Assessment of the Modulus of Elasticity at a Triaxial Stress State for Rocks Using Gene Expression Programming

Journal of Soft Computing in Civil Engineering 4-4 (2020) 47-60
How to cite this article: Atici U. Assessment of the modulus of elasticity at a triaxial stress state for rocks using gene expression
programming. J Soft Comput Civ Eng 2020;4(4):47–60. https://doi.org/10.22115/scce.2020.232778.1223.
2588-2872/ © 2020 The Author. Published by Pouyan Press.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Contents lists available at SCCE
Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Assessment of the Modulus of Elasticity at a Triaxial Stress State
for Rocks Using Gene Expression Programming
U. Atici1*
1. Professor, Department of Mining Engineering, Nigde Omer Halisdemir University, Nigde, Turkey
Corresponding author: uatici@ohu.edu.tr
https://doi.org/10.22115/SCCE.2020.232778.1223
ARTICLE INFO ABSTRACT
Article history:
Received: 27 May 2020
Revised: 21 September 2020
Accepted: 23 September 2020
Rocks were subjected to the deformation test under five
different confining stresses (0, 5, 10, 15, and 20 MPa) using
the Hoek cell to determine changes in the elastic properties
of the rocks under confining stress, and the results were
evaluated based on density, porosity, Schmidt hardness, and
compressive strength. A total of nine different rocks, two
granites, two andesites, two limestones, one tuff, one diorite,
and one marble, were used. When the confining stress was
increased from 0 MPa to 5 MPa and from 5 MPa to 10 MPa,
elasticity increased by approximately 20%. When the
confining stress was increased from 10 MPa to 20 MPa, the
increase was 7% in comparison with the previous value.
Then, to formulate the modulus of elasticity for rocks under
the triaxial stress conditions, a new and intelligent approach
to gene application, gene expression programming was
utilized. The success of the model was thoroughly assessed
based on measurable criteria such as the root mean square
error, mean absolute percentage error, and coefficient of
determination. Furthermore, the success of the model was
comprehensively assessed based on the model testing, and
0.88 and 0.81 R2
values were obtained for training and
validation, respectively. The performance of the gene
expression programming-based formulation was compared
with the formulae previously proposed in the literature. The
gene expression method exhibited the best performance, and
it was identified to calculate the modulus of elasticity under
triaxial stress conditions more effectively.
Keywords:
Triaxial stress state;
Confining stress;
Modulus of elasticity;
Modeling;
Gene expression programming.

48 U. Atici/ Journal of Soft Computing in Civil Engineering 4-4 (2020) 47-60
1. Introduction
The characterization of the engineering properties of rocks plays a vital role in planning the
engineering structure in terms of stability and cost analysis. The modulus of elasticity (E)
represents an essential parameter for understanding the deformation behavior of rocks and is
among the most significant mechanical properties of rocks concerning their areas of usage. The
said parameter is crucial in tunnel projects, slope consistency, pillar configuration, rock
destruction and drilling, dams, and numerous other civil and mining applications. It has been
widely utilized to analyze structural deformations, shrinkage, creep, crack control, etc.
The modulus of elasticity is used for the purpose of determining the deformation properties of
wood, metal, concrete, rocks, and many other materials. The mentioned parameter represents the
ratio of stress to corresponding strain in material under tension or compression. It may be
obtained from unconfined compressive strength (G1) tests and the incline of tensile test stress-
strain diagrams for rocks. The deformation test performed to determine the modulus of elasticity
is difficult, time-consuming, and requires particular specimens and equipment. Many authors
have tried to identify this parameter using more easily identified material properties due to the
difficulties in determining the modulus of elasticity using regression analysis [1–4], genetic
programming applications, e.g. artificial neural networks [4–8], fuzzy logic [9–11], and genetic
algorithms [12–14].
The elasticity test is performed only under uniaxial stresses. Nevertheless, in practice,
underground rocks are also affected by confining stress (G3), assuming that G2= G3≠0 and the
modulus of elasticity (E) may be insufficient to reflect the actual conditions of rocks. In real
situations, rocks are exposed to confining stress, and the effects of their elasticity characteristics
under confining stress vary. In previous studies, it was stated that the strength and the modulus of
elasticity of rocks would increase with the increase in G3. However, there are not many studies in
the literature on the rate of this increase and estimating the respective values. Liang et al. [15]
and Ma et al. [16] investigated the stress-strain relationships for rock salt and anhydrite. On the
contrary, Li et al. [17] reported that the modulus of elasticity was independent of strain rate.
Hoek and Brown [18], Arora [19], and Asef and Reddish [20] developed a different formula for
predicting the modulus of elasticity with confining pressure.
This study's goal was to predict E at a triaxial stress state using a novel approach, gene
expression programming (GEP). The complicated relationship between the parameters affecting
the E value may be easily modeled using the GEP approach, unlike classical methods. Two
hundred fifty-six data sets were collected from experimental studies using nine different rocks
(two granites, two andesites, two limestones, one tuff, one diorite, and one marble). E was used
as an output, and five parameters, including G3 (MPa), G1 (MPa), porosity (g/cm3
) (𝜃), ultrasonic
sound velocity (km/s), and Schmidt rebound hardness (SH), were used as the input parameters.
The model was trained with 192 data sets of the experimental results, following which 64 data
sets were utilized for testing. The result of the obtained GEP-based formulation was tested and
compared to the formulations proposed by Arora [19] and Asef and Reddish [20] using

U. Atici/ Journal of Soft Computing in Civil Engineering 4-4 (2020) 47-60 49
experimental data. It was observed that the findings acquired using the model corresponded to
the experimental data at a high rate.
2. Material and experimental methods
2.1. Rock characteristics
The physical characteristics of the sampled rocks were identified by 𝜃 and ultrasonic P-wave
velocity. The samples were cored from large blocks in the laboratory, and their testing was
carried out following the procedures described in ISRM [21] . Density (𝜌) and 𝜃 were found by
employing the water saturation and caliper techniques on the NX-sized samples, using five
samples every time. The high-frequency ultrasonic pulse technique described by ASTM [22] was
employed to measure the velocities of the P (compression) waves using the 1 MHz nominal
frequency. The wave velocity through the sample was computed using the travel time from the
transmitter to a receiver. A minimum of five measurements was made from every rock type, and
they were averaged.
The measured mechanical properties were UCS and rebound tests, which were determined in
accordance with the standards by ISRM [21]. A minimum of five core samples from every rock
were exposed to strength tests, which were conducted using a fully automatic, servo-controlled,
instrumented, and computer-controlled press machine. Rebound tests were conducted on the
block samples using the Digi-Schmidt 2000 concrete test hammer. At least twenty measurements
at various points of every sample were performed for the rebound hardness of rocks. Table 1
summarizes the physical and mechanical characteristics of the rocks examined. This table shows
that the standard deviation values were high because of working with natural materials.
Table 1
Basic physical and mechanical characteristics of rocks.
Min. Max. Mean Std. Dev.
𝜃 (%) 0.36 18.23 4.36 6.16
𝜌 (gr/cm3
) 1.85 2.86 2.57 0.33
Vp (Km/s) 2.17 5.82 4.28 1.38
SH 43.00 70.60 58.18 8.05
G1 (MPa) 28.00 167.97 81.14 34.95
E (MPa) 1278.20 28260.20 10854.07 5513.98
2.2. Elasticity at a triaxial stress state
Triaxial compression tests with different G3 were carried out according to ISRM [21] using a
hydraulic servo-control test system comprised of three major sub-systems, including a servo-
controlled press machine, automatic axially loading unit, instrumentation, and a personal
computer. G3 was achieved through hydrostatic pressure using an automatic pressure system for
confining stress in a Hoek triaxial cell. A series of G3 at 5, 10, and 15 MPa and at 20 MPa for

only two rocks (one limestone and one granite) were applied as confining pressure. The
maximum confining stress was limited to 20 MPa, which was equivalent to a load of
approximately 800 m depth representing the limit of the depth of most engineering works. Axial
loading and G3 were simultaneously increased to the pre-established values, following which G3
was fixed, and axial loading continued to be increased at a constant speed up to the pre-
determined stresses. The axial strains of the test samples were measured by the linear variable
differential transformer (LVDT) using the software prepared for controlling the press machine. A
PC continuously showed and performed the recording of the axial load and the axial
displacement during the test. The tangential 𝐸 was derived as a tangent to the stress-strain curve
(E=∆G/∆є) at 50% of the maximum stress (G1) derived (Fig. 1). The E values for different G3
values of the nine various rocks used in the study are presented in Fig. 2. It is observed that when
G3 was increased from 0 to 5 and from 5 to 10, the increase in the modulus of elasticity was
20%, and when G3 was increased from 10 to 15, the mean modulus of elasticity increased by 7%
on average in the used rocks.
Fig. 1. Determination of the modulus of elasticity.
Fig. 2. The correlation between E and G3.
The effects of G1 and G3 on the E values are shown in Fig. 3. In particular, when G3 was applied
as 5 and 10 MPa, it was observed that the modulus of elasticity was about 20% higher compared
to the previous value. When G3 was applied as 20 MPa, the modulus of elasticity continued to
increase. However, this rate was only 7% higher than the value at 15 MPa. The most important
reason for this was the fact that the discontinuities in the studied rocks were continuous to trap

up to 10 MPa. After 10 MPa, the closing procedure was continued, but the effect on deformation
was reduced. In case of the low confining stress, the rocks with low G1 and high 𝜃 regarding the
increase in the modulus of elasticity were relatively low. When the confining stress increased, the
increase in the modulus of elasticity was higher. However, in the rocks with high G1 and low 𝜃,
the situation was the opposite (Fig. 4). Likewise, the highest rate of increase in the modulus of
elasticity was observed in rocks with the highest SH as the confining stress increased (Fig. 5).
Fig. 3. The correlation between E and G1, G3.
Fig. 4. The correlation between E and 𝜃, G3.
Fig. 5. The correlation between E and SH, G3.
0
5
10
15
0
50
100
150
0
10
20
30
40
Confining Pressure (MPa)
Compressive Strength (MPa)
Elasticity
(GPa)

3. Gene expression programming (GEP)
Although classical modeling methods such as regression analysis have many advantages, these
methods need a correlation between independent and dependent variables [23]. The significant
weakness of traditional modeling methods is their being not equipped for providing useful
prediction conditions for complicated cases. The soft computing approach surpasses the
problems and disadvantages of regression analysis and yields incredible and accurate results [9].
To overcome this limitation, new methodologies have been proposed. The researchers have
produced rearranged prediction conditions without accepting an earlier type of the current
relationship. Gene expression programming (GEP) is another amazing, transformative AI-based
technique created by Ferreira [24]. It is a groundbreaking developmental calculation that
combines both the basic direct chromosomes of a settled length like the ones utilized in
hereditary calculations and the ramified structures of various sizes and shapes like the parse trees
of hereditary programming. Its assessment arrangement of any information resembles that of the
biological assessment and is a PC program that is encoded in direct chromosomes of a settled
length. In the mentioned methodology, a scientific capacity characterized as a chromosome with
multiple qualities is produced by utilizing the information exhibited to it.
To create a GEP demonstration, five segments, including the function set, terminal set, control
parameters, fitness function, and stop condition, are needed. After encoding the issue for the
competitor arrangement and determining the wellness work, the calculation randomly makes an
underlying populace of reasonable people (chromosomes) and afterward, changes over every
chromosome into an articulation tree related to a scientific articulation. A short time later, the
anticipated target is contrasted, and the real one and the wellness score for every chromosome
are resolved. If it is adequately high, the calculation stops. Nevertheless, a part of the
chromosomes are chosen by utilizing roulette wheel testing, and after this process, they are
transformed to acquire the new ages. This closed cycle proceeds until the wanted wellness score
is achieved, following which the decoding of the chromosomes is performed for the best
arrangement of the issue [25,26].
GEP has two languages, such as the language of qualities and the language of ETs. In GEP, on
account of the basic decisions about the structure of ETs and their associations, it is reasonable to
promptly induce the phenotype presented as the arrangement of quality and vice versa. This
comprehensible bilingual documentation is named the Karva language [24]. For example, a
numerical articulation [𝑎 ∗ (𝑎/𝑏)] + [√𝑎/𝑏
2
] may be expressed by a two-quality chromosome or
an ET, as seen in Fig. 6. The said figure indicates how two qualities are encoded as a straight
string and how they are communicated as an ET. The reader is referred to the study conducted by
Ferreira [24,27] to obtain more detailed information.
3.1. Overview of gene expression programming
Due to the failed linear, nonlinear either single or multiple regression in terms of providing the
suitable outcomes, in this study, the principal point of improvement of GEP-based definitions
was to produce the numerical capacities for the prediction of E at 5, 10, 15, and 20 MPa G3 at a
triaxial stress state. While training and testing the GEP model, G3, density (𝜌), 𝜃, SH, Vp, and G1
were used as the input variables, whereas E was used as the output variable.

Fig. 6. The case of the GEP expression tree.
The GEP model was created with 256 data sets of the rock samples acquired from this
experimental research. The numbers of the experimental data sets utilized for initial practices and
testing/validation in the mentioned model were 191 and 63, respectively. Table 2 summarizes the
parameters utilized in the development of the GEP model.
Many GEP models were tested to find the best model using the GeneXproTools software. Four
basic arithmetic operations (+, −, ∗, /) and several primary mathematical functions (Exp, Ln, X2
,
3Rt, Atan, Min, Max, Avg2, Tanh, Not, Inv) were used, and the RMSE limitation was determined
as the fitness function.
In the GEP analysis, the setting parameter values significantly affected the output model's fitness.
They contained the chromosome number, the number of genes, the head size of the gene, and the
ratio of genetic operators. To select the chromosomal tree, in other words, the head's length and
the gene number, the GEP models primarily utilized a single gene and two lengths of heads and
increased the number of genes and heads, consecutively, in every run, while the testing and
training performance of every model was monitored. Following a number of tests, for the
models, the numbers of genes 3 and eight heads were revealed to yield the best outcomes. The
linking of the sub-ETs (genes) was performed as a result of subtraction.
Table 2
GEP parameters utilized for the models created.
Definition of the parameter GEP Model
Fitness function RMSE
Number of chromosomes 30
Gene number 3
Mutation rate 0.00138
Inversion rate 0.00546
One-point recombination rate 0.00277
Two-point recombination rate 0.00277
Gene recombination rate 0.00277
Gene transposition rate 0.00277
Literals 17
Arithmetic operators + - x /
Mathematical function Exp, Ln, X2
, 3Rt, Atan, Min, Max, Avg2, Tanh, Not, Inv
Head size 8
Tail Size 9
Gene Size 26
Linking function Subtraction

Finally, an association of all genetic operators, including transposition, mutation, and crossover,
was utilized as the set of genetic operators. The chromosome 30 was determined as the best
generation of individuals in estimating E. The accurate formulation after replacing the constants
based on the constructed models for E is given by the following formula:
𝐸 = {68.564 (
10𝑉𝑝
𝑀𝑖𝑛(𝑓𝑥;𝑆𝐻)
∗ (𝜃 ∗ 𝑓
𝑥)} − {𝑀𝑖𝑛(2094.579𝑉
𝑝
2
; ((𝑉
𝑝 + 𝜌) ∗ (𝜎3 ∗ 𝑆𝐻)} −
{[(
(𝑆𝐻+𝜎1)+(𝑉𝑝∗𝜎1
2
) ∗ (𝜌 ∗ 𝑆𝐻)] ∗ 𝐴𝑇𝑎𝑛(−17.39𝜃)} (1)
Fig. 7 presents the expression tree of the formulation for the model, where d0, d1, d2, and d3
denote G3, 𝜌, 𝜃, SH, Vp, and G1. Table 3 contains the list of the constants utilized in the
formulation.
Table 3
Coefficients used in the GEP model.
Constant Gene I Gene II Gene III
C0 8.45 4.09 -2.93
C1 6.19 8.65 6.02
C2 -4.61 -4.79 1.65
C3 5.49 -204.14 7.46
C4 2.65 8.53 -2.04
C5 -0.39 -2.55 -9.48
C6 -6.03 -8.69 -17.39
C7 6.19 -5.13 1.67
C8 9.13 0.27 9.00
C9 -9.63 9.27 -5.84
4. Results and discussion
The current section investigates the findings of the analysis of the created model and a
quantitative evaluation of the model's predictive abilities. In assessing the model, defining the
model's performance and the criteria for estimation accuracy is essential. To produce the GEP
models and demonstrate the prediction ability of the GEP, the database consisting of the test
information was separated into two sets, training and testing. From the 256 data sets, for the
training and testing processes, 192 and 64 experimental data sets, respectively, were selected in a
random way.
To compare the performance of the model with the previous models in the literature, three
previously utilized models were determined. These models are the models of Hoek and Brown
[18], Arora[19], and Asef and Reddish [20].
Hoek and Brown developed a highly accepted formula that uses the modulus of elasticity, strain,
Poisson's ratio, and confining pressure. This is shown by equation (2). However, it is both very
expensive and technically difficult to determine the Poisson's ratio used in this equation under
triaxial stress conditions.

Fig. 7. Expression tree of the GEP approach model.
𝐸𝜀1 = 𝜎1 − 𝜗(𝜎2 + 𝜎3) (2)
Where 𝜗 is the Poisson′s ratio.

Arora suggested a model using three different rock types (plaster of Paris, Jamrani sandstone,
and Agra sandstone) upon the estimation of the modulus of elasticity with confining pressure.
This empirical equation is given below:
𝐸(𝜎3=0)
𝐸(𝜎3=𝜎2≠0)
= 1 − 𝑒𝑥𝑝 (−0.1
𝜎1
𝜎3
) (3)
Asef and Reddish developed the model with its own test data and proposed a new model. It is
stated that this model illustrates a significant improvement by obtaining the coefficient of
determination (R2
:0.69) with the original data of Arora and its own experimental data using
arkose sandstone. This empirical equation is presented below:
𝐸(𝜎3=𝜎2≠0)
𝐸(𝜎3=0)
=
200(
𝜎3
𝜎1
)+𝑏
(
𝜎3
𝜎1
)+𝑏
(4)
Where b=15+60e-0.18G1
, E(G3=0) is the modulus of elasticity at the unconfined compressive
strength (GPa), E(G3= G2≠0) is the modulus of elasticity at a given confining stress (GPa), G3 and
G1 are the confining stress (MPa) and unconfined compressive strength (MPa), respectively.
The results of the created model utilizing GEP and models proposed by Arora and Ased and
Reddish were assessed based on the root mean square error (RMSE) values, mean absolute
percentage error (MAPE), and the coefficient of determination (R2
). The above-mentioned
criteria are presented in equations 5, 6, and 7, respectively.
𝑀𝐴𝑃𝐸 =
1
𝑛
[
∑ |𝑡𝑖−𝑜𝑖|
𝑛
𝑖=1
∑ 𝑡𝑖
𝑛
𝑖=1
∗ 100] (5)
𝑅𝑀𝑆𝐸 = √
1
𝑛
∑ (𝑡𝑖 − 𝑜𝑖)2
𝑛
𝑖=1
2
(6)
𝑅2
= [
(𝑛 ∑ 𝑡𝑖𝑜𝑖−(∑ 𝑡𝑖 ∑ 𝑜𝑖
𝑛
𝑖=1
𝑛
𝑖=1 )
𝑛
𝑖=1 )
2
(𝑛 ∑ 𝑡𝑖
2−(∑ 𝑡𝑖
𝑛
𝑖=1 )
2
𝑛
𝑖=1 )(𝑛 ∑ 𝑜𝑖
2−(∑ 𝑜𝑖
𝑛
𝑖=1 )
2
𝑛
𝑖=1 )
] (7)
Where t denotes the experimental value, o refers to the predicted value, and n refers to the total
number of data.
RMSE indicates the sample standard deviation between the observed and predicted values, while
MAPE represents a measure of the prediction accuracy of the forecasting method in statistics, for
instance, in the estimation of trends, and it generally presents accuracy as a percentage. R2
was
used as the criterion for evaluating the agreement between the experimental and predicted values.
The statistical performances of the model established by GEP and the models proposed by Arora
and Asef and Reddish are summarized in Table 4. As seen in Table 4, for both the training and
validation sets, the GEP model had low MAPE, RMSE and high R2
values. The R2
values related
to the experimental and predicted data from the GEP model for training and testing were 0.88
and 0.81, respectively, implying that the model was capable of explaining 88% and 0.81% of the
variation in the data.

Table 4
Performance statistics of the model.
GEP Model Arora’s Model Asef and Reddish's Model
Training Validation Training Validation Training Validation
MAPE 0.069 0.264 0.173 0.465 1.198 3.80
RMSE 1904.84 2688.59 3991.40 4085.46 27206.39 30446.23
R2
0.88 0.81 0.56 0.60 0,62 0.62
The graphical comparison of the E values predicted by the GEP-based models and models
proposed by Arora and Asef and Reddish in the training and validation phases to their
experimental values is presented in Figs. 8 and 9, respectively. As is observed from the figures in
question, the experimental values were close to the estimated values for the GEP-based models.
They demonstrated the successful performance of GEP models in predicting E at a triaxial stress
state using 𝜃, 𝜌, Vp, SH, and G1. The models proposed by Arora and Asef and Reddish exhibited
a similar performance to each other in the current experimental data. However, the performances
of these models are lower than that of the GEP-based model.
Fig. 8. Comparison of the measured and predicted E results of the training set.
Fig. 9. Comparison of the measured and predicted E results of the validation set.

5. Conclusion
In this study, the triaxial compression test for nine different rocks (two granites, two andesites,
two limestones, one tuff, one diorite, and one marble) was carried out at four confining stress
values (5, 10, 15, and 20 Mpa) to determine the behavior of rocks at a triaxial stress state. The
GEP-based formulation is displayed for predicting the modulus of elasticity (E) from some mass
characteristics of rocks by utilizing the GEP system, which is a novel methodology for this kind
of issues. In the GEP model, confining stress, density, porosity, Schmidt hardness, ultrasonic p-
wave velocity, and uniaxial compressive strength were utilized as the input variables, whereas E
was used as the output variable. For this purpose, 256 data sets were used. The comparison of the
performance of the GEP-based model with the formulae previously suggested in the literature by
Arora [19] and Asef and Reddish [20] was made using the root mean square error (RMSE)
values, mean absolute percentage error (MAPE), and the coefficient of determination (R2
).
From the present study, the following was concluded:
 With the increase in the support pressure, there was a considerable increase in the modulus
of elasticity. However, when the support pressure was further increased, a reduction in the
increasing modulus of elasticity was determined. The reason for this may be the closure of
micro and/or macro fractures under confining pressure. The mentioned finding is parallel
to the findings obtained by Asef and Reddish.
 In the rocks with high Schmidt hardness values, it was observed that the increase in the
modulus of elasticity was higher when the confining stress was increased. However, the
increase in the modulus of elasticity was found to be lower in the rocks with low strength
and high porosity.
 In the present research, a novel and practical approach to formulating the modulus of
elasticity under three-axis loading conditions was applied. The suggested model is
empirical, and experimental results constitute its basis. The verification of the GEP model's
validity was performed by the statistical performance criteria utilized to assess the model,
such as root mean square error (RMSE), mean absolute percentage error (MAPE), and the
coefficient of determination (R2
). These statistical parameters of GEP formulations also
contributed to the excellent performance of GEP models in predicting the modulus of
elasticity for natural stones.
 The models proposed by Arora and Asef and Reddish exhibited a similar performance to
each other in the current experimental data. However, the performance of these models is
lower in comparison with the GEP-based model.
 GEP is an appropriate method for estimating the 𝐸 value of rocks from anisotropic and
heterogeneous materials as well as revealing nonlinear practical connections in which it is
not possible to apply traditional techniques. Moreover, using GEP, the 𝐸 value may be
assessed without conducting complex and tiresome research facility tests. The proposed
GEP formulations were easy since they could be used by anyone, who is not fully familiar
with the GEP method, in comparison with other artificial intelligence methods.

Acknowledgments
The current study did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
Conflicts of Interest
The authors declare no conflict of interest.
References
[1] Z. Liu, J. Shao, W. Xu, Q. Wu, Indirect estimation of unconfined compressive strength of
carbonate rocks using extreme learning machine, Acta Geotech. 10 (2015) 651–663.
https://doi.org/10.1007/s11440-014-0316-1.
[2] E.T. Mohamad, D.J. Armaghani, E. Momeni, A.H. Yazdavar, M. Ebrahimi, Rock strength
estimation: a PSO-based BP approach, Neural Comput. Appl. 30 (2018) 1635–1646.
[3] M. Liang, E.T. Mohamad, R.S. Faradonbeh, D. Jahed Armaghani, S. Ghoraba, Rock strength
assessment based on regression tree technique, Eng. Comput. 32 (2016) 343–354.
[4] S. Charhate, M. Subhedar, N. Adsul, Prediction of concrete properties using multiple linear
regression and artificial neural network, J. Soft Comput. Civ. Eng. 2 (2018) 27–38.
[5] E. Momeni, D. Jahed Armaghani, M. Hajihassani, M.F. Mohd Amin, Prediction of uniaxial
compressive strength of rock samples using hybrid particle swarm optimization-based artificial
neural networks, Measurement. 60 (2015) 50–63.
https://doi.org/10.1016/j.measurement.2014.09.075.
[6] B.Y. Bejarbaneh, E.Y. Bejarbaneh, M.F.M. Amin, A. Fahimifar, D. Jahed Armaghani, M.Z.A.
Majid, Intelligent modelling of sandstone deformation behaviour using fuzzy logic and neural
network systems, Bull. Eng. Geol. Environ. 77 (2018) 345–361. https://doi.org/10.1007/s10064-
016-0983-2.
[7] K. Behzadafshar, M.E. Sarafraz, M. Hasanipanah, S.F.F. Mojtahedi, M.M. Tahir, Proposing a new
model to approximate the elasticity modulus of granite rock samples based on laboratory tests
results, Bull. Eng. Geol. Environ. 78 (2019) 1527–1536. https://doi.org/10.1007/s10064-017-1210-
5.
[8] R.K. Umrao, L.K. Sharma, R. Singh, T.N. Singh, Determination of strength and modulus of
elasticity of heterogenous sedimentary rocks: An ANFIS predictive technique, Measurement. 126
(2018) 194–201. https://doi.org/10.1016/j.measurement.2018.05.064.
[9] C. Gokceoglu, K. Zorlu, A fuzzy model to predict the uniaxial compressive strength and the
modulus of elasticity of a problematic rock, Eng. Appl. Artif. Intell. 17 (2004) 61–72.
https://doi.org/10.1016/j.engappai.2003.11.006.
[10] B. Rajesh Kumar, H. Vardhan, M. Govindaraj, G.S. Vijay, Regression analysis and ANN models to
predict rock properties from sound levels produced during drilling, Int. J. Rock Mech. Min. Sci. 58
(2013) 61–72. https://doi.org/10.1016/j.ijrmms.2012.10.002.

[11] N. Madhubabu, P.K. Singh, A. Kainthola, B. Mahanta, A. Tripathy, T.N. Singh, Prediction of
compressive strength and elastic modulus of carbonate rocks, Measurement. 88 (2016) 202–213.
https://doi.org/10.1016/j.measurement.2016.03.050.
[12] M. Beiki, A. Majdi, A.D. Givshad, Application of genetic programming to predict the uniaxial
compressive strength and elastic modulus of carbonate rocks, Int. J. Rock Mech. Min. Sci. 63
(2013) 159–169.
[13] U. Atici, Modelling of the Elasticity Modulus for Rock Using Genetic Expression Programming,
Adv. Mater. Sci. Eng. 2016 (2016) 1–8. https://doi.org/10.1155/2016/2063987.
[14] G. Hosseini, Capacity Prediction of RC Beams Strengthened with FRP by Artificial Neural
Networks Based on Genetic Algorithm, J. Soft Comput. Civ. Eng. 1 (2017) 93–98.
[15] W. Liang, C. Yang, Y. Zhao, M.B. Dusseault, J. Liu, Experimental investigation of mechanical
properties of bedded salt rock, Int. J. Rock Mech. Min. Sci. 44 (2007) 400–411.
https://doi.org/10.1016/j.ijrmms.2006.09.007.
[16] L. Ma, X. Liu, M. Wang, H. Xu, R. Hua, P. Fan, S. Jiang, G. Wang, Q. Yi, Experimental
investigation of the mechanical properties of rock salt under triaxial cyclic loading, Int. J. Rock
Mech. Min. Sci. 62 (2013) 34–41. https://doi.org/10.1016/j.ijrmms.2013.04.003.
[17] H.B. Li, J. Zhao, T.J. Li, Triaxial compression tests on a granite at different strain rates and
confining pressures, Int. J. Rock Mech. Min. Sci. 36 (1999) 1057–1063.
[18] E. Hoek, E.T. Brown, Empirical strength criterion for rock masses, J. Geotech. Geoenvironmental
Eng. 106 (1980).
[19] V.K. Arora, Strength and deformational behaviour of jointed rocks, (1987).
[20] M.R. Asef, D.J. Reddish, The impact of confining stress on the rock mass deformation modulus,
Géotechnique. 52 (2002) 235–241. https://doi.org/10.1680/geot.2002.52.4.235.
[21] ISRM Rock characterization testing and monitoring. In Brown, E.T. (Ed.), ISRM Suggested
Methods. Pergamon, Oxford; 1981., (n.d.).
[22] ASTM Annual Book of ASTM Standards-Natural Building Stones; Soil and Rock, Part 19. ASTM
Publication Office, Philadelphia, PA; 1980., (n.d.).
[23] U. Atici, Prediction of the strength of mineral admixture concrete using multivariable regression
analysis and an artificial neural network, Expert Syst. Appl. 38 (2011) 9609–9618.
https://doi.org/10.1016/j.eswa.2011.01.156.
[24] C. Ferreira, Gene expression programming: a new adaptive algorithm for solving problems, ArXiv
Prepr. Cs/0102027. (2001).
[25] C. Kayadelen, O. Günaydın, M. Fener, A. Demir, A. Özvan, Modeling of the angle of shearing
resistance of soils using soft computing systems, Expert Syst. Appl. 36 (2009) 11814–11826.
https://doi.org/10.1016/j.eswa.2009.04.008.
[26] I.F. Kara, Prediction of shear strength of FRP-reinforced concrete beams without stirrups based on
genetic programming, Adv. Eng. Softw. 42 (2011) 295–304.
https://doi.org/10.1016/j.advengsoft.2011.02.002.
[27] C. Ferreira, Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence.
Springer, 2006.

Assessment of the Modulus of Elasticity at a Triaxial Stress State for Rocks Using Gene Expression Programming

More Related Content

Similar to Assessment of the Modulus of Elasticity at a Triaxial Stress State for Rocks Using Gene Expression Programming (20)

More from Journal of Soft Computing in Civil Engineering (20)

Recently uploaded (20)

Assessment of the Modulus of Elasticity at a Triaxial Stress State for Rocks Using Gene Expression Programming