THE EFFECT OF SHAPE AND SIZE FOR MARBLE ROCK UNITS UNDER UNIAXIAL COMPRESSION

. To determine UCS values of marbles, cube samples sized 7 cm have been prepared by researches. But it was determined that the UCS values found by using dimensions suggested by ISRM and ASTM were different from each other for core and cube samples. In this study, the core and cube samples different in four marble units were prepared in the laboratory. While the l/d ratio for core samples is selected as 2.5-3, it was decided that the dimensions for cube samples were selected as 50, 70 and 100 mm. 20 core samples and 66 cube samples were tested under uniaxial compression by hydraulic press with servo control which has 3000 kN capacity. A data-base was formed by the uniaxial compressive strength and uniaxial deformability test results. Using the data base, the UCS values of the core and cube samples were compared by graphs. As a result, size and shape effect on uniaxial compression of marbles were discussed and their interpretations were presented in this paper. Keywords: Uniaxial compressive strength (UCS), marble, size effect, shape effect.

INTRODUCTION The natural building stones which have been used for different aims by human being throughout history are continuing to be the main material of modern structures today in addition to testifying to the history. Determination of the physical, mechanical and petrographical properties of these natural stones, which are used widely today in road, pavement, inside and outside of a place, plays an important role in selection of appropriate places for using aims besides of the economical aspect. The distribution of minerals inside the rock is heterogeneous, that is, the mineral distribution in every part of the rock is not the same. According to the Cunha (1993), the difference of the compressive strength of same sized and shaped samples taken from the same rock results from this heterogeneous structure of rock. The factors that increase the heterogeneity of rock are; increase of different mineral components in rock, the great difference in change of characteristics between mineral components, being very different of component sizes and showing highly unsystematic distribution of minerals in the volume (Yavuz, 2001). Also, there exist discontinuities starting with small fissures and then extending as far as big sized fractures and faults in or between mineral components of rocks existing in nature. The strength values of heterogeneous and anisotropic rock units will differentiate depending on the scale. Hoek and Brown (1980) emphasized the influence of the diameter of a specimen on the measured strength of the material. Figure 1 has been improved by the authors which is a combination of a number of strength measurements taken from the literature for different rock types. They report: “These data have been reduced to dimensionless form by dividing individual strength values by the strength of a specimen 50 mm in diameter, the average size of a laboratory specimen. This process not only makes it possible to compare the experimental results but it also eliminates differences due to variation in moisture content, specimen shape, loading rate etc. since these factors are generally the same for a given data set.” Figure 1 proves the sample strength increases as the sample diameter decreases. Hoek and Brown (1980) proposed the formula 1 to convert results measured on different diameter specimens to the strength of a 50-mm diameter sample, based on the “best-fit” line:

where σc50 is the calculated uniaxial compressive strength of a 50 mm diameter sample, σc is the uniaxial compressive strength measured on the specimen, d is the diameter of the specimen in mm. Figure 1. Relationship between UCS strength and height/diameter ratio plotted as dimensionless values (Hoek and Brown 1980). Kogure (2005), has performed uniaxial compressive strength (UCS) tests to explain the effect of specimen size and rock properties on UCS of limestone rock units. Specimens were cubical shaped but different size, having edges of d (1cm, 2.5cm, 5cm,

5cm, and 10cm) and height of 2d. He found that “The UCS has a clear scale effect, decreasing with increasing size of the specimen; the UCS increases with increasing density of the specimen and the UCS increases with increasing secant Young's modulus.” Author determined that the UCS of limestone rock units are strongly influenced by specimen size and rock properties. However, Hawkins (1998) proposed that the diagram and formula commended by Hoek and Brown (1980) (Figure 1) for equating UCS values obtained on different diameter specimens are not applicable for all rocks. According to the author, the assumptions on which their figure is based are questioned. Author declares that the results from testing of sedimentary rock cores of

5–150 mm diameter do not support the Hoek and Brown (1980) best-fit line or formula but he has indicated that a maximum strength is obtained on sample between 38–54 mm in diameter, with a decrease in strength noted for both smaller and larger cores (Figure 2). Figure 2. UCS strengths of seven sedimentary rocks tested on samples at eight different diameters (Hawkins, 1998). Another researcher, Jeremic (1994), who studied shape and size effect on compressive strength, conducted experiments for determination of failure strength in salt rock and coal. He has emphasized that size and shape effects of salt rock and coal material were fairly important. He also stated that strength increased with the increasing size of salt sample and compressive strength stayed almost constant after critical size was reached (Figure 3). According to author, the UCS behaviors based on size of coal materials are similar the best-fit curve proposed by Hoek and Brown (1980)(Figure 1). Figure 3. Size effect on strength of the salt rock and coal materials (Jeremic 1994). A similar study was done by Ozkan and Duzyol (2004) on salt rock samples. Size and shape effect was investigated in the study and a clear strength difference between cubic and core samples were detected by the authors. They have developed the following equation for Çankırısalt rock by considering the size effect as a reason of this difference. [σc]54=0,58[σc]cube [2] Also, the authors have determined the critical cube size as 6 cm for the Cankırı- Turkiye salt rock in the concept of examination of size effect on compressive strength. (Figure 4). 0 5 10 15 20 25 30 35 40

2 4 6 8 10 12 14 16 UniaxialCom pressive Strength,UCS (M Pa) Sample Size, a (cm) UCS=33,5777*(1-exp(-Size)) R2=99,92 Critic Size= 6cm Figure 4. Determination of critical cube size for Tekel Cankiri-Turkiye salt rock (Ozkan and Duzyol, 2004). There exist ISRM and ASTM standards for the determination of above mentioned mechanical properties. According to the ISRM, it is suggested that uniaxial compressive strength tests should be done on by preference 10 core samples with 54 mm diameter while according to the ASTM, 5 cubic samples with 70 mm edge size should be used in these tests. The UCS tests in marble researches are generally conducted on cubic samples. Buyuksagis and Gurcan (2005) concentrated on that taking the size of samples as big as possible would give nearly real results when cavities, fracture and stratification properties of especially natural stones was taken into account. Also, the authors emphasized the importance of conducting two series tests in parallel and perpendicular directions to the layer direction in layered rocks. In their study Yavuz et.al.(2001) prepared cubic samples with 25 to 100 mm edge length and cylindrically shaped samples with 54 mm diameter in NX standards from two natural building stones. They tested them to investigate the effect of sample size on uniaxial compressive strength test. As a result, they determined that compressive strength of building stones were decreased in a great deal with the increase in sample size. The authors developed equations representing the change of uniaxial compressive strength with size by using statistical approaches. They also investigated the shape effect on uniaxial compressive strength by comparing obtained uniaxial compressive strength values of core and cubical samples. Besides of this, they reached results that support the study of Bieniawski (1968). According to Bieniawski (1968) compressive strength of a core sample with a diameter equal to the edge length of a cubic sample would be accepted to be equal with compressive strength of the cubic sample. In spite of some researchers in different view, when looked generally conducted studies show that uniaxial compressive strength value decreases with increasing sample sizes. In addition, researchers emphasize that shape effect is also an important factor. Especially in marble researches, it is evident that conducting evaluation by considering shape effect between cubic and core samples that are used in the determination of compressive strength will produce more reliable results. In this study, uniaxial compressive strength tests were conducted on samples that were prepared in different size and shapes by using four different marble samples. In the direction of obtained values, shape and size effect were investigated in detail and analysis results are presented in this paper.

PREPARATION OF SAMPLES By using the marble blocks obtained from 4 different marble mines in Antalya country, core and cubic samples were prepared in the Mining Engineering Laboratory of Selcuk University. Core samples were prepared according to the ISRM (1981) standard as in NX size having 54 mm diameter and with 2.5 of 1/d rate. Cubic samples on the other side were prepared with 50, 70 and 100 mm edge length. Information about sample sizes and number of samples is given in Table 1. Table 1. Origin, size and number of core and cubic samples used in UCS test. Rock Type Sample Shape Location Diameter, d (mm) Length, l (mm) Edge, a (mm) Number Marble-A Cubic Antalya-Turkiye - - 100 6 Marble-A Cubic Antalya-Turkiye - - 70 6 Marble-A Cubic Antalya-Turkiye - - 50 6 Marble-B Cubic Antalya-Turkiye - - 100 5 Marble-B Cubic Antalya-Turkiye - - 70 7 Marble-B Cubic Antalya-Turkiye - - 50 5 Marble-C Cubic Antalya-Turkiye - - 100 5 Marble-C Cubic Antalya-Turkiye - - 70 6 Marble-C Cubic Antalya-Turkiye - - 50 6 Marble-D Cubic Antalya-Turkiye - - 100 5 Marble-D Cubic Antalya-Turkiye - - 70 6 Marble-D Cubic Antalya-Turkiye - - 50 5 Marble-A Core Antalya-Turkiye 54 135 - 5 Marble-B Core Antalya-Turkiye 54 135 - 5 Marble-C Core Antalya-Turkiye 54 135 - 5 Marble-D Core Antalya-Turkiye 54 135 - 5

UNIAXIAL COMPRESSIVE STRENGTH TESTS The experiments were realized with hydraulic press in the mining engineering laboratory of Selcuk University that has 300 tons loading capacity with a loading speed of 0.7 MPa/sec. The uniaxial compressive strength results obtained after the experiments are given in Table 2 and 3. According to the results in Table 2, it can be seen that strength values decrease with increasing value of size. In addition it is determined from the strength values of both cubic (Table 2) and core (Table 3) samples that Marble D samples have higher strength compared to the other marble samples. Table 2. UCS test values for cubic samples. Rock Type Sample Size, a (mm) UCS, σc (MPa) Standard Deviation (MPa) Marble-A 100x100x100 65.41 10.76 Marble-A 70x70x70 94.85 9.79 Marble-A 50x50x50 106.09 8.59 Marble-B 100x100x100 44.32 4.11 Marble-B 70x70x70 46.59 7.30 Marble-B 50x50x50 51.52 8.42 Marble-C 100x100x100 43.39 11.85 Marble-C 70x70x70 45.83 12.84 Marble-C 50x50x50 47.56 5.41 Marble-D 100x100x100 104.35 12.11 Marble-D 70x70x70 120.49 11.27 Marble-D 50x50x50 121.13 9.85 Table 3. UCS test values for core samples. Rock Type Diameter d (mm) UCS σc (MPa) Standard Deviation (MPa) Marble-A 54 77.74 10.05 Marble-B 54 50.53 5.42 Marble-C 54 37.46 6.32 Marble-D 54 119.20 14.46

SIZE EFFECT ON UNIAXIAL COMPRESSIVE STRENGTH The change of compressive strength of cubic samples prepared from different rock units with respect to sample size is shown in Figure 5-8. It can be seen from the figure that with increasing size of samples, UCS is decreasing. The statistical equation which best represents size dependent change in sample compressive strength was searched and the best equation type which was found with the highest obtained correlation constants was determined to be linear type. [?c] = 149.4-0.826a R? = 0.981 0 10 20 30 40 50 60 70 80 90 100 110 120

10 20 30 40 50 60 70 80 90 100 110 120 UniaxialCom pressiveStrength,? c (M Pa) Sample Size, a (mm) Figure 5.Relation between UCS and size of cubic sample Marble-A. [?c] = 57.65 -0.138a R? = 0.898 0 10 20 30 40 50 60 70 80 90 100 110 120

20 40 60 80 100 120 UniaxialCom pressiveStrength,? c (M Pa) Sample Size, a (mm) Figure 6. Relation between UCS and size of cubic sample Marble-B. [?c] = 51.68-0.083a R? = 1 0 10 20 30 40 50 60 70 80 90 100 110 120

20 40 60 80 100 120 UniaxialCom pressiveStrength,? c (M Pa) Sample Size, a (mm) Figure 7. Relation between UCS and size of cubic sample Marble-C. [?c] = 141.1 -0.351a R? = 0.865 0 10 20 30 40 50 60 70 80 90 100 110 120 130

20 40 60 80 100 120 UniaxialCom pressiveStrength,? c (M Pa) Sample Size, a (mm) Figure 8. Relation between UCS and size of cubic sample Marble-D. The linear equations and correlations obtained as a result of evaluating UCS resulted from experimental studies with respect to the size are given in Figure 5, 6, 7 and 8 respectively. Also by combining the results obtained from four different marble rock units, Figure

is prepared. The mean values of the UCS results of each marble rock units for 100, 70 and 50 mm. sizes are calculated and by using these values, statistical analyses are done. After the analyses conducted with MS-Excel, the following equation for marble rock unit is produced. [σc]= 99.96-0.35a [3] [?c] = 99.96-0.350a R? = 0.979 0 10 20 30 40 50 60 70 80 90

20 40 60 80 100 120 UniaxialCom pressiveStrength,? c (M Pa) Sample Size, a (mm) Figure 9. Relation between UCS and size for all marble units.

SHAPE EFFECT ON UNIAXIAL COMPRESSIVE STRENGTH Besides of cubic samples prepared in four different marble units and three different sizes, twenty NX core samples with 54 mm. diameter are prepared to evaluate shape effect. Uniaxial compressive strength values of these core samples are given in Table 3. In the table, the compressive strength values of cubic samples are compared with the ones of core samples. ASTM proposes that the cubic sample size should be taken as 70 mm but according to the tests conducted on cubic samples it is seen that after a definite value of the sample size, UCS values of marble material don’t continue in a constant behavior (Figure 9). The UCS results of cubic samples with 70 mm edge size proposed by ASTM aren’t compatible with the UCS values of core samples (l/d=2.5) with a diameter 54 mm proposed by ISRM. Unfortunately, this searched systematic hasn’t shown itself for 100 and 50 mm sized cubic samples. It can be said that this difference results from shape effect. If the UCS values of core samples with 54 mm diameter which are frequently used by researchers are taken into consideration, the conversion of UCS values of cubic samples to the UCS values of core samples will provide convenience to the appliers. For the aim of providing this conversion, the UCS values of core samples which ([σc] core) are divided to the UCS values of cubic samples used in this study ([σc] cubic) and the resulted coefficients are compared with cubic size. The mathematical equations reflecting the obtained results from the conducted analyses for each rock unit are presented in the Table 4, respectively. Table 4. Mathematical equations reflecting the obtained results from the conducted analyses for each rock unit. Rock Type Statistical Equation Statistical Correlation Marble-A [σc] core = (0.226+0.009a) [σc] cubic 0.95 Marble-B [σc] core = (0.842+0.003a) [σc] cubic 0.92 Marble-C [σc] core = (0.711+0.001a) [σc] cubic 1.00 Marble-D [σc] core = (0.795+0.003a) [σc] cubic 0.86 Also, the graph showing the results belonging to all marble rock units is presented in Figure 10. The relation which is showing an increasing behavior is; [σc] core = (0.644+0.004a) [σc] cubic [4] where a is cubic size. By the help of this equation, the applier can convert the UCS result tested on cubic sample to the UCS value of a 54 mm core sample. In addition it is seen from the equation 4 that the equal cubic sample size for NX core sample is 90 mm. [σc] core=( 0.644+0.004a)[σc] cubic R? = 0.977 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8

10 20 30 40 50 60 70 80 90 100 110[ σ c ] c o r e / [ σ c ] c u b i c Sample Size, a (mm) Figure 10. Comparing UCS values between cubical and core (NX) marble units.

CONCLUSION In this study, UCS tests have been conducted on NX samples (d=54mm, l/d=2.5) which are suggested by ISRM and frequently used in researches. Also UCS tests have been done on cubic samples (a=50 mm, 70 mm and 100 mm) which are frequently used in marble researches. It has been seen that the experimental results obtained from two different sample types were not confirming with each other because of the shape effect. It has been determined that with the increase in size of cubic samples, UCS values were decreasing in a linear behavior (Figure 5-9). Because it was thought that converting UCS values of cubic samples to the UCS values of core (NX) sample type suggested by ISRM would be important in that the applier engineers could use the same language, the following equation has been proposed; [σc] core = (0.644+0.004a) [σc] cubic With the aid of this equation, the authors have proposed that if tests would be done on cubic samples, the sample size of cubic sample should be 90 mm. ACKNOWLEDGEMENTS This study was supported by The Research Foundation of Selcuk University under Project No. BAP-07701061. REFERENCES

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