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**ABSTRACT:**Although the modulus of deformation of rock masses has crucial importance for geotechnical projects, such as tunnels and dams, the determination of this parameter by in situ tests requires considerable costs and involves difficult operational processes. For this reason, empirical equations for the indirect estimation of the modulus of deformation are an interesting issue for rock engineers and engineering geologists. This study includes assessment of the prediction performances of some existing empirical equations, using in situ plate loading test data and rock mass properties, producing an empirical equation depending on the new data, construction of a fuzzy inference system for the estimation of modulus of deformation, and making a comparison between results obtained from the empirical equations and fuzzy inference system. A series of calculations and statistical analyses were undertaken. It is concluded that the performance of the empirical equations and fuzzy inference system obtained in this study is satisfactory. However, the prediction models developed in this study are limited by the number of the data used and the rock types employed. For these reasons, a cross-check should be performed before using these prediction models for design purposes.International Journal of Rock Mechanics and Mining Sciences 06/2003; 40(4):607. · 1.42 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Subjective judgement normally constitutes an important element in mining geomechanics decision processes. In most instances, subjectivity arises from the imprecise or fuzzy information which in turn results from descriptive data or inaccurate test results. The paper proposes the application of fuzzy set theory in assisting mining engineers in the geomechanics decision processes for which subjectivity plays an important role. In particular, the Bellman-Zadeh optimization procedure is used to synthesize a hazard index for mining excavations. The same procedure is used to evaluate a rock mass classification rating from Bieniawski's system with incorporation of expert knowledge. The extension principle of fuzzy sets is applied to evaluate Barton's quality index Q when information on various contributing indices is fuzzy.Basic principles of fuzzy set theory are described and numerical examples are used to illustrate applications of fuzzy set theory in mining geomechanics.International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts. - SourceAvailable from: sciencedirect.com[Show abstract] [Hide abstract]

**ABSTRACT:**A knowledge-based fuzzy model for performance prediction of a rock-cutting trencher has been developed. A trencher is a machine that uses a rotating cutting chain equipped with bits to excavate trenches in rock and soil. The performance of a trencher, and consequently the cost of a specific excavation project, is determined by its production rate and by the bit consumption (due to wear and breakage). Both these factors depend on the properties of the excavated rock material and on the trencher characteristics. Mathematical modeling of the trencher performance is difficult, since the interactions between the machine tool and the environment are dynamic, uncertain, and complex. The number of available measurements is too small to use statistical methods. Hence, an approach based on expert knowledge was applied to develop a rule-based fuzzy model. The use of fuzzy logic allows for smooth interfacing of the qualitative information involved in the rule base with the numerical input data. The developed model uses six input variables [rock strength, spacing of three joint (discontinuity) sets in the rock mass, joint orientation, and trench dimensions] to predict the production rate and bit consumption in terms of qualitative linguistic values. Numerical predictions are obtained by using a modified fuzzy-mean defuzzification which allows for straightforward adaptation of the consequent membership functions in order to fine-tune the model performance to the data. The expert knowledge is coded as if-then rules, hierarchically organized in four rule bases. The model was validated both qualitatively using dependency analysis and quantitatively using the available data. The results obtained so far are satisfactory.International Journal of Approximate Reasoning 01/1997; 16(1):43–66. · 1.73 Impact Factor

Page 1

International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

Models to predict the uniaxial compressive strength and the modulus

of elasticity for Ankara Agglomerate

H. Sonmez*, E. Tuncay, C. Gokceoglu

Department of Geological Engineering, Applied Geology Division, Hacettepe University, 06532 Beytepe-Ankara, Turkey

Accepted 11 January 2004

Abstract

Determination of the uniaxial compressive strength (UCS) and modulus of elasticity of block-in-matrix rocks (bimrocks) is often

impossible in the laboratory since the preparation of the standard core samples from bimrocks is extraordinarily difficult. For this

reason, some predictive models were developed to estimate the UCS and modulus of elasticity based on the volumetric portion of

blocks in Ankara Agglomerate, which is composed of black and pink andesite blocks in a tuff matrix. The ratio of Eiminof blocks

(5.99GPa) to Eimaxof the tuff matrix (2.83GPa) is 2.2 for Ankara Agglomerate. In addition to this contrast, the minimum ratio of

UCS values of andesite blocks (34.99MPa) to matrix tuff (14.4MPa) is 2.4. In the first stage of the study, fuzzy logic was used as a

tool for the prediction of the UCS of Ankara Agglomerate based on its block and matrix constituents. UCS values for 164

agglomerate cores were evaluated in the prediction model based on fuzzy logic. A triangular chart expressed by ‘‘if-then’’ rules

considers different constituent composition of the agglomerate. Considering the membership functions depending on the portion of

constituents, a Mamdani fuzzy algorithm was constructed and a fuzzy triangular chart was obtained for the estimation of the UCS

of the agglomerate. The ‘variance accounts for’ (VAF) and the root mean square error (RMSE) indices were calculated as 56.9%

and 7.3, respectively, to characterize the prediction performance of the triangular chart. In the second stage of the study, the goal

was to construct a prediction model for the estimation of the modulus of the elasticity. Regression analyses were performed using

103 UCSs and the unit weight data obtained from core samples prepared from tuff matrix, black and pink andesite blocks and

agglomerate. An equation having a correlation coefficient of 0.951 was obtained from the regression analyses. The VAF and RMSE

indices for the multiple regression equation were obtained as 88.8% and 0.84, respectively. Both correlation coefficient and the

performance indices indicated that the prediction capacity of the equation is high.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Ankara Agglomerate; Bimrocks; Fuzzy logic; Regression; Uniaxial compressive strength; Modulus of elasticity

1. Introduction

The measures and estimates of the modulus of

elasticity (Ei) and the uniaxial compressive strength

(UCS) of rock materials are widely used in rock

engineering, being important for rock mass classifica-

tions and rock failure criteria. In addition, analytical

and numerical solutions require both Ei and UCS.

However, NX-sized core samples recommended by

ISRM [1] cannot be generally obtained from geological

mixtures or fragmented rocks, such as agglomerate and

conglomerate, which include strong gravel sized and/or

rock blocks encased within soft cementing matrix

material. At the scale of the laboratory, such rock

mixtures are ‘‘block-in-matrix rocks’’ (‘‘bimrocks’’)

which Medley [2] defined as a ‘‘mixture of relatively

large, competent blocks within a bonded matrix of finer

and weaker texture’’. Coarse pyroclastic rocks, breccia

and sheared serpentinites, melanges and fault rocks are

other examples of bimrocks, as described by Lindquist

and Goodman [3] and Medley and Goodman [4].

The presence of blocks influences the mechanical

properties of the block/matrix mixtures: above a certain

lower volumetric proportion of blocks, the overall

strength of bimrocks tends to be greater than the

strength of the matrix alone [3]. However, the strength

difference between weak matrix and strong blocks in a

bimrock can significantly reduce the quality and the

ARTICLE IN PRESS

*Corresponding author. Tel.: +90-312-297-7700; fax: +90-312-299-

2034.

E-mail address: haruns@hacettepe.edu.tr (H. Sonmez).

1365-1609/$-see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijrmms.2004.01.011

Page 2

number of useable core samples that can be recovered

from drilling, and prepared for laboratory studies, and

as a result determination of the modulus of elasticity

and UCS of bimrocks is extremely difficult. Due to these

reasons, development of predictive models of the

mechanical and deformation properties of rocks has

become an attractive study area in rock engineering,

such as the studies performed by Bell [5], Doberenier

and DeFreitas [6], Hawkins and McConnell [7], Ulusay

et al. [8], Gokceoglu et al. [9] and Gokceoglu [10].

Regression techniques, fuzzy logic and neural networks

are also used for the construction of prediction

models [11–14].

The study described in this paper has the goals of

predicting models for estimating the Ei and UCS of

Ankara Agglomerate by using regression techniques and

fuzzy logic. In the first stage of the study, a fuzzy-based

triangular prediction chart prepared by Gokceoglu [10]

to predict the UCS of Ankara Agglomerates from their

constituents was re-constructed using Gokceoglu’s

approach with 47 additional data to improve his

empirical approach. The second stage of the study was

preparation of a prediction model for Ei of Ankara

Agglomerate based on regression techniques.

2. Properties of Ankara Agglomerate

Ankara Agglomerate is composed of tuff matrix

surrounding pink and black andesite gravels and/or

blocks ranging from few centimeters to about a meter in

size (Fig. 1). The pink and black andesites are known as

Payamlitepe and Huseyingazi andesites, respectively

[15]. Although their origins are somewhat different,

both pink and black andesites are petrographically

classified as trachiandesite on the basis of thin-section

studies [10].

The tuff matrix, black and pink andesite block

samples were collected to determine the UCS, modulus

of elasticity (Ei) and unit weight (g) of the constituents

of the Ankara Agglomerate. The parameters of con-

stituents of Ankara Agglomerate obtained from labora-

tory tests were given in Table 1. The average UCS and

Ei values of both black and pink andesite blocks

(91.1MPa, 8.7GPa and 49.9MPa, 7.4GPa, respec-

tively) are higher than those of tuff matrix (10.6MPa

and 2.1GPa, respectively). The unit weight (g) of tuff

matrix (16.9kN/m3) is also less than the g of both black

and pink andesite blocks (24.3 and 22.7kN/m3, respec-

tively). In other words, the UCS, Eiand g values of the

constituents of Ankara Agglomerate exhibit significant

differences. It is useful to compare the ratios of the

physical and mechanical properties of the matrix and

block constituents of Ankara Agglomerate to the ratios

of the properties of the matrix and block materials used

by Lindquist [16] for physical model melanges that he

prepared Lindquist’s ratio of Eiof block to Eiof matrix

of 2.0 (which Medley [17] suggested as a threshold

criterion for block/matrix contrasts). For the Ankara

Agglomerate the ratio of Eimin of blocks (6.0GPa) to

Eimaxof tuff matrix (2.8GPa) is 2.1. In addition to the

stiffness contrast of the constituents of Ankara Agglom-

erate, the minimum ratio of UCS values of blocks

(34.0MPa) to tuff matrix (14.4MPa) is 2.4. Conse-

quently, based on the significant strength and stiffness

contrasts between tuff matrix and andesite blocks, the

Ankara Agglomerate was considered to be a bimrock,

and thus subject to the general property of bimrocks

ARTICLE IN PRESS

BA

BA

PA

PA

T

0

204080 cm

Fig. 1. View an outcrop of Ankara Agglomerate (BA: black andesite

blocks; PA: pink andesite blocks; T: tuff matrix).

Table 1

Statistical evaluation for the unit weight (g), uniaxial compressive

strength (UCS) and modulus of elasticity (Ei) of constituents of the

Ankara Agglomerate

Statistical parameters

g (kN/m3)UCS (MPa)

Ei(GPa)

Black andesite

Sample size

Average

Standard deviation

Min

Max

35

24.3

0.23

23.8

24.7

33

91.1

11.6

72.2

119.9

23

8.7

0.92

7.1

10.1

Pink andesite

Sample size

Average

Standard deviation

Min

Max

16

22.7

0.94

21.0

23.4

16

49.9

11.4

34.0

78.0

14

7.4

0.84

6.0

8.9

Tuff

Sample size

Average

Standard deviation

Min

Max

23

16.9

0.88

15.2

18.2

21

10.6

1.9

6.4

14.4

19

2.1

0.36

1.6

2.8

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

718

Page 3

discovered by Lindquist and Goodman [3], that the

overall strength of a bimrock mass is simply and directly

related to the volumetric proportion of blocks.

Accordingly, the overall strength of the Ankara

Agglomerate was assumed to be dependent on the

volumetric portion of andesite blocks. The exact

volumetric block proportions of bimrocks at labora-

tory scale in 3D can sometimes be determined by sieve

analysis to separated hard blocks from weak ma-

trix. However, in the case of volcanoclastic Ankara

Agglomerate, separation

mates had to be generated using statistically based

approximations.

To estimate volumetric block proportion, some

previous studies [10,17,18] utilized scan-line surveys in

1D and image analyses in 2D. The correlations between

3D and 2D or 1D showed that block shape and block

orientation controls the uncertainties in estimates of

volumetric block proportion. Medley [19] has also

identified that the amount of measurement and the

actual block volumetric proportion itself are key

contributors to the error between actual 3D volumetric

block proportion and estimates based on 1D and 2D

measurements.

If the dimensions of blocks in 3D are approximately

equal, one source of uncertainty in estimates of

volumetric block portion is lessened. To examine the

possible uncertainties in 2D estimates of the block

proportion (as compared to actual 3D block proportion

the longest and the shortest axes of andesite blocks in

the Ankara Agglomerate were measured in different

directions on photographs of outcrops. The ratios of the

longest to the shortest measured axes of andesite blocks

were evaluated statistically. As shown in Fig. 2, 75% of

the measured blocks have axial ratios less than 1.2 which

prompted the reasonable assumption that the blocks

were equi-dimensional in 2D and 3D. In other words,

the uncertainties in estimates of the 3D block proportion

based on 2D measurements would be less as a result, as

is impossible,andesti-

shown in Fig. 2, which is based on measurements from

photographs taken from different directions. In addi-

tion, the measurements of blocks in 2D revealed that the

block sizes vary between 1 and 69cm while the mean

value is 10.7cm, as indicated in the block-size distribu-

tion graph (Fig. 3).

Image classification and the node-point-counting

methods were applied to the photographs taken from

outcrop exposures of the Ankara Agglomerate to

estimate the block proportion in 2D. These estimates

were further assumed which to be approximately equal

to the actual 3D block volumetric proportion, although

it is known that such assumed equivalence can be

incorrect [19].

The photographs of the exposures were scanned at

high resolutions for the purpose of image analyses. The

overall objective of image classification procedures is to

automatically categorize pixels into classes or themes

[20]. In image processing, there are two types classifica-

tion methods: ‘‘supervised classifications’’ and ‘‘unsu-

pervised classifications’’. The fundamental difference

between these techniques is that supervised classification

involves a training step followed by a classification step.

In the unsupervised approach, the image data are first

classified by aggregating them into the natural spectral

(tonal) grouping or clusters present in the image [20]. In

this study, the supervised image classification method

including three main stages such as training, classifica-

tion and output stages were performed for determina-

tion of the constituents of the agglomerate. In the

training stage, the pixel value ranges of each constituent

were determined within the overall gray-scale tonal

spectrum of 0–255 gray-scale shades. The black ande-

sites have a range of pixel values of 0–61 in gray scale.

The range of pixel values for the tuff and pink andesites

are between 62–115 and 116–255, respectively, accord-

ing to the image analyses performed by Gokceoglu et al.

[9]. It is possible to use the color photographs for the

image classification purposes. However in this study, the

ARTICLE IN PRESS

1

0

20

40

60

80

100

1.21.4 1.61.82.0

Cumulative frequency (%)

The ratio of the longest to the shortest axes of andesite blocks

Fig. 2. Cumulative frequency distribution of the ratio of the longest to

shortest axes of andesite blocks.

100

80

60

40

20

0

1

10100

Cumulative frequency finer (%)

Average block dimension (cm)

Average=10.7

Standard deviation=9.47 cm

Mimimum=1 cm

Maximum=69 cm

Fig. 3. Block-size distribution of using average measured dimension of

andesite blocks in the Ankara Agglomerate.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

719

Page 4

gray-scaled photographs were preferred since the

studied rock surfaces (Fig. 4a) were clear enough and

they were not affected by weathering.

In the second classification stage, each pixel in the

image data set was categorized into the constituents

using a minimum-distance-to-means classifier (Fig. 4b),

since the minimum-distance-to-means strategy is math-

ematically simple and computationally efficient [20].

The third and last stage of classification, we obtained

the percentages of each constituent present in images of

the Agglomerate. In the node-point-counting method, a

mesh having squares of 1cm2was overlaid on the

photographs (Fig. 4c). At each intersection on the mesh,

the underlying material was visually classified as being

tuff, black andesite or pink andesite. (This method is

identical to point-counting performed by mineralogists

and petrologists using rock thin sections viewed through

microscopes in order to determine mineralogical pro-

portions necessary to classify the rock exposed in the

thin section.) (Fig. 4d) The percentages of each

constituent of the Agglomerate exposures were deter-

mined by dividing the number of node intersections for

each constituent by the values for divided and total

number of intersections of the mesh.

Cross-correlation between two methods shows that

the results of the image classification and node-point-

counting methods are similar to each other (Fig. 5).

Therefore, since the image classification method was

quicker to perform, it was preferred for the pre-

laboratory testing estimation of the percentages of the

constituents of Ankara Agglomerate core samples.

Gokceoglu et al. [9] and Gokceoglu [10] performed

some prediction models to determine UCSs of the

Ankara Agglomerate from its constituents. A total of

117 data extracted from the Ankara Agglomerates were

taken from previous studies performed by Gokceoglu

et al. [9] and Gokceoglu [10]. Recognizing the scale

effect of samples relative to in situ and laboratory, scales

they took block samples from outcrops where andesite

blocks were small. The tests of UCS and unit weight

tests performed on the core samples in accordance with

the procedure suggested by ISRM [1]. Furthermore, to

avoid the effect of the degree of weathering on the

mechanical properties of the samples, the agglomerate

ARTICLE IN PRESS

(a) (c)

(d) (b)

PA

BA

Pink andesite

PA: Pink andesite, BA: Black andesite

Black andesite

Fig. 4. (a) Original outcrop view from the Ankara Agglomerate (same as Fig. 1 above); (b) classified image of the same view; (c) original view

covered by a measurement mesh; (d) result of node-point-counting classification.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

720

Page 5

block samples were collected from unweathered or

slightly weathered outcrops as performed by Gokceoglu

et al. [9] and Gokceoglu [10]. In addition to data

published by Gokceoglu et al. [9] and Gokceoglu [10],

this study evaluated new data values obtained from 47

additional core samples from the Ankara Agglomerate.

The total 164 core samples were collected during the

period between 1997 and 2003, which is admittedly a

long period, but was due to the difficulties encountered

in obtaining high quality core samples from the Ankara

Agglomerate.

Modulus of elasticity tests were also performed on the

core samples used for UCS tests. Due to the block-in-

matrix nature of the agglomerates standard strain

gauges were not used to measure the deformation under

load. If standard strain gauges had been used, the

measured strain values would have represented only the

constituent upon which the strain gauge had been

adhered. In other words, the results of the test would

have represented only the local andesite block or area of

tuff matrix instead of the overall deformation of whole

agglomerate sample. To overcome this difficulty, a

strain gauge-type displacement transducer with an

accuracy of 10?4was employed (Fig. 6a). Servo-

controlled loading equipment was used in the modulus

of elasticity and UCS tests. Two strain gauge-type

transducers were located at the lower surface of the core

specimen to measure overall displacement of the core

specimen. The mean value of displacement of two

transducers was assumed to be the axial bulk displace-

ment of the specimen.

It is reasonable to ask ‘‘A series of tests was also

performed to determine if deformation of the loading

device was included in measurement of the sample

deformation?’’. Two transducers were fixed to the upper

and lower surface of the specimen. The displacement

measured from the upper side was generally less than

10?3mm. In other words, the displacement measured

from the lower surface was assumed to be approxi-

mately equal to the bulk displacement of the specimen,

and therefore, the stiffness of the loading equipment had

sufficient stiffness to obtain the overall modulus of the

elasticity of the core specimens.

A typical stress–strain curve obtained from these

experiments is given in Fig. 6b. As shown in Fig. 6, the

average modulus of elasticity for the specimen was

considered throughout the study. The statistical assess-

ments for the mechanical and deformation parameters

are summarized in Table 2. The number of the modulus

ARTICLE IN PRESS

0

10

20

30

40

50

60

70

010203040 506070

1:1

Portion of the constituents in percentage

of agglomerate based on node counting

Portion of the constituents in percentage

of agglomerate based on image analysis

y=1.047 x

r=0.9

Fig. 5. Cross-check between the results of image processing and node

point counting.

0

5

10

15

20

25

30

35

02468

10

Axial bulk strain (x10-6)

Axial stress (MPa)

∆σ

∆ε

(b)

(a)

Fig. 6. (a) The experiment setup used in modulus of elasticity tests and

(b) a typical stress–strain curve obtained from the modulus of elasticity

tests.

Table 2

Statistical evaluation for the unit weight (g), uniaxial compressive

strength (UCS) and modulus of elasticity (Ei) of the Ankara

Agglomerate

Statistical parameters

g (kN/m3)UCS (MPa)

Ei(GPa)

Sample size

Average

Standard deviation

Min

Max

164

21.6

1.53

17.6

24.3

164

25.4

12.11

5.7

55

47

4.1

0.88

2.2

5.9

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

721

Page 6

of elasticity for the Ankara Agglomerate was limited

because the data obtained from the tests performed by

Gokceoglu et al. [9] and Gokceoglu [10] did not include

this parameter.

Image analysis technique was used to determine the

percentage of the block proportion for the core samples.

The photographs of the upper and lower surfaces of the

core samples were taken, and then they were scanned at

high resolution for application of image classification

method before the mechanical testing (Fig. 7a). After

the UCS and Ei tests were performed, failed samples

were covered by plaster of paris and cut perpendicular

to the failure surface to observe the route of the failure

surface (Fig. 7b). The failure surface generally propa-

gates through the tuff matrix (Fig. 7b), but the failure

surface tends to crack andesite grains with the higher

andesite portion.

Additionally, the surface obtained after cutting was

also scanned for the image classification to obtain

additional data for the proportion of constituents of the

core sample. Image classifications were applied in

accordance with the procedure described above relevant

for analyses performed on the photographs of outcrops.

Statistical assessments of percentages of the constituents

of the Ankara Agglomerate core samples are given in

Table 3.

As described previously, the determined percentage of

the constituents for each core samples was obtained

from 2D surfaces. The block proportions determined

from 2D photographs were assumed to be equivalent to

the 3D volumetric block proportion because the

observed block shapes on outcrops were generally

equi-dimensional in 2D (as discussed above). Never-

theless, since there is a large contrast between the unit

weights of the blocks and the matrix constituents, an

additional check was performed by multiplying the

percentage of the constituents and the unit weight of the

both constituents, and comparing the resulting estimate

of the bulk unit weight (g0

weight of the sample (g). Eq. (1) shows the operation

(Fig. 8):

A) with the measured unit

g0

A¼ ðgBA?BAÞ þ ðgPA?PAÞ þ ðgT?TÞ;

ð1Þ

where g0

(kN/m3), gBAthe average unit weight of the black

andesites from laboratory determination (24.3kN/m3),

gPAthe average unit weight of the pink andesites from

laboratory determination (22.7kN/m3), gTthe average

unit weight of the tuff from laboratory determination

(16.9kN/m3), BA the amount of black andesite deter-

minedbyimageclassification

the amount of pink andesite determined by image

Ais the estimated unit weight of the agglomerate

method(%), PA

ARTICLE IN PRESS

Fig. 7. (a) Original agglomerate sample and classified images and (b) cross-section through failed specimens perpendicular to failure surface

(NX-sized core).

Table 3

Statistical evaluations of constituents percentage of the Ankara

Agglomerate

Statistical parametersBlack andesite Pink andesiteTuff

Sample size

Average

Standard deviation

Min

Max

164

32.1

27.1

1

92

164

30.2

27.7

0

82

164

37.6

20.1

3

89

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

722

Page 7

classification method (%) and T the amount of tuff

determined by image classification method (%).

Theoretically, if the volumetric proportion of the

constituents has been correctly estimated by assuming

that 2D measurements are equivalent to the 3D

percentages, the unit weight of each agglomerate core

specimen determined from Eq. (1) (g0

that determined in laboratory for each specimen. Fig. 8

shows a comparison between the unit weight values

estimated from Eq. (1) based on the unit weight of

constituents and those determined in the laboratory (g).

As seen in Fig. 8, the calculated coefficient of correlation

between these values is 0.83, and is close to the line of

1:1. This result indicates that the image classification

method had a sufficient prediction capacity for the

determination of the 2D percentages of the agglomerate

constituents in the core samples and that these values

could be assumed as equal to the volumetric block

portion. It should be noted that this method may not be

applicable to other bimrocks, particularly those contain-

ing asymmetrical blocks.

A) must be equal to

3. Prediction model for uniaxial compressive strength

The notion of fuzzy sets is to model real-life classes in

which there may be a continuum of grades of member-

ship. Fuzzy sets underlie much of our ability to make

decisions based on ambiguous and imprecise informa-

tion, and fuzzy set theory is said to be a generalization of

classical set theory [21]. In addition, the strength of

fuzzy sets and rule-based fuzzy modeling can be

summarized as follows [22]:

(a) Allows explicit expression of the knowledge of the

system via fuzzy ‘‘If-then’’ rules.

(b) Deals with subjective uncertainty (fuzziness, vague-

ness, imprecision) inherent to the way experts

approach their problems.

(c) Numerical and categorical data can be combined.

(d) Provides a sound mathematical basis.

The general descriptions of classical and fuzzy sets are

given in the following paragraphs. In classical set

theory, a membership function, mAðxÞ has two values,

0 and 1, as formulated in Eq. (2).

mAðxÞ ¼

1if xAX;

if xeX:

0

(

ð2Þ

A fuzzy set of X; labeled A, is defined by a

membership function introduced by Zadeh [23], mA;

which associates with each point in x a real number in

the closed interval [0,1] as given in Eq. (3).

A ¼ fðx;mAðxÞÞjxAXg;

ð3Þ

A brief overview on the fuzzy modeling algorithms

was carried out by Alvarez Grima [22]. The Mamdani

fuzzy model, the Tagaki-Sugeno-Kang fuzzy model, the

Tsukamoto fuzzy model and Singleton fuzzy model are

the most commonly used. In this study, the Mamdani

fuzzy algorithm was preferred. Because, the Mamdani

method is perhaps the most appealing fuzzy method to

employ in engineering geological problems [22]. Work-

ers such as den Hartog et al. [24], Gokceoglu [10],

Sonmez et al. [25] and Nefeslioglu et al. [26] have applied

fuzzy approach to the engineering geology and rock

mechanics problems.

For the problem at hand: To transform the percen-

tages of the agglomerate constituents to linguistic terms,

an arithmetic scale was used (Table 4). According to this

scale, the percentages of the constituents were classified

into five linguistic groups. In addition, to express the

range of UCS values, an interval [5.7,55.0] to [0,1], a

linear relationship between UCS and characteristic

membership value was constructed:

mUCS¼ 0:02UCS ? 0:0947;

ð4Þ

where mUCSis membership value for uniaxial compres-

sive strength and UCS is the uniaxial compressive

strength in MPa.

ARTICLE IN PRESS

17

19

21

23

25

17 19212325

Unit weight calculated by

image classification (kN/m3)

Unit weight determined in

laboratory (kN/m3)

1:1

r = 0.83

Fig. 8. Comparisons between unit weights estimated from Eq. (1) and

those determined from laboratory testing.

Table 4

Scale transforming constituent percentage to linguistic terms

Range of constituent

percentage (%)

Linguistic term

0–20

21–40

41–60

61–80

81–100

Very poor in X constituent (VP)

Poor in X constituent (P)

Moderate in X constituent (M)

Rich in X constituent (R)

Very rich in X constituent (VR)

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

723

Page 8

Atriangular petrographicalclassificationchart

for the Ankara Agglomerates was developed by

Gokceoglu [10] (Fig. 9). The classification chart includes

25 sub-triangles and each sub-triangle characterizes

different petrographical compositions expressed in

linguistic terms. Fig. 10 illustrates the distribution of

theUCSsofthe 164

obtained from Gokceoglu [10] and during this study.

It is evident that the rock samples having approximately

the same strength populate many of the same sub-

triangles of the petrographical classification chart except

for some small deviations. While some sub-triangles do

not include agglomerate specimens in the previous study

performed by Gokceoglu [10], the sub-triangles espe-

cially composed of black andesite and tuff were filled by

using the different core samples in this study. Never-

theless, some sub-triangles have none or limited data

(see Fig. 10). It is evident that this situation creates a

negative effect on the accuracy of the results. If the sub-

triangles had included more or less the same number of

samples, the results would have been more accurate.

However, this condition is controlled by the proportion

of the constituents of the samples involved. In order to

eliminate ‘‘such no-data’’ deviations and provide an

interpolation, a total of 15 data-driven membership

functions were formed by simple regression techniques.

When forming the membership functions, the following

agglomeratespecimens

properties for constructing the membership functions

described by Dombi [27] were taken into consideration.

(a) All membership functions are continuous and

linear.

(b) All membership functions map in an interval ½a;b?

to ½0;1?; m½a;b?-½0;1?:

(c) The membership functions are (i) either monotoni-

cally increasing, or (ii) monotonically decreasing, or

ARTICLE IN PRESS

VR,VP,VP

R, P, VP

M, M, VP

P, R, VP

VP, VR, VP

VP, R, P VP, M, M

VP, P, R

VP, VP, VR

VP, R,V P VP, M, PVP, P, M

VP,VP, R

P, M, PP, P, M P, VP, R

P, M, VP P, P, PP, VP, P

M, P, PM, VP, P

M, P, VPM, VP, P

R, VP, P

R, VP, VP

PINK ANDESITE

TUFF

BLACK ANDESITE

20

0

40

60

80

100

0

20

40

60

80

1000

20

40

60

80

100

TUFF

BLACK ANDESITE

PINK ANDESITE

Guide for axes

Order: Black andesite, pink andesite, tuff

VR :Very rich

R :Rich

M :Moderate

P :Poor

VP :Very poor

Fig. 9. Triangular petrographical classification chart for Ankara Agglomerates.

µUCS ≤ 0.19

0.19 < µUCS ≤ 0.39

0.39 < µUCS ≤ 0.59

0.59 < µUCS ≤ 0.79

0.79 < µUCS ≤ 0.99

0.99 < µUCS ≤ 1.00

TUFF

0 20 40 60 80 100

100

80

20

0

60

40

4060

20

0 100

80

PINK ANDESITE

BLACK ANDESITE

Fig. 10. Distribution of the agglomerate specimens on the triangular

petrographical classification chart.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

724

Page 9

(iii) could be divided into a monotonically increas-

ing or decreasing part.

(d) The monotonous membership functions on the

whole interval are (i) either convex functions, or

(ii) concave functions, or (iii) there exists a point c

in the interval ½a;b? such that ½a;c? is convex and

[c;b] is concave (called s-shaped functions).

(e) Monotonically increasing

property uðaÞ ¼ 0;uðbÞ ¼ 1; while monotonically

decreasing functions have the property uðaÞ ¼

1;uðbÞ ¼ 0:

(f) The linear form or linearization of the membership

function is very important.

functions havethe

Five of the membership function graphs characteriz-

ing the axes of the agglomerate constituents are given in

Fig. 11. The general ‘‘If-then’’ rules of these membership

functions are given as follows:

(a) For the black andesite axis on the triangular chart

(BA denotes the percentage of the black andesite).

(a-i) If pink andesite is very poor then mUCS=

0.0095BA+0.0813.

(a-ii) If pink andesite is poor then mUCS=0.011-

BA+0.1013.

(a-iii) If pink andesite is moderate then mUCS=

0.0111BA+0.1288.

(a-iv) If pink andesite is rich then mUCS=

0.009BA?0.1986.

(a-v) If pink andesite is very rich then mUCS=

0.0037BA?0.2708.

(b) For the pink andesite axis on the triangular chart

(PA denotes the percentage of the pink andesite).

(b-i) Iftuffisvery

?0.008PA+0.9521.

(b-ii) If tuff is poor then mUCS=-0.0072PA+0.71.

(b-iii) Iftuffis

?0.0078PA+0.5464.

(b-iv) If tuffis

?0.0095PA+0.3919.

(b-v) Iftuffisvery

?0.0019PA+0.1303.

(c) For the tuff axis on the triangular chart (T denotes

the percentage of the tuff).

(c-i) If black andesite is very poor then mUCS=

?0.0019T+0.3016.

(c-ii) If black andesite is poor then mUCS=

?0.0022T+0.5045.

(c-iii) If black andesite is moderate then mUCS=

?0.0035T+0.7199.

(c-iv) If black andesite is rich then mUCS=

?0.0072T+0.905.

(c-v) If black andesite is very rich then mUCS=

?0.0047T+0.9872.

poor then

mUCS=

moderatethen

mUCS=

richthen

mUCS=

richthen

mUCS=

Membership functions for outputs given at the right

side of the graphs in Fig. 11 were constructed by

employing the 15 membership functions given above.

The numbers of output membership function were 5, 4,

3, 2 and 1 for the linguistic definition of VP, P; M; R and

VR, respectively, as shown in Fig. 11. In other words,

total 45 output membership functions, which are shown

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

OUTPUT

Pink andesite = 10%

Pink andesite = 30%

Pink andesite = 50%

Pink andesite = 70%

Pink andesite = 90%

PERCENTAGE OF BLACK ANDESITE (%)

MEMBERSHIP DEGREE OF UCS, µUCS

FUZZY RULE

VP

PM

RVR

VERY POORVERY RICH

TUFF

PERCENTAGE OF PINK ANDESITE (%)

FUZZY RULE

Tuff = 10%

Tuff = 30%

Tuff = 50%

Tuff = 70%

Tuff = 90%

MEMBERSHIP DEGREE OF UCS, µUCS

0

0.2

0.4

0.6

0.8

1

VP

PM

RVR

VERY POORVERY RICH

BLACK ANDESITE

FUZZY RULE

Black andesite = 10%

Black andesite = 30%

Black andesite = 50%

Black andesite = 70%

Black andesite = 90%

PERCENTAGE OF TUFF (%)

MEMBERSHIP DEGREE OF UCS, µUCS

0

0.2

0.4

0.6

0.8

1

0 20 406080 100

0 204060 80100

0 204060 80100

VP

PM

RVR

VERY POORVERY RICH

PINK ANDESITE

(a)

(b)

(c)

Fig. 11. Membership functions characterizing (a) black andesite, (b)

pink andesite and (c) tuff.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

725

Page 10

on the right side of the graphs given in Fig. 11, were

extracted for three constituents of agglomerates. For

example, if PA is 35, BA is 42 and T is 23, the linguistic

terms of the constituents will be poor for PA, moderate

for BA and poor for T: In this situation, rules (a-ii), (c-

iii) and (b-ii) must be considered. The extraction stages

of outputs of this example are given in the following.

The output fuzzy sets are also illustrated in Fig. 12.

(a) If PA is P and BA is M then mUCSis Output1;

Output1={0/0.5728, 1/0.6838, 0/0.7848}.

(b) If BA is M and T is P then mUCS is Output2;

Output2={0/0.7185,1/0.7189, 0/0.7192}.

(c) If T is P and PA is P then mUCS is Output3;

Output3={0/0.7071, 1/0.7078, 0/0.7086}.

By employing the Mamdani fuzzy algorithm con-

sidering the inputs and outputs produced using the UCS

and petrographic data, a zonation in the classification

triangle for the UCS was constructed with the aid of the

triangulation gridding method. Before completing the

zonation on the classification triangle, a defuzzification

process was performed. Aggregation of two or more

fuzzy output sets gives a new fuzzy set in the basic fuzzy

algorithm. In most cases, a result in the form of a fuzzy

set is converted into a crisp result by the by the

defuzzification process [28]. Some defuzzification meth-

ods such as centroid (center of gravity, center of weights,

center of largest area and center of mass of highest

intersected region) and maxima (means of maximums,

maximum possibility and left–right maxima) exist in the

literature. In this study, the center of gravity method for

defuzzification was employed and a fuzzy-based trian-

gular chart for the prediction of UCS of the Ankara

Agglomerate was obtained (Fig. 13).

4. Prediction model for modulus of elasticity

A triangular chart similar to UCS could not be

constructed for estimating modulus of elasticity since

the number of data including modulus of elasticity value

was limited, and the data were distributed heteroge-

neously on the triangular chart. For this reason,

ARTICLE IN PRESS

INPUTS

PA = 35

OUTPUT (µUCS)

BA = 42

T = 23

Aggregation and

Defuzzification

µUCS=0.576

Fig. 12. An example of the determination of mUCS by the fuzzy

inference system for the case described in the text.

TUFF

0 20 40 60 80 100

100

80

20

0

60

40

40

60

20

0

100

80

PINK ANDESITE

BLACK ANDESITE

Membership value for uniaxial compressive strength

1.00

0.990.79

0.590.39

0.19

0.00

Fig. 13. Fuzzy triangular chart for the prediction of UCS of the Ankara Agglomerate.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

726

Page 11

regression analyses techniques were employed to con-

struct the prediction model for estimating the modulus

of elasticity. Regression analyses were performed using a

total of 102 UCSs and the unit weight data obtained

from core samples prepared from tuff matrix, black and

pink andesite blocks and agglomerate. While the

relationship between the Ei and UCS was determined

as power type (Fig. 14a and Eq. (5a)), a linear relation-

ship (Fig. 14b and Eq. (5b)) was found between the Ei

and g with a high correlation coefficient of 0.949 and

0.921, respectively.

Ei¼ 0:4385UCS0:6759

ðr ¼ 0:949Þ;

ð5aÞ

Ei¼ 0:8463g ? 12:264

ðr ¼ 0:921Þ:

ð5bÞ

Although the relations obtained from simple regres-

sions have high correlations, an equation was also

constructed by using a combined parameter (CP) which

includes both the UCS and g as input parameters. To

construct the CP, the UCS relation between Ei was

multiplied by g (Eq. (6a)) by considering the correlation

of coefficient of Eq. (5a). The regression between the Ei

and CP give power-type equation (Fig. 15 and Eq. (6b))

with 0.951.

CP ¼ 0:4385UCS0:6759g;

ð6aÞ

Ei¼ 0:1223CP0:7981

ðr ¼ 0:951Þ:

ð6bÞ

Although the equation given above requires the UCS

and g (which are obtained from core samples), these

parameters can be estimated by using percentage of

constituents of the Ankara Agglomerate. For this

purpose, the percentage of constituents from outcrop

can be determined using image analyses and the unit

weights ofconstituents

22:7kN/m3, gT¼ 16:9kN/m3) are evaluated in Eq. (1)

to obtain the g of the agglomerate. Then, the UCS

of the agglomerate can be obtained from Fig. 13 based

on the percentage of constituents of the Ankara

Agglomerate.

(gBA¼ 24:3kN/m3,

gPA¼

5. Prediction performances for constructed models

Prediction performance of constructed models were

evaluated by employing both cross-checks and perfor-

mance indices namely the ‘‘variance account for’’ (VAF)

and ‘‘root mean square error’’ (RMSE) given in

Eqs. (7a) and (7b), respectively.

VAF ¼

1 ?varðy ? y0Þ

varðyÞ

??

?100;

ð7aÞ

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

N

1

XN

i¼1ðy ? y0Þ2

r

;

ð7bÞ

where y and y0are the measured and the predicted

values, respectively, and N is the number of data. If the

model has excellent prediction capacity the VAF and

RMSE will be 100% and zero, respectively.

The cross-correlation between measured and pre-

dicted UCS obtained from the fuzzy triangular chart is

illustrated in Fig. 16a and the correlation coefficient was

obtained as 0.8. The VAF and RMSE performance

indices were calculated as 56.9% and 7.3, respectively.

Both cross-correlation and performance indices reveal

ARTICLE IN PRESS

(a)(b)

0

2

4

6

8

10

12

0 20406080100120140

Uniaxial Compressive Strength, UCS (MPa)

Ei = 0.4385(UCS)0.6759

r = 0.949

Modulus of Elasticity, Ei (GPa)

Ei = 0.8463γ − 12.264

r = 0.921

0

2

4

6

8

10

12

151719 21232527

Unit weight, γ (kN/m3)

Modulus of Elasticity, Ei (GPa)

Fig. 14. Relations between the modulus of elasticity and UCS (a), and unit weight (b).

Combined Parameter, CP

0

2

4

6

8

10

12

050 100150200250300

Modulus of Elasticity, Ei (GPa)

E =0.1223CP

r=0.951

0.7981

i

Fig. 15. Relation between combined parameter and modulus of

elasticity.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

727

Page 12

that the prediction performance of the fuzzy triangular

chart is sufficient for practical uses.

The prediction performance of the regression model

constructed to predict the modulus of elasticity was also

controlled by considering both cross-correlation and

performance indices. The correlation coefficient was

determined as 0.942 for correlation between predicted

and measured values of Ei(Fig. 16b). In addition to a

high correlation coefficient, the performance indices

were found to be 88.8% and 0.84 for VAF and RMSE,

respectively. These values reveal that the prediction

capacity of the equation is extremely high.

6. Results and conclusions

While the prediction model for estimating uniaxial

compressive strength was constructed based on fuzzy

membership functions considering the constituents of the

Ankara Agglomerates, a prediction model based on

regression was constructed to predict the modulus

elasticity of the Ankara Agglomerate. The following re-

sults and conclusions can be drawn from the present study.

(a) Utilization of fuzzy logic in indirect determination

of uniaxial compressive strength provided high

performance prediction capacity. The prediction

capacity of the regression equation for estimating

modulus of elasticity is extremely high.

(b) The developed prediction models can be used only

for the Ankara Agglomerates. They should not be

used directly for prediction of uniaxial compressive

strength of other agglomerates or other bimrocks,

for that matter. However, the methodology em-

ployed in the present study could be considered as a

tool for predictive models for other rocks.

(c) Due to the difficulties encountered during the

sampling and sample preparation process, the

number of data especially used in the model of

modulus of elasticity is restricted. The fuzzy trian-

gular chart similar to uniaxial compressive strength

could be developed when sufficient data is available.

(d) The main limitation of the approach described

herein is that the weathering of the outcrops causes

changes in the colors (and thus the gray-scale tones)

of the constituents. For this reason, image proces-

sing must be performed on photographs taken from

fresh surfaces. If the surfaces are affected from

slight weathering, the color photographs can be

used to apply the image classification. In addition, it

is possible to use other conventional techniques

such as scan-line surveys and node point counting

for the determination of block portion.

(e) It is obvious that some generalized equations or

models for predicting the uniaxial compressive

strength and modulus of elasticity of geological

mixtures or fragmented rocks such as conglomerate,

agglomerate, etc. are necessary.

Acknowledgements

This research was supported by TUBITAK (The

Scientific and Technical Research Council of Turkey)

(Project No. 102Y033). The authors are very apprecia-

tive of Dr. Edmund Medley for his great contributions

on the manuscript, and his kind help.

References

[1] ISRM (International Society for Rock Mechanics). In: Brown

ET, editor. ISRM suggested method: rock characterization,

testing and monitoring. London: Pergamon Press; 1981. 211p.

[2] Medley E. Using stereologic methods to estimate the block

volumetric proportion in melanges, similar block-in-matrix rocks

(bimrocks). In: Proceedings of the Seventh Congress of the

ARTICLE IN PRESS

y = 0.96x

r = 0.80

0

10

20

30

40

50

60

0 10 2030 4050 60

1:1

Predicted UCS

Measured UCS

RMSE = 7.3

VAF = 56.9 %

n = 164

Predicted Modulus of Elasticity (GPa)

Measured Modulus of Elasticity (GPa)

RMSE = 0.84

VAF = 88.8 %

n = 103

(a)

(b)

0

2

4

6

8

10

12

02468 1012

1:1

y = 0.99x

r = 0.94

Fig. 16. Cross-correlation between the predicted and measured UCS

(a), and the predicted and measured modulus of elasticity values (b) of

the agglomerate specimens.

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

728

Page 13

International Association of Engineering Geology, Lisbon,

Portugal. Rotterdam: AA Balkema, 1994.

[3] Lindquist ES, Goodman RE. The strength and deformation

properties of the physical model m! elange. In: Nelson PP, Laubach

SE, editors. Proceedings of the First North American Rock

Mechanics Conference (NARMS), Austin, Texas. Rotterdam:

AA Balkema; 1994.

[4] Medley E, Goodman RE. Estimating the block volumetric

portion of m! elanges and similar block-in-matrix rocks (bimrocks).

In: Nelson PP, Laubach SE, editors. Proceedings of the First

North American Rock Mechanics Conference (NARMS), Austin,

Texas. Rotterdam: AA Balkema; 1994.

[5] Bell FG. The physical and mechanical properties of the Fell

sandstone, Northumberland, England. Eng Geol 1978;12:1–29.

[6] Doberenier L, DeFreitas MH. Geotechnical properties of weak

sandstone. Geotechnique 1986;36:79–94.

[7] Hawkins A, McConnell BJ. Influence of geology on geomecha-

nical properties of sandstones. In: Proceedings of the Seventh

International Congress on Rock Mechanics, Aachen, Germany.

Rotterdam: AA Balkema; 1990. p. 257–60.

[8] Ulusay R, Tureli K, Ider MH. Prediction of engineering proper-

ties of a selected litharenite sandstone from its petrographic

characteristics using correlation and multivariate statistical

techniques. Eng Geol 1994;37:135–57.

[9] Gokceoglu C, Kasapoglu KE, Sonmez H. Prediction of uniaxial

compressive strength of Ankara Agglomerates from their

petrographical composition. In: Moore D, Hungr O, editors.

Proceedings of the Eighth International Congress of IAEG and

the Environment, Vancouver, Canada. Rotterdam: AA Balkema;

1998. p. 455–9.

[10] Gokceoglu C. A fuzzy triangular chart to predict the uniaxial

compressive strength of the Ankara Agglomerates from their

petrographic composition. Eng Geol 2002;66:39–51.

[11] Alvarez Grima M, Babu& ska R. Fuzzy model for the prediction of

unconfined compressive strength of rock samples. Int J Rock

Mech Min Sci 1999;36:339–49.

[12] Ali M, Chawathe A. Using artificial intelligence to predict

permeability from petrographic data. Comput Geosci 2000;

26:915–25.

[13] Finol J, Guo YK, Jing XD. A rule based fuzzy model for the

prediction of petrophysical rock parameters. J Petrol Sci Eng

2001;29:97–113.

[14] Kayabasi A, Gokceoglu C, Ercanoglu M. Estimation the

deformation modulus of rock masses: a comparative study. Int

J Rock Mech Min Sci 2003;40:55–63.

[15] Kasapoglu KE. Ankara kenti zeminlerinin jeo-muhendislik

ozellikleri. Thesis for Association of Professorship, Hacettepe

University, Institute of Earth Sciences, 1980. 206p (in Turkish,

unpublished).

[16] Lindquist ES. The strength, deformation properties of melange.

PhD thesis, University of California, Berkeley, 1994. 264p.

[17] Medley E. The engineering characterization of melanges, similar

block-in-matrix rocks (bimrocks). PhD thesis, Department of

Civil Engineering, University of California, Berkeley, 1994. 175p.

[18] Medley E. Orderly characterization of chaotic Franciscan

Melanges. Felsbau Rock Soil Eng-J Eng Geol, Geomech

Tunnelling 2001;19(4):20–33.

[19] Medley E. Uncertainty in estimates of block volumetric propor-

tion in melange bimrocks. In: Marinos PG, Koukis GC,

Tsiambaos GC, Stournas GC, editors. Proceedings of the

International Symposium on Engineering Geology and the

Environment, Athens, Greece. Rotterdam: AA Balkema; 1997.

[20] Lillesand TM, Kiefer RW. Remote sensing and image interpreta-

tion, 2nd ed. New York: Wiley; 1987. 721p.

[21] Nguyen VU. Some fuzzy set applications in mining geomechanics.

Int J Rock Mech Min Sci 1985;22:369–79.

[22] Alvarez Grima M. Neuro-fuzzy modeling in engineering geology.

Rotterdam: AA Balkema; 2000. 244p.

[23] Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–53.

[24] den Hartog MH, Babuska R, Deketh HJR, Alvarez Grima M,

Verhoef PNW, Verbruggen HB. Knowledge-based fuzzy model

for performance prediction of a rock-cutting trencher. Int J

Approx Reason 1997;16:43–66.

[25] Sonmez H, Gokceoglu C, Ulusay R. An application of fuzzy sets

to the geological strength index (GSI) system used in rock

engineering. Eng Appl Artif Intell 2003;16:251–69.

[26] Nefeslioglu HA, Gokceoglu C, Sonmez H. A Mamdani model to

predict the weighted joint density. Lect Notes Artif Intell

2003;2773:I/1052.

[27] Dombi J. Membership function as an evaluation. Fuzzy Sets

Systems 1990;35:1–21.

[28] Berkan RC, Trubatch SL. Fuzzy system design principles,

building fuzzy if-then rule bases. The Institute of Electrical and

Electronics Engineers Inc.: New York; 1997. 496p.

ARTICLE IN PRESS

H. Sonmez et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 717–729

729