Relationships between Volumetric Block Proportions and Overall

SÖNMEZ ET AL.: RELATIONSHIPS BETWEEN VOLUMETRIC BLOCK PROPORTIONS AND OVERALL UCS OF A VOLCANIC BIMROCK

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Block Proportions and OverallUCS of a Volcanic BimrockBy Harun Sönmez, Candan Gokceoglu, Ergün Tuncay,Edmund W. Medley and Hakan A. Nefeslioglu

Beziehung zwischen Gesteinsblockvolumen undDruckfestigkeit von vulkanischem Block-in-MatrixGestein (Bimrock)

Der Beitrag befasst sich mit dem Zusammenhang zwi-schen Gesteinsblockvolumen und Druckfestigkeit von An-kara Agglomerat. Dieser Fels besteht aus überwiegend vul-kanischem Gestein gemischt mit Tuffmatrix; rosafarbeneund schwarze Andesitbblöcke mit harter bis sehr harterKonsistenz werden von einer Tuffmatrix mit relativ gerin-ger Festigkeit umgeben. Das Blockvolumen wurde mithilfeeines Bildanalyseverfahrens näherungsweise ermittelt.Die rosafarbenen und schwarzen Andesitblöcke des soge-nannten Bimrock (Block-in-Matrix Gestein) wiesen erheb-liche Farbunterschiede zur Tuffmatrix auf, was die Bild-analyse von schwarz-weiß und RGB Farbfotoaufnahmenvon Aufschlüssen wesentlich erleichterte. Die mithilfe desBildanalyseverfahrens ermittelten Gesteinsblockabmes-sungen wurde mit den Schätzungen verglichen, die durchAnwendung der nulldimensionalen „Knotenzählmethode“ermittelt wurden. Aufgrund der annähernden Gleichför-migkeit der Vulkangesteinsblöcke kann angenommen wer-den, dass die zweidimensionalen Abmessungen des Ge-steinsblocks, ermittelt durch die Messung anhand von Fo-toaufnahmen, dem Blockvolumen entsprechen. Das ge-messene Gesteinsblockvolumen wurde zur annäherungs-weisen Bestimmung der einachsigen Druckfestigkeit vonAnkara Agglomerat als Funktion des Volumengehalts ei-nes Gesteinsblock verwendet. Hierzu wurden auf Regressi-on basierende Gleichungen entwickelt. Das Verhältnis zwi-schen dem Blockvolumen und der Druckfestigkeit von An-kara Agglomerat hat einen nicht linearen Verlauf, wasdarauf schließen lässt, dass die Gesamtfestigkeit des Bim-

rock von der Festigkeit der einzelnen Gesteinblocktypenabhängt. Diese Abhängigkeit mag bedingt sein durch dieunderschiedlichen Festigkeitseigenschaften zwischen Ge-steinsblöcken und Felsmatrix und erfordert weiterreichen-de Untersuchungen.

This paper describes a study into the relationship betweenvolumetric block proportions and unconfined compressivestrength (UCS) of Ankara Agglomerate, a volcaniclasticblock and tuff matrix mixture containing relatively weaktuff matrix surrounding stronger pink andesite blocks andvery strong black andesite blocks. Volumetric block pro-portions were estimated using image analysis methods.The pink and black andesite blocks in the bimrock exhibit-ed significant color contrasts with the tuff matrix, whichfacilitated image analysis of grayscale and RGB coloredphotographs of outcrops. Estimates of block proportionsfrom image analysis were checked against estimates gen-erated using the zero-dimensional “node-counting meth-od”. 2D block proportions estimated from measurements ofphotographs were considered to be equivalent to the volu-metric block proportions because the volcanic blocks wereapproximately equi-dimensional. The measured volumet-ric block proportions were incorporated into a furtherstage of the study by developing regression-based equa-tions to estimate the overall uniaxial compressive strengthof Ankara Agglomerate as a function of volumetric blockproportions. The non-linear relationship between volumet-ric block proportions and overall UCS of Ankara Agglomer-ate, suggests a dependence between overall UCS of thebimrock and the strengths of the different types of blocks.This dependence may be due to variations in block/matrixstrength contrasts and requires further study.

B lock-in-matrix rocks (bimrocks) are mix-tures of stronger blocks or rock enclosed by

weaker matrix rocks (1). The overall strength ofbimrocks tends to be greater than the strength ofthe matrix alone because the presence of blocksinfluences the mechanical properties above athreshold volumetric proportion of blocks (2).Therefore, the determination of the volumetricblock proportion of bimrocks is of crucial impor-tance for the estimation of their overall mechan-ical properties.

Three measurement methods are commonlyused to determine the volumetric block propor-tion of bimrocks:➮ One-dimensional (scanline and borehole);➮ Two-dimensional (image analyses on photo-

graphs and window mapping); and➮ Three-dimensional (sieve analyses).

Although the sieve analysis method is the mostexact method for laboratory-scale studies, sepa-ration of blocks from the weaker matrix is oftenimpossible, depending on the number and size ofblocks, and the degree of contact strength be-tween blocks and matrix. Accordingly, extensivestudies have been performed on means of deter-mining volumetric block proportions using one-dimensional (boreholes) and two-dimensional(image analyses and physical model) methods.These studies have revealed that the accuratedetermination of 3D volumetric block proportionusing 1D and 2D methods is widely influenced bythe amount of sampling and actual block propor-tion, as well as the shapes, block size distributionand orientation of the blocks (1, 3, 4).

In this study, node-counting and image classifi-cations on grayscale and Red-Green-Blue (RGB)

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Fig. 2 Block-size distribution of andesite blocks in volcanic block and tuff mixture.Bild 2 Blockgrößenverteilung von Andesitblöcken in Vulkangesteinsblock- und Tuff-mischung.

Table 1 Statistical evaluations of uniaxial compressivestrength (UCS) and unit weight (γ) for the constituents ofAnkara Agglomerate (5, 13).Tabelle 1 Statistische Auswertungen von einachsigerDruckfestigkeit (UCS) und Wichte (γ) für Bestandteile desAnkara Agglomerats (5, 13).

Statistical parameter γ [kN/m3] UCS [MPa]

Black Andesite BlocksNumber of samples 35 33Average 24.30 91.09Standard deviation 0.231 11.62Minimum 23.84 72.15Maximum 24.70 119.89

Pink Andesite BlocksNumber of samples 16 16Average 22.66 49.85Standard deviation 0.936 11.44Minimum 21.03 33.99Maximum 23.35 78.03

Tuff MatrixNumber of samples 23 21Average 16.88 10.55Standard deviation 0.883 1.89Minimum 15.17 6.41Maximum 18.23 14.42

colored photographs were performed to deter-mine the volumetric block proportion of AnkaraAgglomerate, a volcanic block and tuff mixture,which according to the criteria of Medley (1, 3), isa bimrock at laboratory and outcrop scales. Thelongest and shortest observable dimensions of theblocks were measured from photographs takenfrom different locations and orientations, in orderto evaluate the shape of the blocks as well as iden-tify the relationships between 2D block propor-tion obtained from photographs, and an estimateof the 3D volumetric block proportions.

The general procedure outlined in this paperwas used to generate estimates of volumetricblock proportions, which were then incorporat-ed into a conceptual approach for the determina-tion of the overall UCS of Ankara Agglomerate (5,13), the preliminary results of which are report-ed in this paper.

Properties of the volcanicblock and tuff mixture

Ankara Agglomerate is a volcaniclastic block andtuff mixture composed of pink (lighter) and black(darker) andesite blocks ranging in size from afew centimeters to about one meter (Figure 1).The blocks are cemented by weak tuff matrix. Todetermine the engineering properties of the con-stituents of the volcanic block and tuff mixture, aseries of laboratory tests were performed onspecimens of the pink and black andesite blocksand tuff matrix, collected from a site in Ankara,Turkey (5). The uniaxial compressive strength(UCS) and unit weight tests on the blocks and tuffwere performed in accordance with the suggest-ed method of ISRM (6). The results of the tests aresummarized in Table 1. The average values ofUCS for pink and black andesite blocks are 49.9and 91.1 MPa, respectively, and that of the tuffmatrix is 10.6 MPa. The minimum and maximumratio of UCS of blocks to the UCS of tuff matrix is2.4 and 18.7, respectively. Furthermore, a totalof 270 NX-size core samples of Ankara Agglom-erate were tested in accordance with the ISRMprocedure (6) to obtain the overall UCS values ofAnkara Agglomerate. The maximum and mini-mum UCS values of Ankara Agglomerate were5.7 and 55 MPa, respectively, and the averageUCS value was 24.9 MPa.

Medley (1) has suggested a threshold value of atleast 2 for the ratio of UCS of block to UCS of ma-trix for a geological rock mixture to be consideredas a bimrock. The UCS data of the constituents ofthe volcanic block and tuff mixture (see Table 1)clearly indicates that there was significant me-chanical contrast between blocks and matrix, andthat the volcanic block and tuff mixture investigat-ed could be evaluated as a bimrock.

In addition, Medley (1) has also suggested thatin a bimrock at the scale of engineering interest,the size and volume of blocks be sufficient to af-fect the overall properties of the mixture. Medley

Fig. 1 Outcrop of volcanic block and tuff mixture (PA: pink andesite blocks, BA:black andesite blocks; T: tuff).Bild 1 Aufschluss einer Vulkangesteinsblock- und Tuffmischung (PA: rosafarbeneAndesitblöcke, BA: schwarze Andesitblöcke; T: Tuff).



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(1) also suggested that at any scale of interestbeing considered (“characteristic engineeringdimension”), blocks would range between 5 and70 % of that scale. The 2D measurements ofblocks in the Ankara Agglomerate revealed thatthe block sizes ranged between 1 and 69 cm(mean value, 10.7 cm), as indicated in the block-size distribution graph shown as Figure 2.Blocks ranged to about 1 m in size in observedoutcrops. Hence, at the characteristic engineer-ing dimensions of the laboratory test specimens(cm scale), the Ankara Agglomerate is a bim-rock, because block sizes occupied the full 5 to70 % range in laboratory scale. However, at thescale of an outcrop (for example, 10 m height)Ankara Agglomerate has “small blocks” limitedto between 0.5 and 1 m in size.

The results of this study, based on laboratoryscale specimens and photographs of outcropsare applicable to larger volumes of the AnkaraAgglomerate if it can be demonstrated that theAnkara Agglomerate has some scale-independ-ence in block size distributions (1, 7).

Determination of volumetricblock proportions by imageanalysis

Previous research (2) revealed that the overallstrength of a bimrock mass having between about25 and 70 % volumetric block proportion is di-rectly related to the volumetric block proportion,with no dependence on the strength of the blocks.Below 25 % the strength can be taken as that ofthe matrix (2). Accordingly, for the purposes ofstudy into the geomechanical behavior of AnkaraAgglomerate (5, 13), the overall strength of thevolcanic block and tuff mixture was assumed tobe primarily dependent on the volumetric propor-tion of andesite blocks. It was thus vital that thevolumetric block proportions be determined.

In some bimrock materials at laboratoryscale, careful sieve analysis can be used to sepa-rate hard blocks from weak matrix to obtain rep-resentative block size distributions and volumet-ric block proportions (8). However in this study,separation of the andesite blocks from weak tuffmatrix was impossible because of the weakwelding (“cementation”) between the volcanicblocks and matrix. Instead of physically separat-ing block and matrix constituents, image analy-sis methods were used to estimate the volumet-ric block proportions. Advantage was taken ofprevious research where 1D scan-line surveys,and node-counting (zero dimensions) and 2D im-age analyses have also been used to estimatevolumetric block proportions (1, 3, 4, 9). Basedon correlations between 1D, 2D and 3D evalua-tions these earlier studies indicate that the quan-tity of sampling, the actual volumetric block pro-portion, and the shape and orientation of blockscontrol uncertainties in estimation of volumetricblock proportion (1, 9).

To define the block dimensions and shapes ofthe volcanic block and tuff mixture, the longestand shortest dimensions of individual block weremeasured from scanlines oriented at differentdirections across photographs taken of differentoutcrops of the Ankara Agglomerate. As shownin Figure 3, 75 % of the measured blocks have2D major:minor dimension ratios less than 1.2,which indicated that the blocks are approxi-mately equi-dimensional in 2D and 3D. The ra-tionale for this assumption was that the frag-mented, volcaniclastic genesis of the bimrock re-sulted in probable random block orientationswithin the rock mass, and that the many meas-urements in 2D sufficiently captured 3D blockdimensions. In that case, the uncertainties in theestimation of 3D block proportions, based on 2Dmeasurements, would be less than those for bim-rocks with ellipsoidal blocks such as melangesstudied by Medley (1, 9) or the idealized mixturesanalyzed by Haneberg (12).

To estimate the volumetric block proportionsof the volcanic block and tuff mixture at fieldscale, image classifications and node-countingmethods were performed on grayscale and RGBcolored photographs of the bimrock. For thispurpose, scaled photographs were taken per-pendicular to nearly planar exposures of AnkaraAgglomerate (see Figure 1). The overall objec-


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tive of image classification procedures is to auto-matically categorize pixels into classes orthemes (10) using “unsupervised classifications”or “supervised classifications”. In the unsuper-vised approach, the image data are first classi-fied by aggregating them into the natural spec-tral (tonal) grouping or clusters present in theimage (10), whereas supervised classification in-

volves a training step followed by a classificationstep. In this study, the supervised image classifi-cation method of training, classification and out-put stages were performed for the determinationof the pink block, black block and tuff matrixconstituents of the Ankara Agglomerate, usingboth grayscale and colored photographs of out-crops exposures.

The colored photographs of agglomerate ex-posures (outcrops and drill core) were scannedin RGB and grayscale with high resolution. Someknown constituents on parts of the colored imag-es were first defined in the training stage of im-age classification (Figure 4a). In the trainingstage of the grayscale photographs, the pixel val-ue ranges of each constituent were determinedwithin the overall grayscale tonal spectrum of 0to 255 grayscale shades. The black andesite hasa range of grayscale pixel values of 0 to 61 in.Except for some small deviations in the un-weathered core samples, the range of grayscalepixel values for the tuff and pink andesite is gen-erally between 62 to 115 and 116 to 255, respec-tively, according to the image analyses per-formed by Gokceoglu and others (11). In the sec-ond stage, whole images were classified on the

Fig. 3 Cumulativefrequency distributionof the ratio of thelengths of longestaxes to lengths ofshortest axes ofandesite blocks.Bild 3 Summen-häufigkeitsverteilungdes Längenverhält-nisses von längster zukürzester Hauptachsevon Andesitblöcken.

Fig. 4 Original andclassified views of

Ankara Agglomerateexposures (a) RGBcolored image and(b) grayscale image

(white scale bar is1 m long).

Bild 4 Original undklassifizierte Ansichten

von Ankara Agglo-merat Aufschlüssen:

a) RGB Farbfoto-aufnahmen und

b) Schwarz-weiß Auf-nahmen (weiße Mess-skala entspricht 1 m).



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basis of the results of the training stage. Eachpixel in the image data set was categorized intothe constituents (Figure 4b) using a minimum-distance-to-means classifier, since the mini-mum-distance-to-means strategy is mathemati-cally simple and computationally efficient (10).

In the node-point-counting method, a meshhaving squares of 1 cm2 was overlaid on the pho-

Fig. 5 Stages of thenode-counting classi-fication.Bild 5 Phasen derKnotenzählzuordnung.

tographs (Figure 5). At each intersection (node)of the mesh, the underlying material was visuallyclassified as being tuff, black andesite or pinkandesite. (This method is similar to point-count-ing performed by mineralogists and petrologistsusing rock thin sections viewed through micro-scopes in order to determine mineralogical pro-portions necessary to petrographically classify


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the rock sampled by the thin section.) The per-centages of each constituent of the agglomerateexposures were determined by dividing thenumber of node intersections for each constitu-ent by the total number of intersections of themesh (see Figure 5).

A cross-check between the proportions of theconstituents obtained from image classificationemployed on grayscale and colored photographsshowed that similar estimations of the propor-tion of the constituents was obtained (Figure 6).As can be seen from Figure 6, there is no signif-icant difference between the two methods, hencethe more practical image classification methodwas used to obtain further estimates of volumet-ric block proportions of the volcanic block andtuff mixture.

Preliminary evaluation of theUCS of Ankara Agglomerate

The volumetric block proportion data describedin this paper are being used to develop relation-ships between volumetric block proportions andoverall UCS for Ankara Agglomerate (13). Theproportion of the constituents of Ankara Ag-glomerate core samples were determined by us-ing grayscale image analysis performed on bothhalves of split core samples.

In a further stage of the study, the relationbetween block proportions and the UCS values ofthe Ankara Agglomerate core samples was eval-uated. It was also of interest to determine if therewas any dependence between overall UCS andstrengths of the two block types. In other words,since black andesite is stronger than pink andes-ite, it was assumed that the mechanical influ-ence of stronger black andesite blocks on theoverall UCS could be higher than the influence ofthe weaker pink blocks.

The sum of the proportions of black and pinkandesite blocks was equal to the total volumetricblock proportion in the specimens. However, acomposite block proportion was used to studythe differences in overall UCS due to differencesin the proportions of the two different blocktypes. A weighted “equivalent block proportion”,or EBP, accommodates two or more types ofblocks differing in individual mechanical proper-ties:

EBP VPUCS

UCSii

blocki

n

= ∑max_

............................ [1]

where VP is the volumetric block proportion of ith

block, UCS is the uniaxial compressive strength,n is the number of different types of blocks, andUCSmax_block is the uniaxial compressive strengthof the stronger block type.

The uniaxial compressive strengths of the An-kara Agglomerate specimens were then normal-ized by dividing them by the average uniaxialcompressive strength of the matrix: this parame-

Fig. 6 Cross-check between proportions of the constituents of agglomerate esti-mated from the image analysis method performed on grayscale images, comparedto estimates based on image analyses performed RGB colored images (red plot),and the node-counting method (green plot).Bild 6 Überprüfung der Größenverhältnisse von Agglomeratbestandteilen, geschätztanhand Bildanalyse von Schwarz-weiß Aufnahmen, im Vergleich zu Schätzungenmithilfe der Bildanalyse von RGB Farbbildern (rote Linie), und Knotenzählmethode(grüne Linie).

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ter is denoted as UCSN. The distributions of thedata between EBP and UCSN are illustrated inFigure 7.

Two exponential type equations were ob-tained by regression analyses of the data ofshown in Figure 7:

UCS xEBPN = ( )1 3361 1 12. exp . ...................... [2]

UCS xEBPN = ( )exp .1 6874 ............................. [3]

As shown in Figure 7, the regression lineforced to intersect to value of “1” on the y axis(data pair of EBP=0, UCSN=1) yields a trend thatis more slightly more representative of the datathan the unforced trend. The relationship of Fig-ure 7 is non-linear, particulary above about 70 %equivalent block proportion, suggesting that thedependence of overall bimrock strength on blockproportion is more complex than previously un-derstood. The plot also indicates that at highequivalent block porportions the overall bimrockbecomes uniformally stronger, with less datascatter. This behavior supports the recommen-dation of Medley (1), based on the work ofLindquist and Goodman (2), that at high volu-metric block proportions (greater than about 70to 75 %), block/matrix rock mixtures should beconsidered as a blocky rock masses with wide in-filled joints, for which fabric conventional rockengineering methods should be applied.

Fig. 7 Relationships between equivalent block portion (EBP) and UCS of Ankara ag-glomerate. Trend forced through intercept (0, 1) is slightly more representative of data.Bild 7 Beziehungen zwischen äquivalentem Blockanteil (EPB) und Druckfestigkeit vonAnkara Agglomerat. Verlauf durch Schnittpunkt (0,1) repräsentiert Daten etwas besser.

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The preliminary work also revealed an appar-ent but unexpected relationship between thestrength of the blocks and the overall UCS for thetwo different types of blocks. Until now it hasbeen assumed that overall bimrock strength is


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not influenced by block strength, as long as thereis sufficient block/matrix strength contrast toforce failure surfaces around blocks (1, 2). How-ever, this study shows that the interaction be-tween blocks and matrix is more complex thanpreviously assumed, and that block/matrixstrength contrasts for two or more mechanicallydiverse blocks may also influence the overall me-chanical behavior of the bimrock. Further workis underway to examine the problem.

Conclusions

Based on the work performed for this study, thefollowing conclusions are presented:➮ If there is sufficient color contrast between

constituent blocks and matrix, image analysismethods are practical for the determination ofthe block proportions of outcrops, laboratoryspecimens and similar exposures of bimrocks.

➮ If the shapes of blocks of a bimrock are ap-proximately equi-dimensional in 3D, the blockproportions obtained from 2D measurementscan be considered equivalent to the 3D volu-metric block proportions.

➮ When using image classification on grayscaleimages, the surfaces must be slightly weath-ered or unweathered. If the surfaces areweathered, image classifications using col-oured photographs should be employed.

➮ The relationship between equivalent blockproportion (EBP) and UCS values of Ankaraagglomerate indicates that the effect of EBP onUCS is small for lower EBP values , but expo-nentially increases with higher values of EBP.

➮ There is an apparent dependence betweenUCS and individual block strengths, which re-quires further study.

References1. Medley, E.W.: The engineering characterization of me-langes and similar block-in-matrix-rocks (bimrocks). PhDthesis, Dept. of Civil Engineering, Univ. of California, Berke-ley, 1994.2. Lindquist, E.S ; Goodman R.E.: The strength and deforma-tion properties of the physical model melange. In: Nelson, P. ;Laubach, S.E. (eds.): Proceedings of the First North Ameri-can Rock Mechanics Conference (NARMS), Austin, Texas.Rotterdam: Balkema,1994.

3. Medley, E.: Orderly characterization of chaotic FranciscanMelanges. Felsbau, Vol. 19 (2001), No. 4, pp. 20–33.4. Gokceoglu, C.: A fuzzy triangular chart to predict theuniaxial compressive strength of the Ankara Agglomeratesfrom their petrographic composition. J. Eng. Geol., Vol. 66(2002), pp. 39–51.5. Sonmez, H. ; Tuncay, E. ; Gokceoglu, C.: Models to pre-dict the uniaxial compressive strength and the modulus ofelasticity for Ankara Agglomerate. Int. J. Rock Mech. Min.Sci., Vol. 41 (2004), No. 5, pp. 717-729.6. ISRM: ISRM suggested method: rock characterization,testing and monitoring. London: Pergamon Press, 1981.7. Medley, E. ; Lindquist, E.S.: The engineering significanceof the scale-indepence of some Franciscan Melanges in Cal-ifornia, USA. In: Daemen, J.K. ; Schultz, E.A.(eds.): 35th USRock Mech. Sym, pp. 907-914. Rotterdam: Balkema, 1994.8. Lindquist, E.S.: The strength and deformation propertiesof melange. PhD thesis, Dept.of Civil Engineering, Univ. ofCalifornia, Berkeley, 1994.9. Medley, E.: Uncertainty in estimates of block volumetricproportions in melange bimrocks. In: Marinos, P.G. ; Koukis,G.C. ; Tsiambaos, G.C. ; Stournaras, G.C., (eds.): Proc. In-ternational Congress, International Association of Engineer-ing Geologists, Engineering Geology and the Environment,Athens, Greece, 1997.10. Lillesand, T.M. ; Kiefer, R.W.: Remote sensing and imageinterpretation. 2nd ed. New York: Wiley, 1987.11. Gokceoglu, C. ; Kasapoglu, K.E. ; Sonmez, H.: Predictionof uniaxial compressive strength of Ankara Agglomeratesfrom their petrographical composition. In: Moore, D. ; Hungr,O. (eds.): Proceedings of the Eighth International Congressof IAEG and the Environment, Vancouver, Canada, pp. 455-459. Rotterdam: Balkema, 1998.12. Haneberg, W.C.: Simulation of 3-D block populations tocharacterize outcrop sampling bias in block-in-matrix rocks(bimrocks). Felsbau, Vol. 22 (2004), No. 5, pp. 19-26.13. Sonmez, H. ; Gokceoglu, C. ; Medley, E. ; Tuncay, E.;Nefeslioglu, H.A.: A conceptual approach for determiningthe uniaxial compressive strength of bimrocks. In prepara-tion, 2004.

AcknowledgementThis research was supported by TUBITAK (The Scientificand Technical Research Council of Turkey: Project No:102Y033). Thanks also to Isabelle Pawlik, PE, of Jacobs As-sociates, San Francisco, USA, for the German translations.

AuthorsDr. Harun Sönmez, Dr. Candan Gokceoglu and Ergün Tun-cay, Department of Geological Engineering, Applied GeologyDivision, Hacettepe University, 06532 Beytepe-Ankara, Tur-key, E-Mail [email protected], [email protected], [email protected]; Dr. Edmund W. Medley,Medley Geoconsultants, 1554 Winding Way, Belmont, Cali-fornia, 94002, USA, E-Mail [email protected]; andHakan A. Nefeslioglu, General Directorate of Mineral Re-search and Exploration, 06520, Ankara, Turkey, [email protected]


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Relationships between Volumetric Block Proportions and Overall