Models for estimating uniaxial compressive strength and elastic

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Građevinar 1/2016
DOI: 10.14256/JCE.1431.2015
Primljen / Received: 31.7.2015.
Ispravljen / Corrected: 24.12.2015.
Prihvaćen / Accepted: 18.1.2016.
Dostupno online / Available online: 10.2.2016.
Models for estimating uniaxial
compressive strength and elastic modulus
Authors:
Subject review
Zlatko Briševac, Petar Hrženjak, Renato Buljan
Models for estimating uniaxial compressive strength and elastic modulus
Zlatko Briševac, PhD. Min.
University of Zagreb
Faculty of Mining, Geology and Petroleum
Engineering
[email protected]
The most significant methods for estimating the uniaxial compressive strength
and Young’s modulus of intact rock material, formulated in the scope of numerous
previous studies, are briefly presented in the paper. The proposal for classification
of these methods, according to which they can generally be divided into simple and
complex methods, is also presented. Simple methods include various diagrams and
tables and the use of simple regression equations, while complex methods comprise
the use of multiple regression equations, fuzzy logic models, neural networks,
evolutionary programming, and regression trees.
Key words:
estimation, uniaxial compressive strength, Young’s modulus, intact rock material
Pregledni rad
Zlatko Briševac, Petar Hrženjak, Renato Buljan
Modeli za procjenu jednoosne tlačne čvrstoće i modula elastičnosti
Assist.Prof. Petar Hrženjak, PhD. Min.
University of Zagreb
Faculty of Mining, Geology and Petroleum
Engineering
[email protected]
U ovom radu ukratko je izložen pregled najznačajnijih metoda za procjenu jednoosne
tlačne čvrstoće i Yangovog modula elastičnosti intaktnog stijenskog materijala koje
su nastale u okviru mnogobrojnih istraživanja. Iznesen je prijedlog podjele metoda
prema kojemu se one u osnovi mogu podijeliti na jednostavne i složene metode.
Jednostavne metode uključuju različite dijagrame i tablice te primjenu jednadžbi
jednostruke regresije, a složene metode uključuju primjene jednadžbi višestruke
regresije, modela neizrazite logike, neuronskih mreža, evolucijskog programiranja i
regresijskog stabla.
Ključne riječi:
procjena, jednoosna tlačna čvrstoća, Yangov modul elastičnosti, intaktni stijenski materijal
Übersichtsarbeit
Renato Buljan, PhD. Geol.
Zlatko Briševac, Petar Hrženjak, Renato Buljan
Croatian Geological Surveyt
Modelle zur Bewertung der einachsigen Druckfestigkeit und des
Elastizitätsmoduls
[email protected]
In dieser Arbeit wird ein Überblick der wichtigsten Methoden zur Bewertung der
einachsigen Druckfestigkeit und Young’s Elastizitätsmodul, die im Rahmen zahlreicher
Untersuchungen entstanden sind, bei intaktem Felsmaterial gegeben. Es wird ein
Vorschlag zur Aufteilung in einfache und komplexe Methoden gegeben. Einfache
Methoden umfassen verschiedene Diagramme und Tabellen, sowie die Anwendung von
Einzelregressionsgleichungen. Komplexe Methoden beziehen sich auf die Anwendung
von Mehrfachregressionsgleichungen, Modelle der Fuzzy-Logik, neuronale Netze,
evolutionäre Programmierung und Regressionsbäume.
Schlüsselwörter:
Bewerung, einachsige Druckfestigkeit, Young’s Elastizitätsmodul, intaktes Felsmaterial
GRAĐEVINAR 68 (2016) 1, 19-28
19
Građevinar 1/2016
Zlatko Briševac, Petar Hrženjak, Renato Buljan
1. Introduction
description is given as shown in Table 1. The strength index
determined by point load test can assist in this estimate.
Various investigations are normally carried out both in the
scope of construction projects, and for making other technical
interventions in the rock mass. These investigations routinely
include realization of laboratory tests aimed at determining
physicomechanical properties of intact rock material. At that,
and in addition to the density of materials, one of the most often
defined properties is the uniaxial compressive strength (UCS),
and the Young’s modulus of elasticity (E). However, it often
occurs that samples of the dimensions required for laboratory
testing can not be extracted from such materials. That is
why the need arose already at early stages of development
of rock mechanics to determine the correlation of various
physicomechanical properties of materials, so that one property
can be estimated based of the value of another one. These
interdependencies have proven to be very useful in preliminary
stages of the planning and design activities. Many researchers
have studied the possibility for estimating the UCS and E
values based on values of other material properties. Although
simple interdependencies were used in the beginning, current
estimation methods are proving to be increasingly complex.
2. Estimate models
2.1. Tables and diagrams
In simplest cases, the UCS estimation can be made according
to an index test based on the method recommended by the
International Society for Rock Mechanics, as presented in Table
1. This table also contains useful additions made by Marinos
and Hoek [1]. In this case, the estimate is made using portable
equipment (nail, knife, geological hammer) and an appropriate
Figure 1. UCS as related to porosity and velocity of ultrasonic waves
for limestones, [2]
Diagrams can also be quite useful in situations requiring rapid
decision making. The diagram showing interdependence
between the density, porosity and velocity of ultrasound waves
Table 1. Determination of uniaxial compressive strength by hand held accessories
Grade
Description
UCS
IS(50)
Field identification
Rock types
R6
Extremely
strong rock
>250
> 10
Specimen can only be pull apart by a geological
hammer
fresh basalt, chert, diabase,
gneiss, granite, and quartzite
R5
Very strong
rock
100 - 200
4 - 10
Specimen requires many blows of geological hammer
to fracture it.
amphibiolite, sandstone,
basalt, gabbro, gneiss,
granodiorite, limestone,
marble, rhyolite, and tuff
R4
Strong rock
50 - 100
2-4
Specimen requires more than one blow by geological
hammer to fracture it.
limestone, marble, sandstone,
and schist
R3
Medium
strong rock
25 - 50
1-2
Cannot be scraped or peeled with a pocket knife;
specimen can be fractured with a single firm blow of a
geological hammer.
phyllite, schist, siltstone
R2
Weak rock
5 - 25
-
Can be peeled by a pocket knife with difficulty; shallow
indentations made by firm blow with a point of
geological hammer.
chalk, rock salt, claystone, marl,
siltstone, schist
R1
Weak rok
1-5
-
Crumbles under firm blows with point of geological
hammer; can be peeled by pocket knife
highly weathered or altered
rock, schist
R0
Extremely
weak rock
0,25 - 1
-
Indented by thumbnail.
stiff fault gouge
UCS – uniaxial compressive strength [MPa]; IS(50) – strength index [MPa]
20
GRAĐEVINAR 68 (2016) 1, 19-28
Građevinar 1/2016
Models for estimating uniaxial compressive strength and elastic modulus
(Figure 1), presented by Price [2], can be used to make a rough
estimate of the UCS value of limestone material.
The best known diagram for the relationship between E and
UCS is the one presented by Deere and Miller [3] (Figure 2).
2.2. Simple regression equations
Simple regression equations comprise relations defined for the
estimation of UCS and E values as dependent variables based
on the tested value of another property that constitutes an
independent variable. Various equations for estimating UCS
and E values have been defined by regression analysis based
on results obtained by testing physicomechanical properties
of intact rock materials. Thus, for instance, Table 2 shows
equations presented by various authors for some rock types,
where UCS and E values are estimated based on a known
porosity value.
Table 2. Uniaxial regression equations with porosity
Equation
Figure 2 UCS to E ratio, [3]
Type of rock
Authors
UCS = 183 - 16,55 n
granite
Turgul and Zarif, 1999. [5]
UCS = 74,4 e
sandstone
Palchik, 1999. [6]
E = 10,10 - 0,109 n
porous rocks
Leite and Ferland, 2001. [7]
UCS = 210,1 e-0,821 n
E = 37,9 e– 0,863 n
shale, claystone,
siltstone
Lashkaripour, 2002. [8]
UCS = 273,1 e -0,076 n
porous chalk
Palchik and Hatzor, 2004. [9]
UCS = 195,0 e -0,21 n
sandstone
Tugrul, 2004. [10]
-0,04 n
UCS - unconfined compressive strength [MPa]; E - elastic modulus [GPa];
n - porosity [%]
Table 3. Simple regression equations with strength index
Equation
Type of rock
Authors
UCS = 15,3 IS(50) + 16,3
all rocks
D’Andrea, and ost.,
1964. [11]
UCS = 16 IS(50)
sedimentary
rocks
Read et all., 1980.
[12]
UCS = (20 to 25) IS(50)
all rocks
ISRM, 1985. [13]
UCS = (od 14,5 do 27) IS(50)
limestone
Romana, 1999.
[14]
UCS = 24,4 IS(50)
hard rocks
UCS = 3,86 I
2
S(50)
Figure 3. Schmidt hardness and UCS ratio, [4]
The diagram published in Miller’s dissertation [4] (Figure 3) can
be used to estimate the UCS value of intact rock based on the
Schmidt hardness (SRH) and unit weight of rock, regardless of
rock type that is being tested.
GRAĐEVINAR 68 (2016) 1, 19-28
+ 5,65 IS(50) weak rocks
UCS = 7,3 IS(50)1,71
limestone,
sandstone, marl
UCS = 24,8 IS(50) - 39,6
rocks with n < 1 %
UCS = 10,2 IS(50) + 23,4
rocks with n > 1 %
UCS = 10,58 IS(50)
all rocks
1,14
UCS = 10,46 IS(50)1,12
sedimentary
rocks
UCS = 6,65 IS(50)1,34
igneous rocks
UCS = 18,15 IS(50)
metamorphic
rocks
Quane and Russel,
2003. [15]
Tsiambaos and
Sabatakakis, 2004.
[16]
Kahraman and ost.,
2005. [17]
Tsallas and ost.,
2009. [18]
UCS - unconfined compressive strength ([MPa]; IS(50) - strength index [MPa]
21
Građevinar 1/2016
Zlatko Briševac, Petar Hrženjak, Renato Buljan
The determination of strength index IS(50) by point load test is
an index test that has been developing from the very start for
the very purpose of estimating the UCS value, so that there
are many papers in which the corresponding equations are
published. The most significant equations for estimating the
UCS value of various rocks similar to those found in Croatia are
presented in Table 3.
It should be noted with regard to Table 3 that exponential
equations are more accurate than linear-form equations for
almost all rock types, except for metamorphic rocks where
linear equation has proven to be slightly more accurate [18].
Other than the strength index, the hardness of materials
determined by Schmidt hammer has also often been used in
these estimations. For instance, the UCS and E values can be
successfully estimated based on Schmidt hardness. Some
examples of equations developed for this purpose are presented
in Table 4. Local experience in the UCS and E estimation [19],
gained during intensive motorway infrastructure development
in Croatia, when 518 boreholes over 7000 m in total length
were analysed, reveal that the dependence between the UCS
and Schmidt hardness greatly deviates in [4, 20, 21] from
correlations presented in this paper.
Table 4. Simple regression equations with Schmidt hardness
Equation
Type of rock
Authors
UCS = 4,29 SRH – 67,52
E = 1,94 SRH – 33,93
33 limestone types
Sachpazis, 1990
[20]
UCS = 2,21e(0,07 SRH)
E = 0,00013 SRH3,09
chalk, two
limestone types,
sandstone, marble,
syenite, granite
Katz and ost.,
2000. [21]
UCS = e(0,818+0,059SRH)
E = e(1,146+0,054 SRH)
gypsum
Yilmaz and Sendir,
2002. [22]
UCS = 0,0028 SRH 2,584
E = 0,0987 SRH 1,5545
travertine,
limestone,
dolomitic limestone
and schist
Yagiz, 2009. [23]
UCS - uniaxial compressive strength [MPa]; E - elastic modulus [GPa];
SRH - Schmidth hardness
Table 5. Simple regression equations with P-wave velocities
Equation
Type of rock
Authors
UCS = 9,95 vP 1,21
dolomite, sandstone,
Kahraman, 2001.
marl, limestone,
[24]
diabase, serpentinite
UCS = 31,5 vP – 63,7
E = 10,67 vP – 18,71
dolomite, marble
and limestone
Yasar and Erdogan,
2004. [25]
E = 2,06 vP2,78
limestone, marble
and sandstone
Moradian and
Behnia, 2009. [26]
UCS - uniaxial compressive strength [MPa]; E - elastic modulus [GPa];
vP - P-wave velocity [km/s]
22
In addition, the study of velocity of passage of ultrasonic
p-waves (vP) through samples of various materials has enabled
establishment of various ratios, and hence also of simple
regression equations that are shown in Table 5.
Simple regression equations that were used to estimate the
UCS and E values were linear and nonlinear in form. It was
established that better estimates were obtained with nonlinear
forms such as general power equations or exponential
equations. It should be noted that simple regression models
are often evaluated in research papers through correlation
coefficients and/or coefficients of determination. Almost
ideal values have been obtained by some authors, and so the
coefficients of determination for E and UCS amount to as much
as 0.99 [21] and 0.98 [22], respectively. This can however be
misleading as the use of more rigorous estimation methods
such as the adjusted R2, root mean square error (RMSE),
Akaike information criterion, or cross-validation, would
certainly show that the models are in fact not so ideal. Practical
usability of a model where complex sample preparation is
required for the independent variable determination, such as
in VP determination, is questionable.
2.3. Multiple regression equations
The multiple linear regression is generally presented with
equation (1):
Y = β0 + β1X1 + β2X2 + ... + βkXk + ε(1)
where is:
- dependent variable
Y
X1, X2, …, Xk- independent variables
βi - denotes contribution of the independent variable Xi
ε
- random error [27].
The linear form of the multiple regression equation completely
dominates in models for estimation of the UCS and E values.
Similarly, multiple regression models are predominantly
developed for comparison with models based on other methods.
Models presented below are the models made for rock material
similar to that prevailing in Croatia, and the UCS estimation is
given in MPa, while E is determined in GPa.
Thus Alvarez Grimaa and Babuška [28] prepared a multiple
regression model (2) based on test results for materials
classified as sandstones, limestones, dolomites, granites, and
granodiorites. The multiple regression model is presented as
follows:
UCS = – 246,804 + 0,386 Ls + 39,268 ρ – 1,307 n
(2)
where is:
Ls - hardness defined with the Equotip hardness tester
ρ - density [kg/m3]
n - porosity [%].
GRAĐEVINAR 68 (2016) 1, 19-28
Models for estimating uniaxial compressive strength and elastic modulus
Građevinar 1/2016
A similar model (3) created by testing the same rock types was
developed by Meulenkamp and Alvarez Grima [29]. This model
is presented with the following equation:
UCS = – 23,859 + 0,48 SRH + 1,863 IS(50) + 0,248 w + 7,972 vP
(3)
where is:
SRH- Schmidt hardness
IS(50) - strength index defined by the point load test [MPa]
w - water content [%]
vP -velocity of ultrasonic P-waves [km/s].
UCS = 0,25 Ls + 28,14 ρ – 0,75 n – 15,47 GS – 21,55 RT
where is:
Ls - hardness defined with the Equotip hardness tester
ρ - density [kg/m3]
n - porosity [%].
GS - grain size
RT - rock type.
Gokceoglu and Zorlu [30] prepared multiple regression models
(4) and (5) for weak, fractured, and thin-bedded rocks that are
represented with the following expressions:
UCS = – 225 + 0,0065 vP + 1,468 BPI + 4,094 IS(50) + 2,418 TS
(4)
E = – 0,038 + 0,003 vP + 0,892 BPI + 3,568 IS(50)
(5)
where is:
vP -ultrasonic velocity of P-waves [m/s]
BPI - block punch strength index [MPa]
IS(50) - strength index defined by the point load test [MPa]
TS - tensile strength [MPa]
Karakus and Tutmez [31] developed the UCS estimation model
(6) based on the testing of marble, limestone and dacite
originating from Malatya and Elazig regions in Turkey. This
model is represented as follows:
UCS = - 35,9 + 0,89 SRH + 13,1 IS(50) – 1,68 vP(6)
where is:
SRH - Schmidt hardness
IS(50) - strength index defined by the point load test [MPa]
vP - velocity of ultrasound waves [km/s].
Kahraman et al [32] developed the models (7) and (8) for tectonic
breccias. These models are represented with the following
expressions:
UCS = – 35,09 – 0,33 VBP + 35,38 vS (7)
E = – 103,88 – 0,16 VBP + 39,65 ρ + 4,2 vP + 4,33 vS(8)
where is:
VBP - the volume percent of fragments [%]
ρ - density [kg/m3]
vS - velocity of ultrasonic S-waves [km/s]
vP - velocity of ultrasonic P-waves [km/s].
Yilmaz and Yuksek developed the models (9) and (10) for natural
gypsum [33]
GRAĐEVINAR 68 (2016) 1, 19-28
(9)
E = 36,315 + 0,64 SRH + 2,254 IS(50)+ 0,935 w + 12,838 vP(10)
Based on the analysis of limestones, marbles, and dolomites
from Iran, Heidari et al [34] developed a nonlinear multiple
regression model (11) for the estimation of E value. This model
is represented with the following expression:
log E =– 0,85448 + 0,91326 log UCS + 0,03198 log n
+ 0,16123 log vp – 0,22327 log ρ(11)
where is:
UCS-uniaxial compressive strength [MPa]
n - porosity [%].
vp - velocity of ultrasonic Pp-waves [km/s]
ρ - density [kg/m3].
Using the most influential petrographic properties of materials,
Manouchehrian et al developed the model (12) for estimating
the UCS of sandstone [35]. The model is represented with the
following expression:
UCS = 38 – 352,26 n – 5,3 Cfc + 10,6 Cf + 93,15 Mp(12)
where is:
n - porosity [%].
Cfc - ferrous carbonate binder percentage [%]
Cf - percentage of iron oxide in cement [%]
Mp - percentage of mica [%].
Equations for estimating the UCS (13) and E (14) values which
use, as independent variables, the frequently determined
physicomechanical properties, were developed based on
testing conducted on 29 types of carbonate rock materials from
19 localities in the Republic of Croatia.
UCS =- 222 + 0,0535 ρ + 0,7801 n + 13,76 IS(50)
+ 1,752 SHRL + 0, 0061 vp
E =
- 182 + 0,0619 ρ + 0,7228 n - 0,459 IS(50)
+ 0,5907 SHRL + 0,0073 vp
(13)
(14)
where is:
ρ - density [kg/m3]
n - porosity [%].
IS(50) - strength index defined by the point load test [MPa]
SRH -Schmidt hardness
vP - velocity of ultrasonic P-waves [m/s].
23
Građevinar 1/2016
The equations (15) and (16) were developed for the mudstone
and wackestone type limestones originating from Croatia [37].
UCS = -106,2093 – 0,04868 ρ + 11,5110 IS(50) + 0,052 vP(15)
UCS = -240,0109 + 1,5087 n + 11,5916 IS(50)+ 0,0522 vP(16)
where is:
ρ - density [kg/m3]
n - porosity [%]
IS(50) - strength index defined by the point load test [MPa]
vP - velocity of ultrasonic P-waves [m/s].
It can be seen from the above presented multiple regression models
that researchers mostly used physicomechanical properties as
independent variables, while petrographic properties of rock
material were less often used. It is interesting to note that the
density and porosity are used together as independent variables
in the models (2), (3), (11), (13), and (14) although, due to great
physical connection of these properties, it can not be claimed that
these variables are independent from one another, as is required by
the mathematical model of multiple regression.
2.4. Models based on fuzzy logic analysis
In the traditional "crisp" logic, a rational claim can be either true or
false. According to fuzzy logic, no claim is fully true or fully false, but
rather a "level of truthfulness" can be attributed to it. Rules have
been set for using this logic, and these rules are a generalisation
of normal Boolean algebra [38]. Although fuzzy logic was initially
used in social sciences, it is now increasingly used in technical
sciences as well, e.g. in risk management on construction projects
[39], but also in various evaluations. Most significant and the most
often used models based on fuzzy logic and fuzzy reasoning are
the Mamdani and Sugeno models.
Based on Mamdani model, Gokceoglu and Zorlu developed a model
for estimation of USC and E values based on the input values
similar to those used in models (4) and (5) [30]. The same type of
fuzzy model was used by Karakus and Tutmez for development of
the UCS estimation model. The investigation was made for nine
different rock types and for a total of 305 samples. The input data
for the model were laboratory testing results as in model (6) [31]. By
comparing fuzzy model with the multiple regression model, these
researchers concluded that a better UCS estimation is obtained
by using the fuzzy model [30, 31]. Alvarez and Babuska developed
the UCS estimation model using theoretical premises from TakagiSugeno model based on results of testing involving 226 intact
rock material samples classified as sandstones, limestones,
dolomites, granites and granodiorites. Input data were the same as
independent variables from equation (2). Estimation results were
compared with the multiple regression modelling results, and also
with the results obtained via neural network that was established
using the same input data. It was concluded that better estimation
was obtained by fuzzy model compared to multiple regression
model, while the neural network provided better estimation
24
Zlatko Briševac, Petar Hrženjak, Renato Buljan
than fuzzy model for low and high UCS values [28]. Subsequent
development of programming techniques resulted in the situation
in which fuzzy models were not often developed on their own, but
rather the fuzzy logic became a part of other methods, e.g. in neural
networks.
2.5. Estimation by neural networks
Neural networks consist of the systems of input and output
values constituting nodes or neurons and the links or synapses
between them, through which attempts are made to artificially
simulate the way the human brain functions. In most cases,
neural networks are not realised as hardware systems but rather
as software programs, i.e. using the programming code. Neural
networks are not programmed as algorithms with accurately
determined relationships, but rather as algorithms capable of
learning through examples. They are presented with examples
and solutions to these examples, and algorithms can then
automatically generate empirical rules. Before such algorithms
can be used for estimating certain values, computer programs
have to be "trained" on as set of known and required values
[38]. Neural networks are increasingly used for solving various
problems and tasks in many areas and so, for instance, they are
used in civil engineering in the design of timber structures [40], in
water management [41], determination of liquefaction potential
[42], design of railway embankments [43], etc.
The possibility of estimating UCS and E values using neural networks
was analysed during study of carbonate rocks from 19 localities in
the Republic of Croatia, from which a total of 425 samples were
taken and tested in laboratory to determine their density, porosity,
strength index, Schmidt hardness, UCS, and E. Thirteen multiple
regression models and 65 neural networks type MLP (multilayer
perceptron) and RBF (radial basis function) were developed in the
program package Statistica 10. An extensive scientific literature was
analysed in the scope of this research [36]. Due to a great number of
neural network types presented by various authors, there are several
possible ways for classifying neural networks for various areas
of application. The authors of this paper consider, based on their
experience, that the most appropriate is the simplified modification
of the Gupta and Rao classification [44] (Figure 4), which primarily
starts from the analysis of artificial neural networks used in the
estimation of UCS and E values, which is presented in paper [36].
Figure 4. Classification of neural networks
GRAĐEVINAR 68 (2016) 1, 19-28
Models for estimating uniaxial compressive strength and elastic modulus
Meulenkamp and Alvarez developed the UCS estimation model
based on the feedforward neural network in which the output
error back propagation algorithm was used for training. Input
values used in this model are similar to those used in model
(3). The network was "trained" using the Levenberg-Marquardt
algorithm [29]. Sonmeza et al described the use of artificial
neural networks in the estimation of E value for different rock
types, based on the input values of UCS and material density
[45]. Kahraman et al used neural networks in the estimation of
the UCS and E values for tectonic breccias. In the UCS estimation
model, the neural network was structured as 2-3-1 (number of
inputs, number of neurons in the hidden layer, output value). The
network input was formed of the volume part of fragments and
the velocities of ultrasonic S-waves. The network of the model
developed for estimating the E value was structured as 4-3-1,
and the input was formed of: volume part of fragments, density,
velocity of ultrasonic S-waves, velocity of ultrasonic P-waves,
roundness of fragments, and the average grain size factor [32].
The use of fuzzy neural networks in the estimation of UCS and
E for natural gypsum was described by Yilmaz and Yuksek. Input
parameters for the model were the following values obtained
by laboratory testing: Schmidt hardness, strength index based
on point load test, water content, and velocity of ultrasonic
P-waves. The network was formed using the ANFIS systems
and the program packages Matlab Version 7.1 and SPSS 10.0.
After comparison of results, these authors concluded that the
best results in the estimation of UCS and E values for gypsum
rocks were obtained by the fuzzy type of ANFIS neural networks,
and by the model with the feedforward type of neural networks,
while results were the worst when the multiple regression
model was used [33]. Heidari et al described the use of the MLP
and RBF structured neural networks in the estimation of the E
value for limestones, dolomites and marls from the Lorestan
area in Iran. The following properties were input parameters
for the model: density, porosity, velocity of ultrasonic P-waves,
and uniaxial compressive strength. Better results were obtained
by the model based on MLP architectures and the LevenbergMarquardt algorithm [34]. Based on the study of travertine
originating from Iran, Dehghan et al developed models for the
estimation of UCS and E values. Laboratory values of porosity,
strength index, velocity of ultrasonic P-waves, and Schimdt
hardness were used as input data. Models were developed
using the principle of generalised regression neural networks
(GRNN) and MLP networks trained using the output error back
propagartion algorithm [46]. Using the set of 95 tests made
for various rock types, Singh et al developed the fuzzy neural
model (ANFIS) for estimating the modulus of elasticity, where
input parameters were the strength index determined by point
load test, density of materials, and water absorption [47].
Manouchehrian et al developed an artificial neural network
model for estimating the uniaxial compressive strength of
sandstone using petrographic properties [35].
Multiple regression models and neural networks were
compared in publications [29, 32-36, 46], and it was concluded
GRAĐEVINAR 68 (2016) 1, 19-28
Građevinar 1/2016
that better estimations are made by neural networks. The
comparison was based on correlation coefficients involving
measured and estimated UCS and E values, and the root
mean square error (RMSE). In addition, the MLP architecture
proved better than the RBF [35, 36] and the generalised
model [46].
2.6. Estimation based on evolutionary programming
Genetic algorithms are inspired by the Darwin’s theory
of natural selection and they even use the corresponding
terminology. Here the problems are solved in several
steps. In this algorithm, "genes" are various programming
instructions. The program that gets more correct results
for a number of specified sets obtains a better grade [38].
Baykasoglu et al applied advanced evolutionary programming
techniques, namely the multi expression programming (MEP),
gene expression programming (GEP), and linear genetic
programming (LGP), in order to estimate the UCS of soft
limestones in the region of Gaziantep in Turkey. The LGP
model of evolutionary programming has proven to be the most
efficient of these estimation tools [48]. Ozbeka et al described
the use of the GEP model for estimating the UCS of basalt and
tuff, and demonstrated that a good correspondence exists
between experimentally determined data and results obtained
through estimation [49]. Based on genetic programming, Beiki
et al developed models for estimating the UCS and E values
of carbonate rocks by testing samples collected at the Asmari
Formation in Iran [50].
In the future, an increasing number of papers is expected to
focus on the comparison of evolutionary programming models
with other complex models. For the time being, evolutionary
programming models have proven to be better than multiple
regression models.
2.7. Estimation based on regression tree
The regression tree method, also known as the Decision Tree,
enables estimation of numerical variables. It is used to create
models that are simple to use and interpret. The parts of the
tree are subsets formed of an input set of data according the
values of one of predictor variables, so that individual predictor
variables are approximately constant in each individual subset.
The regression tree branches out depending on questions that
can be answered with "yes" or "no" and the set of adjusted
values of the variable that is being estimated. Each question
establishes whether the predictor meets the requirement.
Depending on answers to one question, either the next
question is put or it is established that the adjusted value
of the answer (variable being estimated) has been achieved.
The process stops once the stop criterion has been achieved
[51]. The regression tree method has been applied by Tiryaki
for estimating the UCS of intact rock material extracted by
mechanical excavation. Here, the predictor variables applied
25
Građevinar 1/2016
are the density, NCB cone indent, and scleroscope hardness.
He developed the model by testing forty-four samples that
had a wide range of rock strength values, from very soft to
very hard [52]. One of principal problems with regression trees
is that the branching out and calibration of the tree is greatly
dependent on the data that are used to develop the model. In
other words, if the data are randomly divided into two parts,
the results may differ considerably for the same input set.
Various regression tree development procedures aimed at
increasing the model accuracy have been developed over
time. Some of these methods are the bagging method and the
random forest method. In the bagging methods, attempts are
made to reduce variability of estimation results by generating
a great number of samples from the initial set via sampling
with replacement, which is followed by building model on each
sample and calculating the average of individual estimations.
The bagging can greatly improve the estimation accuracy,
but the problem occurs in case a very strong i.e. dominating
predictor variable appears, and then the models look similar to
one another. This problem is avoided in random forests where a
great number of trees is created based on samples chosen from
the initial training set by random selection with repetition but, at
every branching, the subset of predictors is randomly selected,
and a relevant one is then chosen at the branching out step
[53]. The models based on regression trees were developed for
estimating the UCS values for the mudstone and wackestone
type limestones from Croatia, and it was established that the
best model was the one in which the random forest method was
used for improving the estimation. At that, predictor variables
were: density, effective porosity, strength index, Schmidt
hardness, and velocity of ultrasonic P-waves [37].
As the regression tree is just being introduced in the UCS
estimation, the contribution of this method will better be
appreciated only through papers that are yet to be written. A
positive aspect is that regression tree can make use of input
predictors dependent on each other, and that there are no
limitations that are present in multiple regression models.
3. Discussion
Based on papers published by other authors, and according to
papers published by the authors of this study, including the
paper [36] dealing with the UCS and E estimations, and paper
[37] dealing with regression trees, the authors of this subject
review wish to emphasize that most estimations were made in
order to determine the UCS value, and then the E value, while
other features are much less represented. The simplest way
to classify the methods for estimating the UCS and E values
is by complexity of procedure and the technology used in the
estimation process. Consequently, the estimation methods
can be divided into simple and complex ones (Figure 5).
The UCS or E values are estimated using simple methods via
diagrams, tables or based on one type of index testing. On the
other hand, complex methods make use of several types of
26
Zlatko Briševac, Petar Hrženjak, Renato Buljan
test results, which serve as the basis for estimation. Complex
computer programs are needed for implementation of these
methods.
Figure 5. Classification of methods for estimating physicomechanical
properties of intact rock material
The UCS and E estimations of intact rock material based on
various diagrams, and in case of field identification by handheld
accessories, provide rough and general values only, and these
estimations are greatly influenced by subjective impressions
of the assessor.
Simple regression equations provide relatively good estimation
results, but are dependent on the type of rock for which they
have been developed and, even within the same rock type,
they are not able to cover all property variations. All this has
resulted in a great number of equations published in literature,
out of which not all can be used, as some are based on tests
that have been modified in the meantime. In addition, when
these equations are used, care must be taken about the range
of values, both with regard to the value being estimated, and
to the value on the basis of which the estimation is made.
The comparison of complex UCS and E estimation methods, as
developed by numerous researches, shows a certain hierarchy
as to success of estimation. Thus, for instance, multiple
regression models present the biggest error compared to other
complex-method models. However, these models are much
simpler for practical application as they do not require the
use of complex computer programs. In addition, the modelling
using fuzzy logic provides better results in combination with
neural networks, compared to the exclusive use of fuzzy
models. Models based on neural networks have so far proven
to be the most useful tool for the UCS and E estimation. The
evolutionary programming models, and the models based on
regression tree, have a considerable potential with regard to
their estimation capabilities. They have so far proven to be
better than multiple regression models.
Although complex estimation models exhibit more favourable
success parameters compared to simpler models, these
GRAĐEVINAR 68 (2016) 1, 19-28
Models for estimating uniaxial compressive strength and elastic modulus
simpler methods should not be neglected as estimations are
actually made at preliminary design stages. This primarily
concerns development of simple regression equations as
they are simple to use and are also sufficiently accurate for
preliminary stages of design. Although complex methods
for estimating the UCS and E are still not widely used in
engineering practice as they are developed only by researchers
in their research projects, an increasing presence of modelling
based on complex methods may be expected in the future due
to an increasing availability of commercial program packages
(Statistica, Matlab) that enable development of complex
models. As researchers are increasingly programming their
own applications for such estimations, they will very probably
become available through Internet for testing purposes, which
will enable even wider application of complex estimation
methods. In complex modelling, one should be guided by
practical engineering values and avoid the use of input
parameters that are difficult to obtain, as their determination
requires complicated preparation of samples.
Građevinar 1/2016
4. Conclusion
This paper confirms that the real need for estimation of
physicomechanical properties of materials, especially in specific
preliminary phases of engineering design, does not imply
replacement of testing, but rather that these estimations serve
as an extension and verification of some specific data.
Simpler modelling methods should not be neglected in future
modelling activities, which namely concerns simple regression
equations as they are easy to use and are accurate enough in
the preliminary stages of design.
As for complex methods, a particular care should be taken to
avoid the use of input parameters in form of physicomechanical
properties as their determination requires a complex
preparation of samples. It would be advisable to use the values
of density, porosity, strength index, and Schmidt hardness, and
to compare them with other material properties that can easily
be determined, such as the textural and structural description
of intact rock material.
REFERENCES
[1]Marinos, P., Hoek, E.: Estimating the geotechnical properties of
heterogeneous rock masses such as flysch, Bulletin of Engineering
Geology and the Environment, 60 (2001), pp. 85-92, http://dx.doi.
org/10.1007/s100640000090
[2]Price, D.G.: Engineering Geology: Principles and practice, (ur. De
Freitas M.), Springer-Verlag, Berlin Heidelberg, 2009.
[3]
Deere, D.U., Miller, R.P.: Engineering classification and index
properties for intact rock, Technical Report No. AFNL-TR-65-116. Air
Force Weapons Laboratoory, New Mexico, 1966.
[4]Miller, R.P.: Engineering classification and index properties for
intact rock, Ph.D Dissertation, University of Illinois, USA, 1965.
[5]Turgrul, A., Zarif, I.H.: Correlation of mineralogical and textural
characteristics with engineering properties of selected granitic
rocks from Turkey, Engineering Geology 51 (1999), pp 303-317,
http://dx.doi.org/10.1016/S0013-7952(98)00071-4
[6]Palchik, V.: Influence of porosity and elastic modulus on uniaxial
compressive strength in softbrittle porous sandstones, Rock
Mechanics and Rock Engineering, 32 (1999), pp 303–309, http://
dx.doi.org/10.1007/s006030050050
[7]Leite, M.H., Ferland, F.: Determination of Unconfined Compressive
Strength and Young’s Modulus of Porous Materials by Indentation
Tests, Engineering Geology, 59 (2001) 3-4, pp. 267-280.
[8]Lashkaripour, G.R.: Predicting mechanical properties of mudroek
from index parameters, Bulletin of Engineering Geology and the
Environment, 61 (2002), pp. 73-77, http://dx.doi.org/10.1007/
s100640100116
[9]
Palchik, V., Hatzor, Y.H.: Influence of porosity on tensile and
compressive strength of porous chalks. Rock Mechanics and Rock
Engineering, 37 (2004), pp. 331-341, http://dx.doi.org/10.1007/
s00603-003-0020-1
Tugrul, A.: The effect of weathering on pore geometry and
[10]
compressive strength of selected rock types from Turkey,
Engineering Geology, 75 (2004), pp. 215–227, http://dx.doi.
org/10.1016/j.enggeo.2004.05.008
GRAĐEVINAR 68 (2016) 1, 19-28
D’Andrea, D.V., Fisher, R.L., Fogelson, D.E.: Prediction of
[11]
compression strength from other rock properties, Colorado
School of Mines Quarterly, 59 4b (1964), pp. 623 – 640.
[12]Read, J.R.L., Thornton, P.N., Regan, W.M.: A rational approach to
the point load test, Proc. 3rd Australian-New Zealand Geomechanics
Conference 2, pp. 35-39, 1980.
[13]ISRM: Suggested method for determining point load strength,
International Journal of Rock Mechanics and Mining Sciences &
Geomechanical Abstracts, 22 (1985), pp. 51-60, http://dx.doi.
org/10.1016/0148-9062(85)92327-7
[14]Romana, M.: Correlation between uniaxial compressive and pointload (Franklin test) strengths for different rock classes, 9th ISRM
Congress, Paris, pp. 673-676, 1999.
[15]Quane, S.L., Russel, J.K.: Rock strength as a metric of welding
intensity in pyroclastic deposits, European Journal of Mineralogy,
15 (2003), pp. 855-64, http://dx.doi.org/10.1127/09351221/2003/0015-0855
[16]Tsiambaos, G., Sabatakakis, N.: Considerations on strength of
intact sedimentary rocks. Engineering Geology, 72 (2004), pp. 261273, http://dx.doi.org/10.1016/j.enggeo.2003.10.001
[17]Kahraman, S., Gunaydin, O., Fener, M.: The effect of porosity on
the relation between uniaxial compressive strength and point
load indeks, International Journal of Rock Mechanics & Mining
Sciences, 42 (2005), pp. 584-589, http://dx.doi.org/10.1016/j.
ijrmms.2005.02.004
[18]Tziallas, G.P., Tsiambaos, G., Saroglou, H.: Determination of rock
strength and deformability of intact rocks, Electronic Journal of
Geotechnical Engineering, 14 G (2009), pp 1-12.
[19]Pollak, D.: Utjecaj trošenja karbonatnih stijenskih masa na njihova
inženjerskogeološka svojstva, Doktorska disertacija, Rudarskogeološko-naftni fakultet, Zagreb, 2007.
[20]Sachpazis, C. I.: Correlating Schmidt hardness with compressive
strength and Young’s modulus of carbonate rocks, Bulletin of the
International Association of Engineering Geology, 42 (1990), pp. 7584, http://dx.doi.org/10.1007/BF02592622
27
Građevinar 1/2016
[21]Katz, O., Reches, Z., Roegiers, J.C.: Evaluation of mechanical rock
properties using a Schmidt hammer, International Journal of Rock
Mechanics & Mining Sciences, 37 (2000), pp 723–728, http://dx.doi.
org/10.1016/S1365-1609(00)00004-6
[22]
Yilmaz, I., Sendir, H.: Correlation of Schmidt hardness with
unconfined compressive strength and Young’s modulus in gypsum
from Sivas (Turkey), Engineering. Geology, 66 (2002), pp. 211-219,
http://dx.doi.org/10.1016/S0013-7952(02)00041-8
[23]Yagiz, S.: Predicting uniaxial compressive strength, modulus of
elasticity and index properties of rocks using the Schmidt hammer,
Bulletin of Engineering Geology and the Environment, 68 (2009), pp.
55-63, http://dx.doi.org/10.1007/s10064-008-0172-z
[24]Kahraman, S.: Evaluation of simple methods for assessing the
uniaxial compressive strength of rock. International Journal of Rock
Mechanics and Mining Sciences, 38 (2001), pp. 981-994, http://
dx.doi.org/10.1016/S1365-1609(01)00039-9
[25]Yasar, E., Erdogarg Y.: Correlating sound velocity with the density,
compressive strength and Young’s modulus of carbonate
rocks, International Journal of Rock Mechanics and Mining
Sciences, 41 (2004), pp. 871-875, http://dx.doi.org/10.1016/j.
ijrmms.2004.01.012
[26]Moradian, Z.A., Behnia, M.: Predicting the uniaxial compressive
strength and static young’s modulus of intact sedimentary rocks
using the ultrasonic test, International Journal of Geomechanics
(ASCE), 9 (2009), pp.14-19, http://dx.doi.org/10.1061/
(ASCE)1532-3641(2009)9:1(14)
[27]Au-Yong, C.P., Ali A.S., Ahmad F.: Office building maintenance: Cost
prediction model, Građevinar, 65 (2013) 9, pp. 803-809.
Zlatko Briševac, Petar Hrženjak, Renato Buljan
[35]Manouchehrian, A., Sharifzadeh, M., Moghadam, R.H.: Application
of artificial neural networks and multivariate statistics to estimate
UCS using textural characteristics, International Journal of Mining
Science and Technology, 22 (2012), pp. 229–236, http://dx.doi.
org/10.1016/j.ijmst.2011.08.013
[36]Briševac, Z.: Model međuovisnosti fizikalno-mehaničkih značajki
karbonatnih stijena, Doktorski rad, Rudarsko-geološko-naftni
fakultet, Zagreb, 2012.
[37]
Briševac, Z., Špoljarić, D., Gulam, V.: Estimation of uniaxial
compressive strength based on regression tree models., Rudarskogeološko-naftni zbornik, 29 (2014), pp 39-47.
[38]
Dvornik, J.: Numeričke, simboličke i heurističke metode,
GRAĐEVINAR, 55 (2003) 10, pp. 575-582.
[39]Cerić, A., Marić, T.: Određivanje prvenstva pri upravljanju rizicima
građevinskih projekata, GRAĐEVINAR, 63 (2011) 3, pp. 265-271.
[40]Žagar, Z., Janjuš, G.: Primjena neuralnih mreža u projektiranju
drvenih konstrukcija, GRAĐEVINAR, 54 (2002) 10, pp. 577-583.
[41]Vouk, D., Malus, D., Carević, D.: Neuralne mreže i njihova primjena
u vodnom gospodarstvu, GRAĐEVINAR, 63 (2011) 6, pp. 547-554.
[42]
Farrokhzad, F., Choobbasti, A.J., Barari, A.: Determination of
liquefaction potential using artificial neural networks, GRAĐEVINAR
,63 (2011) 9, pp. 837-845.
[43]Tayfur, G., Egeli, I.: Railway embankment design based on neural
networks, GRAĐEVINAR, 65 (2013) 4, pp. 319-330.
[44]
Gupta, M.M., Rao, D.H.: Neuro-Control Systems, Theory and
Applications, IEEE Press, New York, 1994.
[28]Alvarez Grimaa, M., Babuska, R.: Fuzzy model for the prediction
of unconfined compressive strength of rock samples, International
Journal of Rock Mechanics and Mining Sciences, 36 (1999), pp. 339349, http://dx.doi.org/10.1016/S0148-9062(99)00007-8
[45]
Sonmeza, H., Gokceoglua, C., Nefeslioglub, H.A., Kayabasi, A.:
Estimation of rock modulus: For intact rocks with an artificial
neural network and for rock masses with a new empirical equation,
International Journal of Rock Mechanics & Mining Sciences, 43(2006),
pp. 224–235, http://dx.doi.org/10.1016/j.ijrmms.2005.06.007
[29]Meulenkamp, F., Alvarez Grima, M.: Application of neural networks
for the prediction of the unconfned compressive strength (UCS)
from Equotip hardness. International Journal of Rock Mechanics and
Mining Sciences & Geomechanical Abstracts, 36 (1999), pp 29-39,
http://dx.doi.org/10.1016/S0148-9062(98)00173-9
[46]Dehghan, S., Sattari, G., Chehreh, C.S., Aliabadi, M.A.: Prediction
of uniaxial compressive strength and modulus of elasticity
for Travertine samples using regression and artificial neural
Networks, Mining Science and Technology, 20 (2010), pp. 41-46,
http://dx.doi.org/10.1016/s1674-5264(09)60158-7
[30]Gokceoglu, C., Zorlu, K.: A fuzzy model to predict the uniaxial
compressive strength and the modulus of elasticity of a problematic
rock, Engineering Applications of Artificial Intelligence, 17 (2004), pp.
61–72, http://dx.doi.org/10.1016/j.engappai.2003.11.006
[47]Singh, R., Kainthola, A., Singh, T. N.: Estimation of elastic constant
of rocks using an ANFIS approach, Applied Soft Computing, 12
(2012), pp. 40–45, http://dx.doi.org/10.1016/j.asoc.2011.09.010
[31]Karakus, M., Tutmez, B.: Fuzzy and Multiple Regression Modelling
for Evaluation of Intact Rock Strength Based on Point Load,
Schmidt Hammer and Sonic Velocity, Rock Mechanics and Rock
Engineering, 39 (2006) 1, pp. 45–57.
[32]Kahraman, S., Gunaydin, O., Alber, M., Fener, M.: Evaluating the
strength and deformability properties of Misis fault breccia using
artificial neural networks, Expert Systems with Applications, 36
(2008) 3, pp. 6874-6878.
[33]Yilmaz, I., Yuksek, G.: Prediction of the strength and elasticity
modulus of gypsum using multiple regression, ANN, and ANFIS
models, International Journal of Rock Mechanics and Mining Sciences,
46 (2008) 4, pp. 803–810.
[34]Heidari, M., Khanlari, G.R., Momeni, A.A.: Prediction of elastic
modulus of intact rocks using artificial neural networks and nonlinear regression methods, Australian Journal of Basic and Applied
Sciences, 4 (2010) 12, pp. 5869-5879.
[48]
Baykasoglu, A., Gullu H., Canakcı, H., Ozbakır, L.: Prediction
of compressive and tensile strength of limestone via genetic
programming, Expert Systems with Applications, 35 (2008), pp.
111–123, http://dx.doi.org/10.1016/j.eswa.2007.06.006.
[49]Ozbeka, A., Unsal, M., Dikec, A.: Estimating uniaxial compressive
strength of rocks using genetic expression programming, Journal
of Rock Mechanics and Geotechnical Engineering, (2013) 5, pp. 325–
329, http://dx.doi.org/10.1016/j.jrmge.2013.05.006
[50]
Beiki, M., Majdi, A., Givshad, A.D.: Application of Genetic
Programming to predict the uniaxial compressive strength and
elastic modulus of carbonate rocks, International Journal of Rock
Mechanics & Mining Sciences, 63 (2013), pp. 159–169, http://dx.doi.
org/10.1016/j.ijrmms.2013.08.004
[51]Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification
and Regression Trees, Chapman and Hall/CRC, 1984.
[52]
Tiryaki, B.: Predicting intact rock strength for mechanical
excavation using multivariate statistics, artificial neural networks
and regression trees, Engineering Geology, 99 (2008), pp. 51–60,
http://dx.doi.org/10.1016/j.enggeo.2008.02.003
[53]James, G., Witten, D., Hastie, T., Tibshirani, R.: An Intorduction to
Statistical Learning with Applications in R, Springer, New York, 2014.
28
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