗ [11]:  895,6.10 62,3Sh..00066,0 1 J V (11) And equation (12) [12], in where the ultimate compressive strength depends only on the Shore hardness Sh )12Sh.(54,3 1 V (12) Simulation methods such as Monte Carlo or variance reduction techniques such as LHS method can be advantageously used to accelerate and facilitate the transformation and to calculation of statistical moments of PDF assigned to random variables Y in this specific example focused on determination of ultimate compressive uniaxial strength [13]. A random vector X (with description of probability distribution and moments characteristics of its components calculated in the previous paragraphs) serves as the random input for further processing, which consists of the following steps: xDefining the transformation function   YhX ; xCalculating realization of function   YhX for all generated realizations of vector X; 253 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 xAssigning the most suitable probability distribution to the set of data consists of the realizations of the function   YhX and calculating its statistical central moments. Latin Hypercube Sampling technique LHS–mean with 10e5 simulations was used for generating realizations of two-dimensional vector X. The parameters space was described by probability distributions summarized in the paragraph 2.1. The advantage of this method consists in the fact, that it requires less number of simulations while conserving significance estimates of statistical parameters usually [14]. Software tool Freet was used for performing the described analysis [15]. 4. Results Table 5. Results – estimate of uniaxial compressive strength of rock calculated according equation (11). No. Name of locality Rock type Mean [MPa] Std COV 1 Dolní Kounice Granodiorite 156 26 0.164 2 Ústi nad Labem Trachyte 134 18 0.132 3 Velké Opatovice Sandy marlite 61 7 0.110 5 ýertovy schody Limestone 91 11 0.121 7 VlastČjovice Orthogneiss 152 28 0.182 8 Hanušovice Amphibolite 150 43 0.285 10 Vrané nad Vltavou Tuffite 185 33 0.180 11 ŠtČchovice Shale 153 21 0.138 Table 6. Results – estimate of uniaxial compressive strength of rock calculated according equation (12). No. Name of locality Rock type Mean [MPa] Std COV 1 Dolní Kounice Granodiorite 197 23 0.118 2 Ústi nad Labem Trachyte 193 21 0.106 3 Velké Opatovice Sandy marlite 90 18 0.206 5 ýertovy schody Limestone 119 18 0.148 7 VlastČjovice Orthogneiss 196 28 0.144 8 Hanušovice Amphibolite 168 36 0.216 10 Vrané nad Vltavou Tuffite 220 25 0.113 11 ŠtČchovice Shale 188 19 0.100 The result of above presented analysis is the set of probability distributions described by their moment parameters summarized in Table 5 and Table 6. The design parameters (e. g. lover 5 % quantile, the upper 95 % quantile, the mean etc. according to the actual design situation) can be determined from these distributions via common statistical methods. Data in Table 5 and Table 6 were approximated by the Lognormal (3 par) probability distribution. Additionally, the relative effect of each basic variable on the response of model function was measured using the partial correlation coefficient between each basic input variable and the response variable (uniaxial strength). The nonparametric rank-order statistical correlation was expressed by the Spearman correlation coefficient. A high positive correlation coefficient, in range < 0.9; 1.0 >, was observed for variable Sh for both model alternatives – equation (11) and equation (12). Opposite, the response of model equation (11) seemed not to be so sensitive on variable Ȗ, which correlation coefficients was close to zero in case of every tested rock types. 254 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 Fig. 1. Sensitivity of Equation (11) response on Shore hardness Sh (left) and on Dry unit weight of rock Ȗ (right) for rock No.1 Granodiorite Dolní Kounice. Adapted from Freet graphical results. Figure 1 (left picture) shows in cartesian coordinates the strong positive sensitivity of response of equation (11) on the Shore hardness Sh. The right picture illustrates the small sensitivity of the same model function on the dry unit weight of rock Ȗ. Both pictures are related to the rock samples No. 1 – Granodiorite from Dolní Kounice. Fig. 2. Chart comparing results of Mean UCS [MPa] obtained from various methods. 5. Discussion Stochastic simulation techniques were employed to make an estimate of the uniaxial compressive strength of variety of rock types originating from Czech Republic area. Two empirical equations formulating relationship between shore hardness and uniaxial compressive strength that are commonly used in practice were examined. Values of the uniaxial strength resulting from those relationships were compared to the values of uniaxial strength measured directly on rock samples tested in hydraulic press. There can be found some conclusions coming from comparison (see Fig. 1) of directly and indirectly assessed uniaxial strength of rock material: xBoth commonly used equations overestimate the uniaxial strength of rock. The relationship (12) according to [12] overestimates the strength more than the equation (11) presented in [11]. xBigger difference can be seen between the directly and indirectly determined strength in case of rock materials with the higher heterogeneity (e. g. Granodirite loc. 1) and orthotropy (e. g. Shales loc. 11). xAs the dynamic - elastic Shore scleroscope hardness test procedure was originally developed for testing of steel in metallurgy, the values of strength of relatively homogeneous materials (e. g. limestone loc. 5 or sandy marlite loc. 3) determined by this technique are apparently in good accordance with those measured directly. 255 Martin Závacký et al. / Procedia Engineering 191 ( 2017 ) 248 – 255 Based on results of conducted measurement and calculations it can be recommended to use the shore scleroscope for estimating the uniaxial strength of rock with highest caution and with taking the information about origin of the tested rock into account. The correlations between shore hardness and uniaxial compressive strength should be continuously revised and improved to achieve their usability for practice. Results of analysis summarized in this paper can also be used for this purpose. Secondary outcome of completed work is the set of fully described probability distributions of dry unit weight of rock that can be eventually used in fully probabilistic design methods. Acknowledgment This research was financially supported by the research grant No. FAST-S-15-2743 and within sustainability research project of Ministry of Industry and Trade of the Czech Republic No. FR-TI4/329. References [1] T. Hólmgeirsdóttir, P.R. Thomas, Use of the D-762 shore hardness scleroscope for testing small rock volumes, International Journal of Rock Mechanics and Mining Sciences 35 (1998) 85-92. http://dx.doi.org/10.1016/S0148-9062(97)00317-3. [2] E. Yaúar, Y. Erdo÷an, Estimation of rock physicomechanical properties using hardness methods, Engineering Geology 71 (2004) 281-288. http://dx.doi.org/10.1016/S0013-7952(03)00141-8. [3] F.I. Shalabi, E. Cording, O. 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