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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 crosscheck 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 BellmanZadeh 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 knowledgebased fuzzy model for performance prediction of a rockcutting 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 rulebased 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 fuzzymean defuzzification which allows for straightforward adaptation of the consequent membership functions in order to finetune the model performance to the data. The expert knowledge is coded as ifthen 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
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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 BeytepeAnkara, Turkey
Accepted 11 January 2004
Abstract
Determination of the uniaxial compressive strength (UCS) and modulus of elasticity of blockinmatrix 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 ‘‘ifthen’’ 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, NXsized 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 ‘‘blockinmatrix 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.: +903122977700; fax: +90312299
2034.
Email address: haruns@hacettepe.edu.tr (H. Sonmez).
13651609/$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 fuzzybased
triangular prediction chart prepared by Gokceoglu [10]
to predict the UCS of Ankara Agglomerates from their
constituents was reconstructed 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 thinsection
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 scanline 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 equidimensional 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 blocksize distribu
tion graph (Fig. 3).
Image classification and the nodepointcounting
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 grayscale tonal
spectrum of 0–255 grayscale 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. Blocksize 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
grayscaled 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 minimumdistancetomeans classifier (Fig. 4b),
since the minimumdistancetomeans 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 nodepointcounting 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 pointcounting 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.
Crosscorrelation between two methods shows that
the results of the image classification and nodepoint
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 nodepointcounting 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 blockin
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 gaugetype 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 gaugetype
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. Crosscheck between the results of image processing and node
point counting.
0
5
10
15
20
25
30
35
02468
10
Axial bulk strain (x106)
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
equidimensional 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) crosssection through failed specimens perpendicular to failure surface
(NXsized 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 reallife 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 rulebased fuzzy modeling can be
summarized as follows [22]:
(a) Allows explicit expression of the knowledge of the
system via fuzzy ‘‘Ifthen’’ 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 TagakiSugenoKang 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 subtriangles and each subtriangle 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 subtriangles do
not include agglomerate specimens in the previous study
performed by Gokceoglu [10], the subtriangles espe
cially composed of black andesite and tuff were filled by
using the different core samples in this study. Never
theless, some subtriangles 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 nodata’’ deviations and provide an
interpolation, a total of 15 datadriven 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 sshaped 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 ‘‘Ifthen’’ 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).
(ai) If pink andesite is very poor then mUCS=
0.0095BA+0.0813.
(aii) If pink andesite is poor then mUCS=0.011
BA+0.1013.
(aiii) If pink andesite is moderate then mUCS=
0.0111BA+0.1288.
(aiv) If pink andesite is rich then mUCS=
0.009BA?0.1986.
(av) 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).
(bi) Iftuffisvery
?0.008PA+0.9521.
(bii) If tuff is poor then mUCS=0.0072PA+0.71.
(biii) Iftuffis
?0.0078PA+0.5464.
(biv) If tuffis
?0.0095PA+0.3919.
(bv) Iftuffisvery
?0.0019PA+0.1303.
(c) For the tuff axis on the triangular chart (T denotes
the percentage of the tuff).
(ci) If black andesite is very poor then mUCS=
?0.0019T+0.3016.
(cii) If black andesite is poor then mUCS=
?0.0022T+0.5045.
(ciii) If black andesite is moderate then mUCS=
?0.0035T+0.7199.
(civ) If black andesite is rich then mUCS=
?0.0072T+0.905.
(cv) 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 (aii), (c
iii) and (bii) 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 fuzzybased 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 powertype 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 crosschecks 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 crosscorrelation 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 crosscorrelation 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 crosscorrelation 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 grayscale 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 scanline 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.
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ARTICLE IN PRESS
y = 0.96x
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Predicted UCS
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ARTICLE IN PRESS
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729
Supplementary to (1)

Models to predict the uniaxial compressive strength and the modulus of elasticity for Ankara Agglomerate