(PDF) Investigating Relationships between Engineering Properties of Various Rock Types

Global Journal of Earth Science and Engineering, 2018, 5, 1-25 1 Investigating Relationships between Engineering Properties of Various Rock Types Mehmet Sari* Dept. of Mining Engineering, Aksaray University, Adana Road, 68100 Aksaray, Turkey Abstract: In the literature on rock mechanics several authors have proposed different constants and empirical equations for the physical and mechanical properties of various rock types. This is because researchers usually base their results on the limited number of test samples and/or lithological units. The research presented here has been primarily performed to solve this variation problem. It makes sense to have a general expression for all types of rocks. To accomplish this, a considerable amount of data comprising 4,991 datasets was collected through an extensive literature review. The results were statistically analyzed to determine the range, mean, standard deviation for each investigated rock property. The physical and mechanical properties of the rocks were also plotted against each other in order to estimate one property from the other. The correlation coefficients and best-fit equations were determined by the least squares curve fit method. Usually good correlations were found between index tests, and physico-mechanical properties. At the end, various regression equations are proposed particularly applicable to the whole rock types. Keywords: Intact rock, Physical and mechanical properties, Uniaxial compressive strength, Brazilian tensile strength, Point load strength, Young’s modulus, Schmidt rebound hardness. 1. INTRODUCTION require minimum sample preparation, and testing time is short. Researchers have been developed a number It has been widely acknowledged that the of correlations for the interpretation of rock properties knowledge of rock properties is one of the prime obtained from indirect tests. For instance, some requirements for understanding the rock material. researchers have investigated the relationship between Rocks have been clearly defined according to their unconfined compressive strength (UCS) and point load significant physical and mechanical properties. In strength (PLS) and considerably different constants addition, the physical and mechanical properties of have been found. This is because; they generally base intact rocks are very important in civil, mining, their results on a limited number of observations and petroleum and geological engineering works that lithological units. interact with rock such as underground structures, dams and foundations, tunneling and slopes. It is generally recognized that natural materials like Mineralogy, grain size, density and porosity are rock tend to show a considerable variety of properties. assumed to be the intrinsic properties determining the However, not only does it vary widely with the property rock strength. Mechanical properties such as hardness investigated but it also shows extensive deviation from and strength, however, are not intrinsic material test to test. Therefore, when defining a mechanical properties which mostly depend on the type of testing property it is extremely important to specify the test instrument and the test procedure adopted. procedure adopted. For instance, various techniques of Index tests have traditionally been used to estimate recording Schmidt rebound hardness have been one or more of the mechanical properties of rock. To consistently proposed in the literature [1]. It was seen evaluate rock strength and deformation usually direct that the rock property correlations reported in the tests are performed, but they are generally expensive technical literature often have a limited database and and require considerable time, especially in the should be considered with caution. Also, there is no preparation of rock cores for testing. Instead, various generally accepted empirical equation or approach in indirect testing methods were developed and used to the literature to estimate different rock properties. Most interpret the engineering properties of rocks. The of the researchers state that their model might be a indirect tests i.e., point load, Schmidt rebound useful tool for the rocks specific to their study, but their hardness, Shore scleroscope hardness, p-wave work was not conclusive and more data is required to velocity are relatively easy to perform, are not costly, improve the validity of the proposed models. Therefore, an attempt should be made to develop empirical equations essentially applicable to the whole rock *Address correspondence to this author at The Dept. of Mining Engineering, Aksaray University, Adana Road, 68100 Aksaray, Turkey; Tel: +90 382 range. Then, it will be possible to predict and estimate 2150953; Fax: +90 382 2150592; E-mail:

which property needs high quality core samples and E-ISSN: 2409-5710/18 © 2018 Avanti Publishers 2 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari sophisticated test equipment which property can be 2. THE DATABASE easily determined using simple and quick tests. The database comprised a wide variety of physical In this study, an extensive research on the literature properties such as dry density (γd), specific gravity (Gs), was performed to assemble data on the physical and water content (w), porosity (n) and wave velocity (p- mechanical properties of intact rock from a wide array wave) and mechanical properties such as uniaxial (σc) of different national and international published and triaxial compression tests, elastic modulus (Et), sources. This information will facilitate an assessment Poisson’s ratio (ν), Brazilian tests (T0). Various index of physical and mechanical characterization of intact tests namely point load (Is), Schmidt hammer (R), rock with a single expression. Instead having been Shore scleroscope (Sh), and slake durability (Id) were based solely on the sample statistics that involve also included. The data compiled embraces various mostly a limited number of tests and rock samples; this types of rock from coal to granite, various compressive work aims to find the population parameters by strength values from 0.4 MPa standing for “very weak collecting all necessary information. The main objective rock” to 436 MPa falling in “extremely strong rock” and of this study is, therefore, to derive simple empirical various testing conditions. Hence, the data is relationships between various physico-mechanical considered to be highly representative of all the properties of intact rocks for a wide range of rock types variations that could be anticipated in rock engineering at different origins. A second aim of the study is to practices. In total, 4,991 rock types were sampled in acquire a single but still functional relationship for most the database and they are all employed for the of the rocks by presenting frequency distributions of statistical analysis. Names, types and classes of rocks several rock characteristics and well-known constants, are presented in Table 1 and the total sample numbers which shows considerable variations from study to of each rock represented in the database are given in study in the literature. the brackets. * Table 1: Primary Rock Types in the Database Classified by Geologic Origin Sedimentary Metamorphic Igneous Clastic Organic Chemical Foliated Massive Intrusive Extrusive Pyroclastic Agglomerat Conglomerate (27) Limestone (640) Gneiss (71) Marble (197) Granite (432) (8) e Breccia (12) Argillite (3) Syenite (49) Calcarenite (11) Pegmatite (3) Coarse Tonalite (4) Monzonite (10) Dunite (13) Norite (14) Sandstone (711) Travertine (58) Hornfels (5) Granodiorite (57) Andesite (173) Breccia (6) Medium Siltstone (103) Dolostone (139) Schist (77) Migmatite (3) Diorite (38) Dacite (16) Tuffite (5) Greywacke (30) Caliche (13) Phyllite (15) Diabase (23) Trachyte (2) Dolerite (35) Shale (177) Coal (126) Chalk (101) Slate (14) Quartzite (43) Gabbro (29) Basalt (165) Tuff (288) Mudstone (98) Anhydrite (10) Granulite (5) Amphibole (16) Epidiorite (5) Rhyolite (12) Ignimbrite (17) Fine Marlstone (144) Gypsum (46) Serpentine (24) Pyroxenite (5) Pumice (19) Claystone (11) Rock salt (18) Talc (4) Mylonite (4) Charnockite (11) Total (1314) (126) (1062) (229) (254) (750) (367) (342) * Numbers in parenthesis refer to total number of rocks sampled in the database. Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 3 The following steps are carefully taken into point load were taken into consideration in the consideration during the construction of the database. database. • Each rock in the sample data is given a code A summary of the physical and mechanical representing the rock type and rock class. properties covered in the database is given in Table 2, presenting the range, mean, standard deviation for • The correction for scale and shape effects was each investigated rock property. It is understandable made using the suggestions of Hoek and Brown that the most represented rock property in the [2], Turk and Dearman [3] for UCS, Broch and database is the uniaxial compressive strength with a Franklin [4] and ISRM [5] for PLS, if it was not total of 3,511 data points. Dry density and total porosity already taken into account in the original study. come after this property with a total representation of 3,068 and 2,901 values, respectively. The least • A new conversion factor has been proposed and represented tests in the database are the slake subsequently used in this study to convert the durability index with a total of 245 tests and Los Angles Schmidt hammer readings from N-type to L-type, abrasion index with a total of 212 tests. since notable differences were reported between the two types of hammer values on the same 3. RESULTS AND DISCUSSION rock specimens [1, 6, 7]. In order to propose an indirect estimation by • The values for static and dynamic elasticity, empirical equations, statistical methods are traditionally effective and total (absolute) porosity, dry and used. Bivariate correlation provides a means of saturated property, normal and parallel to summarizing the relationship between two different foliation strength, bulk and grain density, fresh variables. Linear regression (y=ax+b) is the most and weathered property, axial and diametric Table 2: Descriptive Statistics for the Investigated Properties of Rocks in the Database N Min. Max. Mean Std. Dev. 3 Dry density, gr/cm 3068 0.62 4.04 2.37 0.43 Specific gravity 2746 1.65 4.05 2.70 0.11 Effective porosity, % 1587 0.01 70.00 9.41 11.66 Total porosity, % 2901 0.01 74.92 13.14 13.65 UCS, MPa 3511 0.40 436.00 80.60 66.28 BTS, MPa 1864 0.05 45.10 7.32 5.80 PLS, MPa 1190 0.05 20.80 5.12 3.94 Shore scleroscope hardness 537 3.00 107.00 46.12 21.92 Schmidt rebound hardness 1002 9.50 72.00 41.43 12.31 P-wave velocity, km/s 1277 0.38 7.19 4.19 1.49 Cohesion, MPa 492 0.03 74.00 13.21 11.57 o Internal friction angle, 481 9.00 69.00 39.63 11.57 Young’s modulus, GPa 2031 0.05 137.40 28.52 24.51 Poisson’s ratio 1024 0.03 0.65 0.25 0.07 Slake durability index, % 245 29.75 99.80 90.17 14.22 Los Angles abrasion index, % 212 10.20 99.90 30.36 18.57 1/2 Fracture toughness, MPa.m 267 0.03 3.21 1.30 0.71 UCS/BTS ratio 1674 2.02 51.89 11.77 5.63 UCS/PLS ratio 1036 2.58 55.00 18.01 7.45 PLS/BTS ratio 659 0.50 7.50 1.79 0.85 Modulus ratio, Et/σc 1769 12.08 2057.7 347.09 244.6 4 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari common statistical procedure for fitting a straight line to a set of experimental data and is based on the least- square curve estimation. In addition to linear regression analysis, power (y=axb), logarithmic (y=a+Inx) and exponential (y=aebx) relationships between variables are also investigated [8]. The regression line equations could mostly be used to predict one property from the results of other empirical tests. Many researchers have proposed different empirical equations concerning the rock properties. A supplementary list of the expressions proposed by different authors for the index, physical and mechanical properties of intact rocks is presented in Appendix Table A1. At a glance, one can easily discern that there are enormous variances between the proposed Figure 1: Relationship between total porosity and dry density. relationships for the various rock types by different authors. Although the number of tests for the indices is 2 γdry = 2.751-0.029ntot (r =0.946) (1) usually large enough for statistical inferences, the number of property tests is seldom sufficient to draw 3 where γdry in gr/cm and ntot in %. necessary conclusions for the whole rock spectrum. Usually, most of the researchers based their analysis 3.1.1. Porosity on a limited number of rock types and testing range. Total or absolute porosity is a measure of the total 3.1. Significance of Porosity and Density Data void volume and can be obtained from the following formula, The presence of pores in the fabric of a rock material decreases its strength, and increases its ntot = (1-γdry/Gs)100% (2) deformability [9]. A small change of volume fraction of pores can produce considerable mechanical effects. where ntot is total porosity (%), γdry is dry density Since sandstones and carbonate rocks, in particular, 3 (gr/cm ) and Gs is specific gravity of rock. occur within a wide range of porosities, they exhibit a highly variable mechanical character; igneous rocks The effective porosity, on the other hand, is a weakened by weathering processes also have typically measure of the apparent void volume and is high porosities. Most rocks have similar grain densities determined by the saturation and air porosimeter and therefore, have porosity and dry density values methods [11]. Unaltered rocks typically have porosity that are highly correlated. A low-density rock is usually that is less than 20% which may increase due to highly porous. It is often sufficient, therefore, to quote weathering to values of 50% or even higher [12]. values for porosity alone but a complete description requires values for both porosity and density [10]. Porosity is the single most important physical property that influences rock strength. The pores in a It is clearly evident from Figure 1 that there is a rock which are prone to water saturation are principally highly significant linear relationship between the total known as effective porosity. With increasing effective porosity and dry density, as the former increases the porosity, penetration of water and consequently, the later decreases. Here, it is necessary to clarify that negative effects increase. The physical explanation of since coal and evaporate rocks such as rock salt and this is that high porosity contributes the networking gypsum, have considerably lower grain densities (1.7, (propagation) of stress-induced micro fractures [13]. It 2.2 and 2.35, respectively) compared to rocks in is established that the uniaxial compressive strength general (2.7), they are instantaneously excluded in the (σc) of porous rocks such as sandstones, granites, following regression analysis between dry density and dolerites, basalts, dolomites, limestones, and chalks total porosity. The simple linear equation that relates depend on porosity (n) (Appendix Table A1). dry density (γdry) dependent on total porosity (ntot) is: Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 5 Figure 2: Total porosity vs. a. UCS, b. Brazilian tensile, c. Point load, and d. Young’s modulus. Figure 2 illustrates the relationship between total compressive and tensile strengths considerably porosity (ntot) and uniaxial compressive strength (σc) for decrease as the porosity increases, this effect is the 2 all rocks. Clearly, the uniaxial compressive strength is most pronounced for UCS (r =0.679) then for BTS 2 2 inversely related to porosity. Large void space (high (r =0.596) and then for Young’s modulus (r =0.477) 2 porosity) has a negative effect on rock strength. and PLS (r =0.421). Similarly, there is a non-linear relationship of a hyperbolic nature between tensile and point load 3.1.2. Density strengths, and Young’s modulus and total porosity, as can be seen in the same figure. The plots indicate a One of the basic properties of a rock is that its sharp decrease in strength with an increase in the total density is influenced, primarily, by the specific gravities porosity. The following formula was derived to relate of the minerals it contains and the amount of uniaxial compressive strength to total porosity: unoccupied void space within it. The plots of four properties as a function of dry density are presented in 2 σc =126.5exp(-0.071ntot) (r =0.679) (3) Figure 3. As the dry density increases, so does the unconfined compressive, Brazilian tensile, point load where σc is the uniaxial compressive strength (MPa) strengths, and Young’s modulus increases and ntot is the total porosity (%). exponentially. The relationship is significant in the case 3 of the densities lower than 2.4 gr/cm but less The unconfined compressive strength, Brazilian pronounced and scattered around the band of 2.4 and strength and the point load strength had highly 3 2.9 gr/cm owing to fact that most rocks typically have significant relationships with total porosity. The dry densities around this range as shown in Figure 4. relationships with effective porosity were not as good On the other hand it is noticed that there was a wide but were still significant (Table 3). In other words, the range in dry density values in the database. The lowest 6 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari Table 3: A Complete List of the Proposed Equations for all Rock Types in the Database Equation R Equation R σc = 10.04T0 0.77 T0 = 10.31exp(- 0.061ntot) 0.77 - 0.506 σc = 8.54T0 + 18.21 0.79 T0 = 11.83ntot 0.67 0.967 σc = 11.22T0 0.91 T0 = 0.049exp(1.922γdry) 0.76 σc = 15.15Is(50) 0.84 T0 = 0.173γ dry 3.903 0.75 σc = 13.36Is(50) + 14.05 0.86 T0 = 0.524exp(0.055RL) 0.75 0.879 0.90 T0 = 0.003R L 2.064 0.75 σc = 19.22Is(50) 0.80 T0 = 6.78KIC 0.77 σc = 0.259exp(2.217γdry) 4.543 0.79 T0 = 5.82KIC + 1.64 0.79 σc = 1.07γ dry 0.78 T0 = 6.88KIC 0.948 0.87 σc = 1.819Sh 0.79 0.82 T0 = 1.517Is(50) σc = 2.123Sh – 17.23 0.81 0.83 T0 = 1.361Is(50) + 1.23 1.203 σc = 0.705Sh 0.77 T0 = 2.038Is(50) 0.835 0.85 σc = 13.76exp(0.032Sh ) 0.79 T0 = 2.403Vp – 2.15 0.67 σc = 102.6exp(-0.085neff) 0.74 T0 = 0.856exp(0.459Vp) 0.73 σc = 98.37neff -0.551 0.82 T0 = 0.647Vp 1.629 0.77 σc = 126.8exp(-0.071ntot) 0.74 Is(50) = 6.286exp(-0.056ntot) 0.64 -0.605 0.75 Is(50) = 8.428ntot -0.507 0.63 σc = 154.0ntot 0.76 Is(50) = 0.039exp(1.84γdry) 0.65 σc = 3.50R L – 67.68 0.76 3.67 0.63 σc = 4.97exp(0.058RL) Is(50) = 0.136γ dry 0.67 0.71 2.123 Is(50) = 0.222R L – 3.94 σc = 0.022RL 0.76 0.73 Is(50) = 0.324exp(0.057RL) σc = 29.57Vp – 32.45 0.80 2.123 0.74 Is(50) = 0.0015RL σc = 7.215exp(0.514Vp) 0.66 0.68 Is(50) = 1.567Vp – 1.145 1.741 σc = 5.912Vp 0.66 0.79 Is(50) = 0.613exp(0.449Vp) σc = 58.14KIC + 35.34 0.72 1.261 0.77 Is(50) = 0.751Vp σc = 88.37K 0.69 0.72 0.60 IC RL = 23.44γd ry– 14.68 0.68 0.60 σc = 43.95exp(0.586KIC) RL = 8.33exp(0.65γ dry) 0.78 0.59 Et = 0.309σc RL = 12.54γdry 1.322 0.79 0.77 Et = 0.280σc + 4.05 RL = 0.424Sh + 21.63 1.072 0.84 0.76 Et = 0.204σc RL = 23.2exp(0.011Sh) 0.62 0.79 0.435 Et = 3.326T0 0.79 RL = 7.88Sh 0.62 0.988 Et = 3.20T0 RL = - 0.721ntot + 50.35 0.69 0.63 Et = 39.74exp(-0.079ntot) RL = 50.07exp(-0.02ntot) 0.62 0.59 - 0.616 – 0.154 Et = 47.41ntot RL = 52.8ntot 0.70 0.63 Et = 0.032exp(2.562γdry) 0.68 Vp = 0.078RL + 0.54 0.61 5.382 Et = 0.146γdry 0.62 Vp = 1.16exp(0.025RL) 0.62 0.934 Et = 0.68exp(0.069*RL) 0.62 Vp = 0.109RL 0.75 Et = 0.0012RL 2.515 0.75 Vp = -0.101ntot + 5.122 0.77 Et = 11.65Vp – 16.09 0.77 Vp = 5.176exp(-0.032ntot) 0.71 –0.245 Et = 1.578exp(0.623Vp) 0.83 Vp = 5.459ntot 0.76 2.073 Et = 1.314Vp 0.94 Vp = 2.857γdry – 2.854 0.76 γdry = 2.673 - 0.033neff 0.95 Vp = 0.46exp(0.861γ dry) 0.74 1.79 γdry = 2.715exp(-0.017neff) 0.97 Vp = 0.785γdry 0.82 γdry = 2.751 - 0.029ntot σc: Uniaxial compressive strength (MPa), T0: Brazilian tensile strength (MPa), Et: Elastic tangent modulus (GPa), Is(50): Point load strength index (MPa), γdry: Dry 3 density (gr/cm ), n tot: Total porosity (%), neff: Effective porosity (%), R L: L-type Schmidt rebound hardness, Sh : Shore scleroscope hardness, Vp: P-wave velocity 1/2 (km/s), KIC: Fracture toughness (MPa.m ). Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 7 Figure 3: Dry density vs. a. UCS, b. Brazilian tensile, c. Point load, and d. Young’s modulus. relationship between uniaxial compressive strength (σc) and dry density (γdry) is given as: 2 σc = 0.261exp(2.213γdry) (r =0.637) (4) 3 where σc in MPa and γdry in gr/cm . 3.2. Uniaxial Compressive Strength Uniaxial compressive strength (UCS) is one of the most important mechanical properties of rocks, which is mainly used for the design of engineering structures and characterization of intact rock materials. In rock engineering, the UCS value is typically determined by an unconfined compression test where a cylindrical core sample is loaded axially to failure, with no confinement (lateral support). Conceptually, the peak value of the axial stress is taken as the UCS of the sample. The test requires good quality test specimens Figure 4: Histogram of dry density data. of right circular cylinders having a height to diameter 3 ratio 2.5-3.0 and a diameter of preferably not less than value recorded from pumice is 0.62 gr/cm and the 3 NX core size, approximately 54 mm [11]. highest value from quartzite at 4.04 gr/cm . The 8 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari The rocks in the database exhibit an average UCS purpose, the histograms of uniaxial compressive, of 80.60 MPa while they are extending from the lowest Brazilian tensile and point load strengths, and Young’s of 0.4 MPa for tuff to the highest of 436 MPa for basalt. modulus for all rocks are given in Figure 5 with the According to ISRM [11] classification scheme, the associated descriptive statistics and a hypothetical strength of the rocks covered in the database normal distribution. A simple examination of the corresponds to 3.3% “very weak” (1-5 MPa), 18.0% histograms reveals that the strength values mostly “weak” (5-25 MPa), 19.6% “medium strong” (25-50 distributed in specific ranges and are also fairly skewed MPa), 29.3% “strong” (50-100 MPa), 27.5% “very towards the higher values. This type of pattern is strong” (100-250 MPa), and 2.3% “extremely strong” mostly characterized by a lognormal distribution [14] (>250 MPa). Besides presenting the mean and range the only exception is the Young’s modulus, which of collected data, it seems good to know how the data shows a negative exponential character. is distributed over the complete strength range. For this Figure 5: Histograms of a. UCS, b. BTS, c. PLS and d. Young’s modulus data. Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 9 3.3. Brazilian Tensile Strength and there is a strong correlation (R=0.77). In literature, many different ratios were recommended for UCS and The tensile strength of rocks is one of the least BTS varying from 2 to 50 (Appendix Table A1). Based investigated rock strength characteristics. In part, these on all the data collected in this study it can be formally are due to the use of compressive and shear stresses stated that rocks, in general, tend to have a tensile rather than tensile stresses in the design of rock strength of 1/10 of their compressive strengths. The structures. Rock is relatively weak in tension, and thus, linear equation relating UCS (σc) to Brazilian tensile the tensile strength (T0) of an intact rock is strength (T0) is: considerably less than its compressive value (σc) [9]. The Brazilian tension test, also known as the splitting σc = 10.04T0 2 (r =0.590) (6) tensile test, is widely used to evaluate the tensile strength of rocks, as it is easy to prepare and test where both T0 and σc is in MPa. specimens. Compression-induced extensional fracturing generated in this test is also more Furthermore, tensile strength to compressive representative of the in situ loading conditions and strength ratio is one of the important fundamental failure of rocks. In the Brazilian tension test, a circular properties of rock. Figure 7 shows the frequency disk placed between two platens is loaded in distribution of this ratio for all rocks in the database. compression producing a nearly uniform tensile stress The ratio has a mean of 11.77 with a standard distribution normal to the loaded (vertical) diametric deviation of 5.63 and ranges between 2.02 and 51.9. plane, leading to the failure of the disk by splitting [15]. The overall distribution of UCS/BTS ratio seems to be The equation for Brazilian test is: truly skewed towards the higher values. It can be seen graphically in Figure 8 that the ratio between UCS and Τ0 = 2P/πLD (5) BTS has an average value of 11.77, which slightly differs from the constant found by linear regression as where P is the failure load, and L and D are the length 10.04 in Figure 6. This discrepancy is basically caused and diameter of the disk. by the techniques used in the calculations, the first one is simply the arithmetic mean of all ratios and the second one is the slope of the best fitting linear equation. Figure 6: Relationship between Brazilin tensile and UCS. The next step in the study is to relate the tensile and the uniaxial compressive strengths of the rocks. It is assumed that a fixed ratio exists between the tensile strength and compressive strength of the rocks. The plot of UCS as a function of the Brazilian tensile strength (BTS) is shown in Figure 6. Since, physically, a zero BTS also implies a zero UCS, it is, therefore, required that the best-fit line passes through the origin in the linear regression analysis. The slope of the best fitting line passing through origin is found to be 10.04 Figure 7: Histogram of UCS/BTS ratio data. 10 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari relationship differs considerably from the models found by Broch and Franklin [4], Bieniawski [16], and Cargill and Shakoor [18]. Again, since physically a zero Is(50) also implies a zero UCS, therefore, it is necessary for the best fit line pass through the origin in the linear regression analysis. The following equation allows the estimation of UCS as a function of the point load index for all rocks: 2 σc = 15.15Is(50) (r =0.712) (8) where Is(50) is the point load index of the 50 mm diameter core. In Figure 9, it can be seen that the ratio between Figure 8: Relationship between UCS and point load strength. and has an average value of 18.01, which slightly differs from the constant found by linear regression as 3.4. Point Load Index 15.15 in Equation 8. As noted before, this is due to the different techniques used in the calculations, the first The point load test may be used as an index value is a simple arithmetic mean of all ratios and the property in rock engineering applications where the second value is the slope of linear line passing through true UCS is sought. This is because obtaining point the origin. The distribution of UCS/PLS ratio is slightly load strength data is by far simpler than obtaining skewed towards the right. If the ratios greater than 30 actual uniaxial compression test data, which requires are not taken into account, the histogram in Figure 9 sophisticated testing techniques and strict adherence can be said fairly symmetrical in shape. to sample preparation standards. Therefore, it has become standard practice to rely on published correlations for predicting uniaxial strength from point load data. An index: strength ratio (UCS/Is) has been suggested by Broch and Franklin [4], ISRM [5], and Bieniawski [16] who propose conversion factors (K) ranging from 20 to 25 for intact rocks. However, there are many different constants mentioned in the literature ranging from 2-55 for different rock types (Appendix Table A1). The test can be applied to rock samples with irregular or regular shapes. Three test methods; diametrical, axial, and block are available. For rocks possessing horizontal bedding or foliation, the diametric test is an unreliable indicator of the rock strength and axial testing perpendicular to the bedding is required to give a consistent rock strength index [17]. The equation for the diametrical test is: 2 Is = P/D (7) where P is the failure load and D is the diameter of the core sample. Figure 9: Histogram of UCS/PLS ratio data. It is obviously evident from Figure 8 that all data Figure 10 presents the relationship between collected from uniaxial compressive strength and point Brazilian and point load tests, which are considered as load strength testing indicate a strong correlation two kinds of indirect tensile strength of the rocks. There between these parameters, although the proposed Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 11 is a common tendency in all XY-graphs correlating rock a prescribed energy level and impacts a mass against strengths up to now that the higher the strength values the plunger. The distance of rebound of the mass is the more scattered the data points and it is also valid measured on a scale and is taken as a measure of for this case. When the histograms of the strength hardness [19]. Therefore, the harder the surface, the properties of rocks are considered, they are typically higher the rebound distances. The Schmidt hammer skewed towards the right suggesting a lognormal models are designed with different levels of impact distribution. If two log-normally distributed variables in energy, but the types L and N are more commonly scatter plots are compared the result will be a adopted for the testing of rock and concrete with impact scattering pattern around the best-fit line at larger energy levels of 0.735 and 2.207 Nm, respectively [11]. values. The linear equation relating Brazilian tensile strength (T0) to point load strength (Is(50)) is as follows: The significant correlations have been found between the rebound values of two models in field 2 T0 =1.517Is(50) (r =0.670) (9) applications [6] and it has been also stated by Buyuksagis and Goktan [1] and Aydin and Basu [7] that the correlations found between rebound values and UCS of rocks by using the N-type hammer are consistently higher than those of the L-type. In order to overcome the discrepancy between readings of the two models and harmonize the data for a single Schmidt rebound number, a new correction factor has been proposed specifically for this study which is presented in Figure 11. According to the data collected from [1], [7], [20], the following linear relationship is typically acquired between N-type and L-type Schmidt hammer rebound numbers: 2 RL = 0.84RN (r =0.945) (10) Figure 10: Relationship between point load and Brazilian tensile strengths. The constant of the linear equation is somewhat interesting. The only difference between Brazilian tensile strength (Eq. 5) and point load index (Eq. 7) equations will the factor of 2/π in the former if it is provided that an equal dimension is used in the both experiments. The inverse of this factor is 1.57, which is very close to the constant of linear equation. Actually, the both tests are an indirect measurement of tensile strength of the rocks. 3.5. Schmidt Rebound Hardness Figure 11: Relationship between N-type and L-type Schmidt The Schmidt hammer is one of the widely used rebound hardness. portable instruments for estimating rock strength indirectly. In the civil engineering and mining industries, There is a strong non-linear relationship between it is used for non-destructive testing of the quality of the SHV and the UCS, BTS, PLS and Young’s modulus concrete and rock, both in the laboratory and in the of rocks as shown in Figure 12. The non-linear field. It measures the surface rebound hardness of the equation exhibited between UCS and Schmidt rebound tested material. The plunger of the hammer is placed hardness can be written as: against the specimen and the specimen is depressed 2 by pushing the hammer against the specimen. Energy σc = 4.969exp(0.058RL) (r =0.575) (11) is stored in a spring, which is automatically released at 12 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari Figure 12: SHV vs. a. UCS, b. Brazilian tensile, c. Point load, and d. Young’s modulus. where σc is the uniaxial compressive strength in MPa The distribution (Figure 13) of the Schmidt hammer and RL the L-type Schmidt rebound value. values obtained from the collected data shows a noteworthy difference from the other property histograms. Most of the distributions evaluated in this study are usually skewed towards the larger values; the only exception is the distribution of dry density, which is skewed towards the smaller values. SHV histogram, however, shows a normal distribution. This may be attributed to the suggested methods for Schmidt rebound recording [11, 19]. In standard tests it is recommended to average certain readings whilst disregarding outliers. This may lead to this specific type of distribution. It is well established in statistics that the distribution of samples produced by averaging certain values can hypothetically produce bell-shaped (normal) distributions in accordance with the Central Limit Theorem whatever the distribution of the parent population from which the samples are drawn. Theoretically, if the parent population is normal, or the sample size is large (often n=10 or 20 will be large enough), then in either case the sampling distribution of average values has an approximately normal shape [21]. Also, there is a natural threshold at lower value (0) in SHV readings and it is limited on higher values. On Figure 13: Histogram of Schmidt rebound hardness data. Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 13 the other hand, for the rest of the tests evaluated in this The velocities of the P and S waves are calculated work, there is only a natural lower bound at 0; however, from the measured travel time and the distance it is unlikely to extent an upper bound, which, between transmitter and receiver. In order to measure theoretically, goes to infinity. a SV index value, the Pundit testing machine is generally used. The Pundit has a pulse generator, 3.6. P-Wave Velocity transducers, and an electronic counter for time internal measurements. Ultrasonic techniques are non-destructive and easy to apply, both at site and laboratory conditions. The There are statistically important correlations sound velocity of a rock mass is closely related to the between P-wave velocity and both uniaxial intact rock properties. The P-wave velocity, as a natural compressive, Brazilian tensile, point load strengths and characteristic of rocks and different materials, depends modulus of elasticity. The type of relationship obeys the on their micro and macro structure, the existence of law of either exponential or power as can be seen in minor cracks, porosity and the characteristics of their Figure 14. The relationship between p-wave velocity mineralogical components, such as elastic parameters, (Vp) and UCS (σc) can be formulated as: density and micro-porosity [22]. In rock engineering, 1.741 2 sound velocity (SV) techniques have increasingly been σc = 5.912Vp (r =0.645) (12) used to determine the dynamic properties of rocks [23]. The SV testing method determines the velocity of where Vp in km/s and σc in MPa. propagation of elastic waves in laboratory conditions. 3.7. Effect of Water on Strength ISRM [11] describes three methods, the high and low frequency ultrasonic pulse techniques, and the It is well known that moisture content may influence resonant method. The velocities of longitudinal waves the mechanical properties of rocks. Even igneous rocks were determined using the pulse transmission method. are affected by the amount of water content. The work Figure 14: P-wave velocity vs. a. UCS, b. Brazilian tensile, c. Point load, and d. Young’s modulus. 14 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari of many authors has conclusively shown that moisture modulus can be formulated in terms of their dry has a significant effect on the strength of rocks [24-26] equivalents, as follows: finding varying degrees of reduction in compressive 2 strength, ranging from 6 to 85%, with increasing σcwet = 0.757σcdry (r =0.904) (13) moisture content. Colback and Wiid [24] provided an 2 T0wet = 0.789T0dry (r =0.911) (14) example of the change in the UCS of quartzitic sandstone relative to the moisture content. The UCS of Is(50)wet = 0.823Is(50)dry 2 (r =0.876) (15) the rock decreases as the material becomes saturated. 2 Dyke and Dobereiner [25] indicated a 25-35% loss in Etwet = 0.828Etdry (r =0.888) (16) strength between UCSdry and UCSwet for British sandstones. The presence of liquids, most particularly It is obvious from Figure 16 that most of the rocks water, substantially reduces the strength of the rocks. exhibit a marked decrease in their measured strengths The lower strength was attributed to the decreasing of when tested wet. There are different viewpoints as to the surface free energy of the solid due to physical what causes strength loss in saturated rocks. A adsorption from the surrounding liquid [13]. noteworthy explanation is that the filling of pores inside the rock probably leads to the strength reduction due to pore water pressure. Figure 15: Relationship between dry and saturated samples of UCS. To investigate the effect of moisture content on the strength of all rocks, first, the influence of water content on the UCS (σc) is investigated. The measured strength under saturated conditions is plotted as a function of the strength under dry conditions in Figure 15. It appears that the saturated strength is linearly related to the dry strength and also, the saturation of test samples can promote a considerable loss of compressive strength in the rocks. These reductions in Figure 16: Histogram of % strength loss in saturated samples of UCS. strength vary between –5.56% and 85.9%, the average strength reduction being 34.14% for all rocks (Figure 3.8. Modulus Ratio 16). The minus sign in reduction refers to an increase in saturated strength. The effect of water content on The relationship between compressive strength and some other mechanical properties of rocks can be modulus of elasticity has been discussed by many briefly described in a similar way in the following researchers and they found a significant linear equations and, for practical purposes, the complement correlation between UCS and modulus of elasticity for of constants of linear equations between saturated and different rock types (Appendix Table A1). The dry values can be loosely taken as the average percent relationship between the elastic tangent modulus and reduction in strength. Saturated uniaxial compressive, UCS is graphically shown in Figure 17. It can be easily Brazilian tensile, point load strengths and Young’s seen that the value of the elastic modulus rises with Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 15 increasing UCS. A linear relationship characterizes the elastic tangent modulus to UCS [27] (MR=Et/σc). Also, correlation between the UCS and static modulus of the same figure shows the modulus ratio of rocks in the elasticity of rocks in the database. The following database extending from a minimum of 12.08 for coal formula relates the UCS to elastic Young’s modulus: to a maximum of 2,058 for marlstone. The majority of 2 the data (51.7%) have modulus ratios in the range of Et = 0.309σc (r =0.612) (17) 200 to 500 (medium modulus ratio), 29.2% are less than 200 (low modulus ratio) and 19.1% have a high where Et is the static modulus of elasticity in GPa and modulus ratio (>500) according to the engineering σc is in MPa. classification of intact rock proposed by Deere and Miller [27]. 3.9. Shore Scleroscope Originally designed for use on metals the Shore scleroscope is a non-destructive, hardness-measuring device. In this test, a diamond tipped indenter drops freely from a fixed height onto the surface of a specimen. The height of rebound indicates relative values of hardness (Shore hardness index, SHI), which may be correlated to the material strength. It is measured on a calibrated scale ranging from 0 to 140. The disadvantages of this test are that a large number of tests are required to give a good measure of the Figure 17: Relationship between UCS and Young’s modulus. average hardness [28] and the measured hardness is sensitive to roughness of the specimen surface being tested [11]. Wuerker [29] showed plots of Shore hardness values against the UCS of more than 100 rock groups. Deere and Miller [27] published extensive studies on the relation between the Shore hardness and compressive strength of 28 different rocks, using the C-2 Shore scleroscope model. A linear relationship is observed between Shore scleroscope hardness and both UCS and Schmidt rebound hardness as shown in Figure 19. The equations that relate UCS and SHV in terms of SHI are given as: 2 σc = 2.11Sh – 16.23 (r =0.626) (18) 2 RL = 0.424Sh + 21.73 (r =0.592) (19) where σc is uniaxial compressive strength in MPa, RL and Sh are the Schmidt rebound and Shore scleroscope values, respectively. 3.10. Fracture Toughness Figure 18: Histogram of modulus ratio data. Fracture toughness, KC, is the resistance of a The constant of the equation, if multiplied by a material to failure from fracture starting from a factor of one thousand for unit conversion, is very close preexisting crack. In the case of testing on fracture to the average modulus ratio (MR) of 347.1 given in toughness (mode I) two methods exist, the Chevron Figure 18. The modulus ratio is defined as the ratio of bend specimen and the short rod specimen, 16 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari Figure 19: Shore scleroscope hardness vs. a. UCS, and b. Schmidt rebound hardness. 2 respectively. Both methods allow two levels of testing T0 = 6.777KIC (r =0.597) (20) Level 1 requires only the recording of maximum load 1/2 during bending and is supposed to be suitable for where KIC is in MPa.m and T0 is in MPa. screening purposes. Level 2 testing needs load and A collection of the best-fitting equations for the displacement measurements requiring more properties of rocks in the database is given in detail in sophisticated testing apparatus. Level 1 testing on Table 3. It includes all of the selected equations that fracture toughness KIC may serve to obtain index are found to be significant whether they have been values for intact rock with respect to its resistance to mentioned directly in the text. The correlation crack propagation [30]. In recent years there is a coefficient (R) measures the extent to which two growing research into the measurement of the fracture variables are related to each other. A quick evaluation toughness of intact rock. As such, fracture toughness of Table 3 indicates that for all rock types in the values for rock do not exist in a large extent in the database moderate to strong correlations (0.59 to 0.97) database for the comparison with the other tests. are found between different engineering properties of However, the practical usefulness of this test has been intact rocks. suggested by Gunsallus and Kulhawy [31]. CONCLUSIONS A linear relationship is found between fracture toughness and Brazilian tensile strength as shown in An extensive review of rock mechanics literature reveals that many studies are performed to investigate Figure 20. The following equation gives the estimation the relationships between rock strength, deformation of Brazilian tensile strength (T0) as a function of the and rock hardness. Many simple and complex fracture toughness (KIC): empirical models are increasingly proposed from these studies. In the current research a comprehensive database was accomplished to provide the basis for the detailed analysis of rocks at different origin and it is aimed to gain insight on existing studies looking for the relationships between rock physical and mechanical properties. Based on the results of the regression analysis and frequency histograms of rock properties evaluated in the present study the following conclusions can be derived: • It was seen in the frequency histograms that most of the rock properties and ratios tend to Figure 20: Relationship between fracture toughness and skew in the direction of higher values except for Brazilian tensile strength. dry density and Schmidt rebound hardness. Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 17 • There is a common tendency in all XY-plots • The current work clearly confirms that although relating rock strength properties that the higher the limited test data for a specific rock type is the values the more scattered the data points. more likely to produce good correlations, it is still This is typically due to correlating two log- possible to obtain some significant relationships normally distributed variables, which generally for different kinds of rocks. produce this type of pattern [32]. • The relevant relationships are considered to be • The constant of 10.04 found in linear equation the best suited for the prediction of engineering relating UCS to Brazilian tensile strength properties of all rock types since the data statistically validates the customary statement included in the analyses cover a wide range of “generally rocks have tension strength of one property and rock lithology. tenth of their compression strength” [9]. • A final remark can be pointed out here that the • The problem in establishing a single factor data are scattered in the most of the plots. A relating Is(50) and UCS values is highlighted. The possible reason for the scattering is that the test constant 15.15 found from the linear equation values in the plots were collected from a wide and the average ratio of 18.01 between the UCS range of scientific sources covering different rock and the point load strength for all rocks are both units and testing procedures. On the other hand, comparatively less than the 24 quoted by Broch if some data could be re-plotted for the similar and Franklin [4]. rock classes or rock groups, this procedure may be increase the reliability of relevant correlations. • The results show that moisture substantially The relationships between the properties of the reduces the strengths of rocks. The wetting of same rock type and similar rock units should be the rock samples causes an average unconfined investigated in the future. compressive strength loss of 34%. APPENDIX Table A1: A Supplementary List of the Empirical Equations Proposed by Various Authors Reference Equation R Rock unit Wuerker [29] σc = 2.76Sh More than 100 rock groups σc = 5…22T0 Fairhurst [15] σc = 11.5T0 Hobbs [33] T0 = 0.25σc + 4.6 0.88 Massive and laminated rocks σc = 2.84T0 – 3.34 D’Andrea et al. [34] σc = 15.3Is(25) + 16.3 0.95 49 lithologic units Deere & Miller [27] σc = 9.97exp (0.02RL*γ dry ) 0.94 28 different lithology σc = 28.75exp(0.009Sh*γ dry ) 0.92 σc = 31.19γ dry – 36.27 0.60 σc = 3.54Sh - 42.85 0.90 σc = 8.59RL - 240.62 0.88 σc = 20.7Is (5 4 ) + 29.6 0.92 2 Et = 0.19RL*γ dry - 7.87 0.88 Et = 0.042γ dry*Sh + 12.62 0.80 Et = 0.094γ dry*RL - 20.28 0.85 Et = 0.74Sh + 11.52 0.75 Et = 1.786RL – 29.59 0.73 Smorodinov et al. [35] σc = 0.0864exp(0.291γdry) Carbonate rocks σc = 254exp(-0.091neff) Broch & Franklin [4] σc = 23.7Is(50) 0.88 15 different rocks Szlavin [36] σc = 20NCB + 12.4 0.88 σc = 2.1Sh – 35.2 0.84 18 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari σc = 3.6T0 + 15.2 0.76 T0 = 0.37Sh – 3.9 0.81 T0 = 0.16σc + 4.4 0.76 T0 = 3.1NCB + 5.8 0.73 Bieniawski [16] σc = 24Is(54) Sandstone, quartzite, norite 3. 4 7 Dearman & Irfan [37] σc = 0.00016RL 0.86 Granite Et = 1.89RL - 60.55 0.93 Hassani et al. [38] σc = 29Is(50) 0.94 Limestone, siltstone , sandstone σc = 10.5T0 + 1.2 0.85 Kidybinski [39] σc =0.477exp(0.045RN *γ dry ) 0.82 Coal, shale, mudstone, sandstone Singh et al. [40] σc = 2.00RL 0.72 Sandst., siltst., mudst., seatearth T0 = 0.23RL – 0.81 0.72 Sheorey et al. [41] σc =0.40RN - 3.60 0.94 Coal Gunsallus & Kulhawy [31] σc = 16.5Is(50) + 51.0 0.69 Sandstone, limestone, dolostone σc = 12.4T0 – 9.0 0.76 KIC = 0.0044σc + 1.04 0.72 KIC = 0.0736T0 + 0.76 0.73 KIC = 0.0995Is(50) + 1.11 0.67 Huang & Wang [42] KIC = 0.65Vp – 1.68 0.90 Haramy & DeMarco [43] σc =0.994RL - 0.383 0.70 Coal 1.3 3 σc = 0.287RL 0.85 Ghose & Chakraborti [44] σc = 0.88RL - 12.11 0.87 Coal T0 = 0.06RL – 0.92 0.81 1. 5 6 0 Van Heerden [23] Et = 0.075E d 0.98 10 different rocks Singh & Eksi [45] σc = 23.31Is ( 5 0 ) 0.95 Gypsum, marlstone σc = -1.14 + 27.2NCB 0.94 σc = 2.5Sh 0.94 Is(50) = 1.1NCB 0.94 Sh = 10.2NCB 0.95 Vallejo et al. [46] σc = 12.5Is (5 0 ) 0.62 Shale σc = 17.4Is (5 0 ) 0.38 Sandstone O’Rourke [47] σc = 4.85RL - 76.18 0.77 5 different rocks σc = 21.8Is(50) + 43.2 0.77 Ojo & Brook [13] σc = 3.54(Sh – 12) Sandstone, mudstone Xu et al. [48] σc = 2.98exp(0.06RL) 0.95 Mica-schist Et = 1.77exp(0.07RL) 0.96 σc = 2.99exp(0.06RL) 0.91 Prasinite Et = 2.71exp (0.04RL) 0.91 σc = 2.98exp(0.063RL) 0.94 Serpentinite Et = 2.57exp(0.03RL) 0.88 σc = 3.78exp(0.05RL) 0.93 Gabro Et = 1.75exp(0.05RL) 0.95 σc =1.26exp(0.52RL*γ dry ) 0.92 Mudstone Et = 0.07exp(0.31RL*γ dry ) 0.89 Cargill & Shakoor [18] σc = 3.32exp(0.043RL*γ dry ) 0.93 Sandstone σc = 18.17exp(0.018RL*γ dry ) 0.98 Carbonates σc = 23.0Is (5 0 ) + 13 0.94 13 lithologic units Sachpazis [49] σc = 4.29RL - 67.52 0.96 33 different carbonates Et =1.94RL - 33.93 0.88 Ghosh & Srivastava [50] σc = 16.0Is(50) 0.75 Granitic rocks Christaras [51] γ dry = 0.75 + 0.30V p 0.92 Marly limostone Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 19 σc = 6.202exp(0.48Vp) 0.97 Grasso et al. [52] σc = 9.30Is(50) + 20.04 0.71 Mudstone T0 = 1.53Is(50) – 0.21 0.89 0.57 σc = 25.67Is(50) 0.73 T0 = 1.01exp(0.47Is(50)) 0.93 σc = 9.68exp(0.045RL) 0.75 Et = 1.28exp(0.033RL) 0.72 σc = 2.83exp(1.14Vp) 0.64 Et = 0.29exp(1.08Vp) 0.80 Whittaker et al. [53] T0 = 9.35KIC – 2.53 0.77 KIC = 0.708 + 0.006σc KIC = 0.27 + 0.107T0 KIC = 0.336 + 0.026Et 2 Vernik et al. [54] σc = 254(1- 0.027ntot) 0.96 Arenite Singh & Singh [55] σc = 23.37Is(50) 0.98 Quartzite 0.983 Arioglu & Tokgoz [56] T0 = 0.081σc 0.85 20 different rocks Ulusay et al. [57] σc = 19.5Is(50) + 12.7 2.4 6 Kahraman [58] σc = 0.00045(RN*γ dry ) 0.96 10 different lithology 3.2 7 Gokceoglu [59] σc = 0.0001RL 0.84 Marl Chau & Wong [60] σc = 12.5Is(50) 0.73 Granite, tuff Zhixi et al. [61] KIC = -0.332 + 0.361Vp 0.96 Sandstone KIC = 0.054Vp + 0.388 0.75 Shale -0.47 Karpuz & Pasamehmetoglu Vp = 6.05neff 0.95 Andesite [62] Vp = 6.03 – 0.194neff 0.90 neff = 23.2exp(-0.04R L) 0.96 neff = 25.2 – 0.28RL 0.88 Is(50) = 871neff - 1.48 0.80 Is(50) = 98.8neff - 1.18 0.91 Vp = 4.33 + 1.22In(σc) 0.91 Vp = 0.48 + 0.069RL 0.95 2.18 Is(50) = 0.0465Vp 0.94 0.47 RL = 9.51Is(50) 0.97 RL = 17.26In(σc) - 67.25 0.94 Brown & Reddish [63] KIC = 3.21γ dry - 6.95 0.95 17 different rocks KIC = 3.35γ dry - 6.87 0.84 Holmgeirsdottir & Thomas σc = 4.65Sh – 40.46 15 different rocks [64] σc = 3.0Sh – 22.8 0.62 Zhang et al. [65] T0 = 8.88KIC 0.97 Tugrul & Zarif [66] σc = 162.9γdry – 362 0.93 Sandstone σc = 140.16exp(-0.19neff) 0.97 Tugrul & Zarif [67] σc = 8.36RL - 416 0.87 Granitic rocks γ dry = 2.644 - 0.025ntot 0.86 V p = 6.52 - 0.36ntot 0.81 σc = 577.2γ dry – 1347 0.82 σc = 35.54V p – 55 0.80 σc = 201 - 78.22neff 0.81 σc = 183 - 16.55ntot 0.83 σc = 15.25Is ( 5 0 ) 0.98 T0 = 0.15σc – 0.73 0.92 Et = 0.35σc – 12 0.94 Koncagul & Santi [68] σc = 0.895Sh + 41.98 0.57 Shale 20 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari σc = 0.658I d 2 + 9.081 0.63 Sh = 0.37I d 2 – 5.23 0.56 Starzec [69] γdry = 0.2Vp + 1.73 0.74 Crystalline rocks Et = 0.48Ed – 3.26 0.91 Bearman [30] KIC = 0.209Is(50) 0.95 12 different rocks Katz et al. [70] σc = 2.208exp(0.067RN) 0.96 Limestone, sandstone 3.0 9 Et = 0.00013RN 0.99 Syenite, granite γdry = 1.308In (RN) -2.874 0.96 Altindag [71] KIC = -0.221+ 0.003σc 0.96 Marble, limest., sandst., andesite KIC = -0.957 + 0.281T0 0.90 KIC = -0.916 + 0.163Et 0.81 KIC = 0.632 + 0.325Is(50) 0.70 KIC = -0.820 + 4.731log (NCB) 0.75 0.98 Gupta & Rao [12] Et = 0.286σc 0.93 Igneous rocks 1.91 Et = 0.080σc 0.77 Sedimentary rocks 1. 11 Et = 0.150σc 0.91 All rocks Tugrul & Zarif [72] γdry = 2.70 – 0.033ntot 0.88 Limestone ntot = 0.62Vp+ 5.37 0.78 σc = 538.9γdry – 1309 0.89 σc = 16.73Vp + 21.25 0.94 σc = 144 – 17.29ntot 0.77 σc = 14.38Is(50) + 42 0.92 T0 = 0.56σc – 15 0.90 Et = 0.512σc – 20.41 0.90 Sulukcu & Ulusay [73] σc = 15.31Is(50) 0.83 23 different rocks T0 = 2.30Is(50) 0.80 Kahraman [74] σc = 6.97exp(0.014RN *γdry) 0.78 Carbonates 1. 2 1 σc = 9.95V p 0.83 σc = 23.6Is (5 0 ) - 2.69 0.93 Coal measure rocks σc = 8.41Is (5 0 ) + 9.51 0.85 Other rocks Chaterrejee & Mukhopadhyay σc = 55.57γ dry – 100.75 0.94 Sandstone, siltstone, limest., shale 0.89 [75] σc = 10.33T0 0.97 σc = 64.23exp(-0.085neff) 0.96 Et = 0.73σc + 0.17 0.96 Yilmaz & Sendir [76] σc = 2.27exp(0.059RL) 0.98 Gypsum Et = 3.15exp(0.054RL) 0.91 Zhang [77] T0 = 6.88KIC 0.97 Lashkaripour [78] σc = 21.43Is(50) 0.93 Mudrock -0.821 σc = 210.12ntot 0.82 1.086 Et = 0.103σc 0.90 -0.863 Et = 37.9ntot 0.83 Vasarhelyi [79] Et = 0.178σc 0.86 Sandstone σcsat = 0.759σcdry 0.95 Quane & Russel [80] σc = 24.4Is(50) Strong rocks 2 σc = 3.86Is(50) + 5.65Is(50) Weak rocks Alber & Brardt [81] KIC = 0.0654exp(0.681Vp) 0.94 KIC = 0.015exp(1.74γ dry ) 0.87 Hudyama et al. [82] σc =-49.36In(ntot) + 189.35 0.79 Tuff Tugrul [83] σc =195exp(-0.21ntot) 0.89 Sandst., basalt, limest., granodior. σc =125exp(-0.20neff) 0.89 Investigating Relationships between Engineering Properties of Various Rock Global Journal of Earth Science and Engineering, 2018, Vol. 5 21 γdry = 2.713 - 0.033ntot 0.97 γdry = 2.684 – 0.151log(neff) 0.92 ntot = 4.36log(neff) + 1.17 0.91 Yasar & Erdogan [22] γdry = 1.1623In(Sh ) -2.093 0.89 Limest., sandst., marble, basalt γd ry = 0.9377ln(RL) - 1.03 0.92 RN = 56.883ln(Sh ) -181.38 0.91 neff = -0.2RL + 11.21 0.89 neff = -9.06ln(Sh ) + 38.042 0.64 -8 5. 5 5 5 σc = 1x10 S h 0.91 -6 4. 2 9 2 σc = 4x10 RL 0.89 Jeng et al. [84] σc = 133.7exp(-0.107ntot) 0.89 Sandstone 1.71 Tsiambaos & Sabatakakis [85] σc = 7.3Is(50) 0.91 Sedimentary rocks σc = 23.0Is(50) 0.87 Yasar & Erdogan [86] Vp = 0.032σc + 2.02 0.89 Carbonate rocks Vp = 0.094Et + 1.75 0.93 Vp = 4.32γ dry – 7.51 0.90 Palchik & Hatzor [87] Is(50) = 7.74exp(-0.039ntot) 0.92 Chalk σc = 273.2exp(-0.076ntot) 0.81 σc = 8…18Is(50) Dincer et al. [88] σc = 2.75R L – 36.83 0.97 Andesite, basalt, tuff Et = 0.47R L – 6.25 0.92 Et = 0.17σc + 0.28 0.92 Basarir et al. [89] σc = 10.96Is(50) 0.79 Dacite 0.69 σc = 4.72R N 0.81 2.69 σc = 0.68γ dry*Vp 0.81 Aydin & Basu [7] σc =1.45exp(0.07RL) 0.92 Granite Et =1.04exp(0.06RL) 0.91 σc = 0.92exp(0.07RN) 0.94 Et = 0.72exp(0.05RN) 0.92 ntot = -0.43RL + 30.4 0.89 neff = -0.32RL + 21.15 0.90 γ dry = 0.01RL + 2.00 0.92 Kahraman et al. [90] σc = 10.91Is(50) + 27.41 0.78 38 different rocks Vasarhelyi [91] γ dry = -0.0268ntot + 2.71 0.99 Limestones T0 = 0.129σc 0.86 σc = 0.056exp(2.75γ dry ) 0.80 Fener et al. [92] σc = 9.81Is(50) + 39.32 0.85 11 different rocks σc = 4.24exp(0.059RN) 0.81 -0.56 Sousa et al. [93] σc =124.28neff 0.81 Granite -0.42 Vp=4.083neff 0.89 1.247 σc = 4.0Vp 0.85 Kahraman & Alber [94] σc = 17.91Is(50) + 7.93 0.89 Fault breccia Aydin & Basu [95] T0 = 0.00004exp(4.6γ dry ) 0.83 Igneous rocks T0 = 8.3exp(-0.14neff) 0.83 T0 = 10.74exp(-0.111ntot) 0.86 Kolay & Kayabali [96] γ dry = 0.239Is(50) + 1.535 0.69 Coarse grained rocks Id2 = 29.0exp(0.412γ dry ) 0.75 Id2 = 10.48Is(50) + 44.5 0.80 -2.05 Palchik [97] σc = 7164ntot 0.99 Sandy shale c = 0.55exp(0.088σc) 0.97 -1.1 c = 43.9ntot 0.92 22 Global Journal of Earth Science and Engineering, 2018, Vol. 5 Mehmet Sari φ = 93.53 -1.24ntot 0.99 Buyuksagis & Goktan [1] σc = 2.482exp(0.073RL) 0.94 Granite, marble, limest. travertine Shalabi et al. [98] σc = 3.201RL - 45.6 0.76 Dolostone, limestone σc = 3.326Sh -79.76 0.80 Et = 0.971Sh -26.907 0.92 Shale σc = 1.581Sh - 62.2 0.85 σc = 73γ dry - 110.32 0.62 Et = 0.531σc + 9.57 0.84 Sharma & Singh [99] σc = 0.0642Vp – 117.99 0.90 Seven rock types Id2 = 0.0069Vp + 78.577 0.78 ISI = 0.0118Vp + 58.105 0.81 2.584 Yagiz [100] σc = 0.0028RL 0.92 Travertine, limestone, schist Et = 1.233RL -17.8 0.85 0.562 Vp = 0.537R L 0.77 0. 3 4 8 γ dry = 6.434RL 0.78 neff = 344.3exp(-0.115RL) 0.71 Mishra & Basu [101] σc = 5BPI 0.93 Granite, schist, sandstone σc = 14.63Is(50) 0.94 σc = 2.38exp(0.065RL) 0.93 Fereidooni [102] Id2 = 12.89γdry + 62.61 0.95 Hornfels -0.2 3 RL = 48.48 neff 0.96 -6 3.80 T0 = 3.5x10 RL 0.96 T0 = 2.28Is(50) – 4.66 0.97 σc = 24.36Is(50) – 2.14 0.99 2.28 σc = 0.02R L 0.96 σc = 10.03T0 + 55.19 0.96 Hebib et al. 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Engng 2006; 39: 77-85. https://doi.org/10.1007/s12517-017-3144-4 https://doi.org/10.1007/s00603-005-0069-0 Received on 21-02-2018 Accepted on 17-04-2018 Published on 30-04-2018 DOI: http://dx.doi.org/10.15377/2409-5710.2018.05.1 © 2018 Mehmet Sari; Avanti Publishers. This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.