Ferrocement composites are widely used as a novel method for many different structural purposes recently. The uniform distribution and the high surface area-to-volume ratio of the reinforcement of such composites would improve the crack-arresting mechanism. Given these properties, ferrocement is an ideal option as a replacement for some traditional structures methods. In members with axially loaded reinforced concrete ferrocement composite, it would be the best alternative to use ferrocement members. Lack of sufficient research in this approach is the cause of not well defining this field for RC structures. This study has aimed to evaluate the moment capacity of ferrocement members using the GMDH method. Mechanical and geometrical parameters including the width of specimens, total depth specimens, compressive strength of ferrocement, ultimate strength of wire mesh and volume fraction of wire mesh are considered as inputs to predict the moment capacity of ferrocement members. For evaluating this model, mean absolute error (MAE), root mean absolute error (RMAE), normalized root mean square error (NRMSE) and mean absolute percentage error (MAPE) were carried out. The results conducted that the GMDH model is significantly better than some previous models and comparable to some other methods. Moreover, a new formulation for moment capacity of ferrocement members based on GMDH approach is presented. Finally, Sensitivity analysis is operated to understand the influence of each input parameters on moment capacity of ferrocement members.

1. Full Length Article
A new proposed approach for moment capacity estimation of
ferrocement members using Group Method of Data Handling
Hosein Naderpour ⇑
, Danial Rezazadeh Eidgahee, Pouyan Fakharian, Amir Hossein Rafiean,
Seyed Meisam Kalantari
Faculty of Civil Engineering, Semnan University, Semnan, Iran
a r t i c l e i n f o
Article history:
Received 14 August 2018
Revised 16 April 2019
Accepted 23 May 2019
Available online 30 May 2019
Keywords:
Moment capacity
Ferrocement members
Group Method of Data Handling (GMDH)
Combinatorial algorithm
Concrete
a b s t r a c t
Ferrocement composites are widely used as a novel method for many different structural purposes
recently. The uniform distribution and the high surface area-to-volume ratio of the reinforcement of such
composites would improve the crack-arresting mechanism. Given these properties, ferrocement is an
ideal option as a replacement for some traditional structures methods. In members with axially loaded
reinforced concrete ferrocement composite, it would be the best alternative to use ferrocement members.
Lack of sufficient research in this approach is the cause of not well defining this field for RC structures.
This study has aimed to evaluate the moment capacity of ferrocement members using the GMDH
method. Mechanical and geometrical parameters including the width of specimens, total depth speci-
mens, compressive strength of ferrocement, ultimate strength of wire mesh and volume fraction of wire
mesh are considered as inputs to predict the moment capacity of ferrocement members. For evaluating
this model, mean absolute error (MAE), root mean absolute error (RMAE), normalized root mean square
error (NRMSE) and mean absolute percentage error (MAPE) were carried out. The results conducted that
the GMDH model is significantly better than some previous models and comparable to some other meth-
ods. Moreover, a new formulation for moment capacity of ferrocement members based on GMDH
approach is presented. Finally, Sensitivity analysis is operated to understand the influence of each input
parameters on moment capacity of ferrocement members.
Ó 2019 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
According to American Concrete Institute Committee 549
(1997), ferrocement is a thin layer of reinforced concrete, which
is made of hydraulic cement mortar; condensed layers of relatively
small-pore mesh were also used to reinforce it [1,2]. Many
researches and advancements were recently achieved in ferroce-
ment, which most of them were accepted and utilized. Different
applications are innovated and designed for ferrocement across
the world [3]. Since early 19600
s, it is extensively used in Australia,
New Zealand, and the United Kingdom, and numerous vessels and
structures were created for ferrocement in different countries.
Ferrocement dwelling houses in Bangladesh, Indonesia, and Papua
New Guinea benefit from wood, bamboo or bush sticks as low-cost
replacements for steel. Ferrocement precast elements were used
for roofs, wall panels, and fences in India, Philippines, Malaysia,
Brazil, Papua New Guinea, Venezuela and the Pacific. Sri Lanka
introduced a ferrocement house resistant to cyclones. Corrugated
ferrocement roofing sheet reinforced with local materials are
similar to asbestos cement and iron sheets used in Singapore, India,
Indonesia, Peru, and Zimbabwe [4].
Moreover, in a number of practical applications such as repair of
shear damaged reinforced concrete beams, beams and slab with
excessive deflection, joints, repair or strengthening of channels
[5], brick masonry columns as well as plain concrete column
[6,7] ferrocement is beneficial due to increased capacity in load
carrying [8–10], better cracking behavior, ductility, energy absorp-
tion properties [9,10], stiffness [11,12] and flexural capacity [11].
To strengthen axially loaded reinforced concrete (RC) members,
ferrocement composites are used as a jacketing material. Strength-
ening concrete structures is of great importance in construction
activities since such structures are often exposed to damage due
to environmental factors [13]. Ductile RC frames may fail if a beam
is placed above the balanced reinforcements [14]. Ferrocement can
be extensively applied in low-cost housing systems. Nevertheless,
https://doi.org/10.1016/j.jestch.2019.05.013
2215-0986/Ó 2019 Karabuk University. Publishing services by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
⇑ Corresponding author at: Faculty of Civil Engineering, Semnan University,
Semnan 3513119111, Iran.
E-mail address: naderpour@semnan.ac.ir (H. Naderpour).
Peer review under responsibility of Karabuk University.
Engineering Science and Technology, an International Journal 23 (2020) 382–391
Contents lists available at ScienceDirect
Engineering Science and Technology,
an International Journal
journal homepage: www.elsevier.com/locate/jestch

2. since ferrocement is thin, using it for roofing and exterior walls
purposes raised questions about indoor thermal status [15]. There
are some experimentally formulated equations to calculate the
moment capacity of the ferrocement member, although some
investigations could not support it practically [3].
Ferrocement members have complicated modeling. The analyt-
ical models of ferrocement members for flexural capacity rely on
different assumptions, approximations, and simplifications [16].
An analytical method used to calculate moment capacity is a plas-
tic analysis relied on the equilibrium of forces [17]. Based on this
method [17] ferrocement is an excellent homogeneous elastoplas-
tic material. According to this method, there is a bilinear relation-
ship between stress-strain and mortar. The mechanism which
simplifies the plastic analysis is another method to calculate
moment capacity [18]. Based on this approach, the neutral axis is
tightly close to the surface; therefore, the reinforcements are under
pressure [18]. However, it is not correct to use such a way to sim-
plify calculating the moment capacity, as it may cause errors [16].
Under such circumstances, experimental datasets are used to
develop an empirical model, which reduces modeling uncertainty
[19].
By the use of input-output data, the system can identify and
model the complex processes; it has dragged the attention of many
researchers. In order to model and predict the behaviors of
unknown and very complex systems, such techniques are applica-
ble in different fields using the obtained data [20]. Modeling a sys-
tem theoretically, the clear mathematical input-output
relationships should be understood meticulously. However,
although it is challenging to model such clear mathematical sys-
tems, it cannot be avoided in poorly understood systems. These
materials strength characterization can be considered as a complex
system which is frequently simplified using linear and homoge-
neous rules and have independent variable assumptions. The soft
computing methods such as artificial neural networks (ANNs),
adaptive neuro-fuzzy inference system (ANFIS), and Gene Expres-
sion Programming (GEP) were used in different studies [3,19,21–
23]. Furthermore, particular attention is paid to soft-computing
methods [24] designed to compute an ambitious environment.
Many studies emphasized the use of evolutionary methods as prac-
tical tools to identify a system [3,25].
ANNs were introduced in the 1940s, owing to the biological
neural system. ANNs are a kind of mathematical methods that
are widely used in signal and figure recognition, financing, and mil-
itary settings. The application of neural network in civil engineer-
ing was reported for the first time in 1989 in a specialized journal
[26] and accordingly, many authors emphasizing this method in
their studies on civil/structural engineering. ANN is usually used
to identify structural damages [27,28], optimize structures [29],
control structural vibration [30], model structural materials
[31–33], and predict building cost and properties of concretes
[34–41]. The creep or shrinkage of concrete is predicted in a few
articles [42–45].
Fuzzy inference system is another method of prediction, which
is extensively employed in automatic control, medical diagnosis,
analyzing systems, artificial intelligence, processing information,
recognition of the pattern, geological exploration, and forecasting
the weather [46]. This civil engineering theory was recently intro-
duced, mainly to control structural vibration and predict the
behaviors of materials [47–50]. By the use of fuzzy inference sys-
tem, researchers developed several prediction models to reinforce
normal [51], high-performance [52,53], recycled aggregate, and
self-compacted concrete [54,55]. The predicting models results
were precisely analyzed [56,57].
Genetic programming (GP) is a new specialty in genetic algo-
rithms (GAs); according to this program, the solutions instead of
binary strings are computer-based. This approach, which is relied
on GP, is a predictive tool to generate predictive equations regard-
ing the association among input variables. The equations formu-
lated based on GP are simple and easy to implement. GEP [58] is
an achievement in GP. GEP solutions are the computer-based, dif-
ferent sized and shaped programs, which can encode fixed-length
linear chromosomes. Some other studies have applied GEP to
model civil engineering [59–62], particularly in concrete structures
[63,64].
In this study after introducing existing models for predicting
the moment capacity, a new formulation for this purpose is pre-
sented using Group Method of Handling (GMDH). For more evalu-
ation of the effect of each input parameters which influences the
moment capacity, a sensitivity analysis was performed. Based on
many models derived from the GMDH approach, the best model
with high accuracy and empirical equation is presented.
2. Existing models for predicting the moment capacity
There are some equations to measure the ferrocement moment
capacity. The typical cross section of ferrocement is shown in Fig. 1.
In this figure, b and h are respectively width and total depth of
specimens. Additionally, Table 1 presents the used parameters
and the corresponding descriptions.
2.1. Plastic analysis method
Mansur and Paramasivam [65] considered equilibrium of forces
conditions in an innovative method.
Mu ¼ rtu b h x1
ð Þ
h
2
ð1Þ
2.2. Mechanism approach method
Based on the plastic analysis, Paramasivam and Ravindrarajah
[17] proposed a simple method; according to this method, the neu-
tral axis from the surface has small size, which in turn imposes
pressure to reinforcements.
Mu ¼ rtu
bh
2
2
ð2Þ
2.3. Simplified method
A non-dimensionalized regression equation was suggested by
Naaman and Homrich [66] to calculate the ferrocement moment
capacity.
y ¼ 0:005 þ 0:422x 0:0772x2
ð3Þ
where
x ¼
vf ry
f
0
c
ð4Þ
y ¼
Mu
gof
0
c bh
2
ð5Þ
2.4. GEP models
A Gene Expression Programming (GEP) based model to formu-
late the ferrocement members moment capacity was developed
by Gandomi et al. [19].
Mu;GEP ¼
b h 11
ð Þ h þ fcu
ð Þ
5184
fumf
0:6
ffiffiffiffiffiffi
fcu
p ð6Þ
H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391 383

3. 3. Model development
3.1. Group Method of Data Handling (GMDH)
GMDH model is an algorithm to find a linear parameter com-
plex polynomial function. The combinatorial model is a branch of
polynomial function [67,68]. For instance, if a data-set of two input
variables (x1 and x2) and an output variable (y) is being modeled,
the quadratic polynomial function for which the optimization of
constants, ap in where, must be performed is as follow:
y ¼ a0 þ a1x1 þ a2x2 þ a3x1:x2 þ a4x2
1 þ a5x2
2 ð7Þ
The maximum power of polynomial function is user-defined,
and the problem complexity would be increased in case higher
orders are chosen. Combinatorial GMDH uses an optimally com-
plex model; for instance, y ¼ a0 þ a3x1x2 is a subset of a complete
polynomial function. Data preprocessing stage lets different oper-
ators to apply x1 and x2 variables; for example, exponent, a sigmoid
function, time series lags, etc. However, the final model is linear in
the parameters. It is time-consuming to investigate a full combina-
tion of model components; hence, the search can be limited in a
way that no more than n terms are included in the model. The
two-term models, for example, provide search opportunity among
thousands possible combination of variables and probably more
massive sets might be assembled. Moreover, models with 25
polynomial or linear terms are not suitable candidates for a full
search. For a linear combination of three input variables, seven dif-
ferent possibilities exist (2m
1 is the number of possibilities for
linear combination, in which m is the number of input variables).
Combinatorial GMDH is a time-consuming algorithm. However, it
is capable of providing a closed-form solution which can straight-
forwardly provide the target, if proper parameters, such as the
appropriate fitness function, are chosen before running the algo-
rithm. Further information about combinatorial GMDH approach
can be found in the references [67,69–72].
3.2. Database
The selected database contains 75 test result that Mashrei et al.
[3] put together from 9 literature [3,17,18,65,73–77]. Mechanical
and geometrical properties of the collected specimens are pre-
sented in Table 2. Input parameters consist of the width of speci-
mens (b), total depth specimens (h), the compressive strength of
ferrocement (fcu), the ultimate strength of wire mesh (ful) and vol-
ume fraction of wire mesh (vf). The output value is moment capac-
ity of ferrocement members (Mu). In addition, the statistical
properties of inputs and output are shown in Table 3.
Normalization of all data was made before using them in GMDH
model. In order to scale the data from 0.1 to 0.9, minimum and
maximum values were taken to use a linear relationship between
those values. Table 4 shows equations for each parameter.
3.3. Proposed GMDH models
Eqs. (8) was driven using combinatorial GMDH method for esti-
mating moment capacity of ferrocement elements after so many
trials and the highest regression value and the least model error
was achieved. Trials consist of the input parameters which are b,
h, fcu, ful, vf and Mu with different orders and combinations with
constants. The best capturing model can be presented as follow.
C1 þ
C2hfcuful
b
C3h
2
vf
þ
bvf C4h
2
ful C5vf
ful
ð8Þ
Herein, Ci (i ranging between 1 and 5) are the model constants
and presented in Table 5. Figs. 2 and 3 shows the comparison of
measured and predicted Mu values by GMDH model after normal-
ization, as it can be seen, 60 sets of the database (80% of all data)
are considered for training and others for testing purpose. The con-
sistency of the measured and predicted values for Mu can be vividly
seen in Fig. 2, specifically for the testing group.
Closed-form formulation for predicting moment capacity Mu,
can be finally written in terms of inputs (width of specimens (b),
total depth specimens (h), the compressive strength of ferrocement
(fcu), the ultimate strength of wire mesh (ful) and volume fraction of
wire mesh (vf)) as below.
Mu;GMDH ¼ 0:091 þ
0:092hfcuful
b
0:042h
2
vf
þ
bvf 10:37h
2
ful 0:021vf
ful
ð9Þ
The statistical values of mean absolute error (MAE), root mean
absolute error (RMAE), mean absolute percentage error (MAPE),
normalized root mean square error (NRMSE) are shown in Table 6.
It can be seen that the errors have acceptable values and the real
Fig. 1. Cross section of ferrocement specimen.
Table 1
List of used parameters.
Parameter Description
h Overall depth of the section
b Width of the beam
x1 Depth of the neutral axis
fcu Cube compressive strength of the matrix
fu The ultimate tensile strength of wire mesh
As The cross-sectional area of steel
rtu
As :f u
b:h
vf Total volume of the fraction
go Global efficiency factor of mesh reinforcement
ry Yield tensile strength of wire mesh
f
0
c
Cylinder compressive strength of the matrix
384 H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391

4. Table 2
Mechanical and geometrical properties of the collected database.
Test no. b (mm) h (mm) f cu (MPa) ful (MPa) vf (%) Mu Exp (N.m) Reference
1 200 50 42.7 600 0.25 393.5 [3]
2 200 50 42.7 600 0.5 500.3
3 200 50 46.2 600 0.97 901
4 200 50 48 600 1.3 1434.2
5 200 80 50.8 600 0.164 483.6
6 200 80 50.28 600 0.33 684
7 200 80 49.1 600 0.65 1584
8 200 80 50 600 1.03 2168
9 300 50 46.2 600 0.28 451
10 300 50 49.1 600 0.5 717
11 300 50 49.1 600 0.99 1701
12 300 50 50 600 1.3 2001
13 300 80 40.4 600 0.164 850.5
14 300 80 50 600 0.34 1300
15 300 80 50.8 600 0.63 3101
16 300 80 50 600 1.1 3752
17 100 100 29.9 500 3.92 3937 [17]
18 100 25 50 371 2.1 137.5
19 100 35 50 371 1.48 190
20 100 25 50 371 3.01 202.5
21 100 35 50 371 2.22 285
22 100 25 50 371 4.18 237.5
23 100 20 29.9 533 3.62 176 [18]
24 100 30 29.9 533 3.62 303
25 100 40 29.9 533 3.62 690
26 100 60 29.9 533 3.62 1240
27 100 100 29.9 533 3.62 3375
28 100 20 29.9 500 3.92 171
29 100 30 29.9 500 3.92 351
30 100 40 29.9 500 3.92 616
31 100 60 29.9 500 3.92 1463
32 100 35 50 371 2.96 355 [65]
33 100 25 56 371 2.28 155
34 100 25 45 371 1.72 125
35 100 25 45 371 2.28 145
36 100 25 45 371 2.86 170
37 100 25 38 371 2.28 135
38 200 25 28.3 979 2.43 323.6 [73]
39 76 50 36 628 0.62 241.3
40 76 50 36 628 1.85 734.5
41 76 50 36 628 2.5 969.5
42 76 50 36 628 3.12 1051
43 76 50 36 628 4.98 1367.3
44 76 50 36 628 2.52 870.1
45 76 50 36 628 5.04 1096.1
46 76 50 36 628 0.32 144
47 76 50 36 628 0.94 312
48 76 50 36 628 1.54 502.3
49 76 50 36 628 0.34 176.4
50 76 50 36 628 1.02 474.6
51 76 50 36 628 1.68 734.5
52 76 50 36 628 2.36 842
53 76 50 36 628 3.4 1011.4
54 130 13 62 562 4.62 93.2 [74]
55 100 26 24.2 382.6 0.7 76.1
56 100 26 24.2 382.6 1.38 89.7
57 100 26 24.2 382.6 2.85 168.4
58 100 26 24.2 382.6 4.09 209.7
59 100 26 24.2 382.6 5.48 250.4
60 100 26 24.2 382.6 6.82 270.2
61 130 13 62 513 2.22 43 [75]
62 130 13 62 513 4.44 89.5
63 130 13 62 513 6.64 130.6
64 130 13 62 714 2.22 50.1
65 130 13 62 714 4.44 106.7
66 130 13 62 714 6.64 155.3
67 130 13 62 562 1.54 33
68 130 13 62 562 3.1 65.9
(continued on next page)
H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391 385

5. deviations (MAPE) of the presented model from the experimentally
measured values are 7.06 and 6.49 percent for training and testing
datasets, respectively.
Fig. 4 outlines the situation of error versus Mu observations. It
can be seen that absolute error values are negligible, for which
the minimum and maximum normalized error values of 0.0002
and 0.077 were acquired.
4. Results and discussion
The comparison between experimental and theoretical predic-
tions of ferrocement members results are discussed in this section.
The results of Back-Propagation Neural Networks (BPNN), Adaptive
Neuro-Fuzzy Inference System (ANFIS), Gene Expression Program-
ming (GEP) and traditional equations are presented in Table 7 in
order to provide a direct comparison between the presented model
and available literature.
Among conventional benchmark model performance checking
variables, MAE, RMSE, MAPE, NMAE and R2
are implemented
herein. These parameters formulation are presented in Eqs. (10)–
(14). Based on the available results and the introduced model
checking variables, Table 8 can be presented in which different sta-
tistical parameters along with the model performance variables for
BPNN, ANFIS, GEP, GMDH and traditional methods for prediction of
moment capacity calling plastic, mechanism and simplified
approaches are provided for ferrocement members.
MAE ¼
1
n
X
n
i¼1
Mu model
ð Þ Mu actual
ð Þ
ð10Þ
Fig. 2. Measure and predicted moment capacity of ferrocement elements for training and testing in GMDH.
Table 3
Moment capacity of ferrocement elements statistical data.
Variables b (mm) h (mm) fcu (MPa) ful (MPa) vf (%) Mu Exp (N.m)
Mean 151 43 39.948 543.096 2.425 819.789
Minimum 76 13 12.6 371 0.164 33
Maximum 400 100 62 979 8.25 5393
Standard deviation 91.030 22.238 13.180 140.258 1.803 1054.078
Coefficient of variation 0.604 0.518 0.330 0.258 0.744 1.286
Table 4
Moment capacity of ferrocement elements statistical data.
Variables Equation
b bscaled ¼ 0:9 0:1
ð Þ b bmin
ð Þ= bmax bmin
ð Þ
½ þ 0:1
d hscaled ¼ 0:9 0:1
ð Þ h hmin
ð Þ= hmax hmin
ð Þ
½ þ 0:1
f cu f cuscaled ¼ 0:9 0:1
ð Þ fcu fcumin
ð Þ= fcumax f cumin
ð Þ
½ þ 0:1
f ul f ulscaled ¼ 0:9 0:1
ð Þ f ul f ulmin
ð Þ= ful fulmin
ð Þ
½ þ 0:1
vf vfscaled ¼ 0:9 0:1
ð Þ vf vfmin
= vfmax vfmin
þ 0:1
Mu Muscaled ¼ 0:9 0:1
ð Þ Mu Mumin
ð Þ= Mumax Mumin
ð Þ
½ þ 0:1
Table 2 (continued)
Test no. b (mm) h (mm) f cu (MPa) ful (MPa) vf (%) Mu Exp (N.m) Reference
69 400 75 12.6 371 0.8 3450 [76]
70 400 75 12.6 371 1.2 5393
71 400 50 12.6 371 0.6 955
72 400 50 12.6 371 1.2 1935
73 100 26 24.2 382.6 8.25 293.5 [77]
74 200 25 28.3 979 0.81 102.5
75 200 25 28.3 979 1.62 191.2
Table 5
Model constants.
Model Constants
C1 C2 C3 C4 C5
0.0912964 0.0919207 0.0418427 10.3667 0.0210076
386 H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391

6. RMSE ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N
X
N
i¼1
Mu model
ð Þ Mu actual
ð Þ
2
v
u
u
t ð11Þ
MAPE ¼
1
M
PM
i¼1 Mu mod el
ð Þ Mu actual
ð Þ
PM
i¼1Mu actual
ð Þ
100
#
ð12Þ
NMAE ¼
1
n
Pn
i¼1 Mu model
ð Þ Mu actual
ð Þ
Mu model
ð Þ Mu actual
ð Þ
ð13Þ
R2
¼ 1
PN
i¼1 Mu actual
ð Þ Mu actual
ð Þ
2
P
N
i¼1
Mu actual
ð Þ Mu actual
ð Þ
2
ð14Þ
It can be seen that the mean absolute percentage error (MAPE)
of ANFIS and BPNN models are slightly better than the GMDH
model but compared with GEP model and other traditional
method, the GMDH model provides lower MAPE. However, the
considerable advantage of GMDH model is introducing an explicit
and closed form equation (presented as Eq. (9)) which eases the
calculation of Mu for ferrocement members.
Developed models based on soft computing approaches includ-
ing GMDH combinatorial, can be evaluated using different criteri-
ons such as linear regression coefficient(R) or R2
to find out the
desirability of the predicted target values. In addition, error values
can be directly examined using different available benchmarks
such as mean absolute error (MAE), root mean squared error
(RMSE). Moreover, predicted parameter error values can be pre-
sented in the form of mean absolute percentage error (MAPE)
which provides the exact deviation of the predicted data from
actual experimentally achieved ones. As stated in Table 8, MAE,
RMSE and MAPE are respectively 0.0157, 0.0232 and 6.8% for the
introduced model. These values are less than that of achieved for
Plastic, Mechanism, Simplified method and GEP. Thus, between
the models which are able to provide a mathematically
Fig. 3. Experimentally measured values versus GMDH approach predicted moment capacity of ferrocement elements (a) Training and (b) Testing datasets, in the form of
normalized values.
Fig. 4. Error values according to the acquired GMDH based correlation.
Table 6
Training and testing evaluation.
MAE RMSE MAPE (%) NRMSE (%) NMAE (%) R2
Correlation Coefficient (R)
Training 0.0164 0.0241 7.06% 3.01% 2.05% 0.9779 0.9889
Testing 0.0134 0.0192 6.49% 2.41% 1.68% 0.9804 0.9902
H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391 387

7. Table 7
A comparison between results of experimental and predicted moment capacity datasets.
Mu (N.m) BPNN [3] ANFIS [3] Plastic [61] Mechanism [16] Simplified method [62] GEP [18] GMDH
0.1000 0.1031 0.0999 0.1013 0.1020 0.0989 0.0992 0.1069
0.1015 0.1048 0.1014 0.1032 0.1043 0.1005 0.0999 0.1059
0.1026 0.1069 0.1021 0.1059 0.1080 0.1019 0.1010 0.1160
0.1049 0.1074 0.1056 0.1069 0.1094 0.1024 0.1014 0.1100
0.1064 0.1039 0.1053 0.1013 0.1018 0.0986 0.1077 0.1050
0.1084 0.1090 0.1091 0.1096 0.1136 0.1053 0.1024 0.1086
0.1085 0.1105 0.1098 0.1067 0.1086 0.1014 0.1140 0.1025
0.1090 0.1101 0.1095 0.1112 0.1163 0.1055 0.1030 0.1120
0.1104 0.1127 0.1092 0.1254 0.1316 0.1143 0.1394 0.1255
0.1110 0.1129 0.1114 0.1138 0.1208 0.1079 0.1041 0.1203
0.1137 0.1101 0.1115 0.1084 0.1098 0.1017 0.1153 0.1080
0.1146 0.1135 0.1142 0.1148 0.1228 0.1098 0.1045 0.1094
0.1152 0.1164 0.1138 0.1121 0.1150 0.1035 0.1186 0.1109
0.1156 0.1129 0.1158 0.1114 0.1132 0.1031 0.1183 0.1120
0.1166 0.1208 0.1182 0.1079 0.1084 0.1036 0.1245 0.1291
0.1167 0.1151 0.1163 0.1125 0.1150 0.1037 0.1191 0.1123
0.1182 0.1140 0.1208 0.1129 0.1150 0.1039 0.1200 0.1145
0.1183 0.1206 0.1182 0.1197 0.1337 0.1135 0.1065 0.1233
0.1202 0.1227 0.1192 0.1159 0.1231 0.1073 0.1244 0.1152
0.1204 0.1193 0.1209 0.1164 0.1202 0.1056 0.1226 0.1161
0.1206 0.1190 0.1206 0.1147 0.1242 0.1077 0.1174 0.1169
0.1213 0.1187 0.1214 0.1144 0.1237 0.1075 0.1172 0.1177
0.1214 0.1208 0.1182 0.1079 0.1084 0.1036 0.1256 0.1307
0.1234 0.1233 0.1290 0.1186 0.1205 0.1070 0.1316 0.1200
0.1236 0.1264 0.1260 0.1468 0.1681 0.1311 0.1623 0.1438
0.1253 0.1193 0.1228 0.1173 0.1210 0.1060 0.1239 0.1180
0.1264 0.1273 0.1265 0.1217 0.1346 0.1118 0.1315 0.1233
0.1305 0.1257 0.1323 0.1246 0.1314 0.1098 0.1302 0.1242
0.1311 0.1321 0.1340 0.1199 0.1218 0.1095 0.1389 0.1494
0.1324 0.1309 0.1339 0.1273 0.1482 0.1168 0.1384 0.1307
0.1354 0.1342 0.1385 0.1315 0.1607 0.1213 0.1445 0.1365
0.1376 0.1401 0.1351 0.1283 0.1324 0.1115 0.1417 0.1337
0.1389 0.1393 0.1390 0.1355 0.1752 0.1261 0.1505 0.1414
0.1403 0.1467 0.1406 0.1386 0.1595 0.1231 0.1512 0.1485
0.1416 0.1466 0.1495 0.1309 0.1351 0.1153 0.1513 0.1657
0.1434 0.1394 0.1422 0.1627 0.2047 0.1467 0.1808 0.1602
0.1475 0.1474 0.1461 0.1391 0.1606 0.1236 0.1517 0.1490
0.1481 0.1559 0.1419 0.1387 0.1459 0.1166 0.1505 0.1455
0.1538 0.1427 0.1532 0.1170 0.1175 0.1115 0.1595 0.1283
0.1624 0.1617 0.1624 0.1440 0.1454 0.1270 0.1982 0.1804
0.1659 0.1520 0.1545 0.1345 0.1396 0.1172 0.1542 0.1693
0.1673 0.1676 0.1673 0.1514 0.1524 0.1410 0.2095 0.1392
0.1697 0.1675 0.1682 0.1483 0.1510 0.1241 0.1927 0.1623
0.1700 0.1803 0.1791 0.1509 0.1619 0.1267 0.1707 0.1890
0.1870 0.1851 0.1880 0.1734 0.2115 0.1457 0.1959 0.1928
0.1972 0.2059 0.1999 0.1788 0.1810 0.1515 0.2694 0.2043
0.1981 0.1846 0.1752 0.1725 0.2096 0.1450 0.1950 0.1902
0.2021 0.2031 0.2020 0.1754 0.1790 0.1404 0.2410 0.2218
0.2047 0.2011 0.1977 0.1629 0.1797 0.1343 0.1795 0.1991
0.2047 0.1911 0.1885 0.1570 0.1708 0.1305 0.1748 0.1937
0.2207 0.2219 0.2193 0.1765 0.2019 0.1435 0.1928 0.2139
0.2220 0.2223 0.2220 0.1792 0.1810 0.1565 0.2723 0.2458
0.2249 0.2254 0.2233 0.1791 0.2064 0.1453 0.1967 0.2183
0.2296 0.2442 0.2313 0.1971 0.2070 0.1456 0.2400 0.2179
0.2376 0.2378 0.2375 0.1685 0.1781 0.1312 0.2980 0.2900
0.2398 0.2254 0.2233 0.1791 0.2064 0.1453 0.1962 0.2177
0.2460 0.2494 0.2550 0.2000 0.2465 0.1615 0.2167 0.2405
0.2519 0.2425 0.2455 0.1934 0.2331 0.1561 0.2106 0.2337
0.2587 0.2793 0.2786 0.2299 0.3177 0.1888 0.2491 0.2777
0.2801 0.2985 0.2785 0.2694 0.3528 0.2073 0.3122 0.3056
0.2891 0.2889 0.2892 0.2207 0.2240 0.1795 0.3614 0.3443
0.2991 0.2793 0.2786 0.2299 0.3177 0.1888 0.2480 0.2764
0.3091 0.2990 0.3080 0.2249 0.2406 0.1583 0.2677 0.2524
0.3134 0.2978 0.3147 0.2713 0.3570 0.2091 0.3142 0.3154
0.3315 0.3308 0.3313 0.2581 0.2670 0.1832 0.3576 0.3106
0.3490 0.3244 0.3123 0.2490 0.2630 0.1717 0.3150 0.3012
0.3839 0.3838 0.3840 0.2265 0.2612 0.1617 0.4026 0.4101
0.3937 0.3938 0.3938 0.2906 0.3134 0.1906 0.3540 0.3478
0.4187 0.4194 0.4187 0.3581 0.3816 0.2263 0.4404 0.4218
0.5579 0.5581 0.5577 0.3400 0.3530 0.2285 0.4800 0.4800
0.5988 0.5985 0.5992 0.5792 0.8110 0.4069 0.6648 0.6653
0.6100 0.6100 0.6102 0.3070 0.3443 0.1995 0.6488 0.6542
0.6551 0.6545 0.6552 0.5255 0.5679 0.3078 0.6339 0.6758
0.6827 0.6828 0.6823 0.5845 0.8227 0.4117 0.6702 0.7006
0.9000 0.8998 0.8998 0.3909 0.4688 0.2451 0.8013 0.8398
*Note that testing sets are shown in bold format.
388 H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391

8. closed-form solution, the error values are acceptable. ANFIS and
BPNN approaches are not able to provide a simple and closed-
form formulation for moment capacity and merely the best opti-
mized structures were previously reported.
Additionally, entire achieved results are depicted in Fig. 5 which
provides information about the diversity of the predicted values
using different approaches versus the experimentally measured
Mu for ferrocement members. Assuming 5 to 75 error percentage
zones with 5 percent increments, Table 9 can be presented. The
result of this discretization is aimed to provide the error for all of
the predictions in each model. Putting BPNN and ANFIS aside,
GMDH based model is the most accurate approach among other
available models. As it was previously pointed out, BPNN and
ANFIS do not provide a closed form mathematical formulation.
The estimated results of GMDH versus measured values are
compared with the BPNN, ANFIS and other traditional methods
in Fig. 5. Among the models, it seems GMDH has the best perfor-
mance after ANFIS and BPN, better than GEP, BPNN and traditional
methods.
5. Sensitivity analysis
In order to calculate importance of variables, their mean values
are replaced in the model one by one, and the root mean squared
error (RMSE) is then calculated. The original model error is consid-
ered a zero percent impact on RMSE and 100% impact is a case
where all variables are replaced with their mean values. The
impact can easily exceed 100% in the case the variable in a model
is multiplied by another variable or squared. A small negative per-
centage can also happen if a variable is merely useless for the
model. Impact on RMSE can be defined as a percentage value that
aids to compare variables and reference values which can be
calculated as
Impact on RMSE ¼ RVar Rori
ð Þ= Rall Rori
ð Þ
½ 100% ð15Þ
where RVar is the RMSE of the considered variable, Rori is zero-
impact RMSE, and Rall is RMSE of a model where all variables are
replaced with mean values. As it is shown in Fig. 6, vf with
Fig. 5. Comparison of predicted values of Mu versus experimental data for proposed
GMDH equation and other existing models.
Table 9
Distribution of errors for different Mu models versus GMDH model relative to experimental values.
Error
Deviations
Number of data Percentage of data
BPNN ANFIS Plastic Mechanism Simplified
method
GEP GMDH BPNN ANFIS Plastic Mechanism Simplified
method
GEP GMDH
5 63 66 28 23 11 32 37 84.0 88.0 37.3 30.7 14.7 42.7 49.3
10 75 73 39 40 15 50 56 100.0 97.3 52.0 53.3 20.0 66.7 74.7
15 75 75 52 52 31 62 68 100.0 100.0 69.3 69.3 41.3 82.7 90.7
20 75 75 57 58 38 66 74 100.0 100.0 76.0 77.3 50.7 88.0 98.7
25 75 75 66 65 42 68 75 100.0 100.0 88.0 86.7 56.0 90.7 100.0
30 75 75 71 68 50 73 75 100.0 100.0 94.7 90.7 66.7 97.3 100.0
35 75 75 71 69 56 74 75 100.0 100.0 94.7 92.0 74.7 98.7 100.0
40 75 75 72 72 64 75 75 100.0 100.0 96.0 96.0 85.3 100.0 100.0
45 75 75 73 74 66 75 75 100.0 100.0 97.3 98.7 88.0 100.0 100.0
50 75 75 74 75 68 75 75 100.0 100.0 98.7 100.0 90.7 100.0 100.0
55 75 75 74 75 71 75 75 100.0 100.0 98.7 100.0 94.7 100.0 100.0
60 75 75 75 75 73 75 75 100.0 100.0 100.0 100.0 97.3 100.0 100.0
65 75 75 75 75 73 75 75 100.0 100.0 100.0 100.0 97.3 100.0 100.0
70 75 75 75 75 74 75 75 100.0 100.0 100.0 100.0 98.7 100.0 100.0
75 75 75 75 75 75 75 75 100.0 100.0 100.0 100.0 100.0 100.0 100.0
Table 8
Evaluation of the exiting and proposed model.
Statistical Parameter BPNN ANFIS Plastic Mechanism Simplified method GEP GMDH
Mean 0.2167 0.2162 0.1791 0.2026 0.1444 0.2196 0.2179
Standard Deviation 0.1569 0.1570 0.1019 0.1371 0.0598 0.1521 0.1567
MAE 0.0045 0.0034 0.0403 0.0388 0.0734 0.0204 0.0157
RMSE 0.0072 0.0072 0.0842 0.0774 0.1309 0.0298 0.0232
MAPE 2.40% 1.70% 12.65% 12.69% 23.50% 8.88% 6.81%
NRMSE 3% 3% 39% 36% 60% 14% 11%
NMAE 2% 2% 19% 18% 34% 9% 7%
R2
0.9979 0.9980 0.8427 0.7637 0.7570 0.9641 0.9780
Correlation Coefficient (R) 0.9990 0.9990 0.9180 0.8739 0.8701 0.9819 0.9890
H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391 389

9. 172.29% and fcu with 2.66%, have the maximum and minimum
influence on RMSE, respectively.
6. Conclusion
In this study, GMDH was applied in order to evaluate the
moment capacity of ferrocement members based on key pre-
determined input variables. The regression values of the chosen
network for training and testing were 0.9779 and 0.9804 respec-
tively. MAPE, RMSE, MAE and NMAE were calculated as 6.81%,
0.0232, 0.0157 and 7%, respectively. The results of the proposed
equation indicated that it can be used for prediction with a high
level of accuracy. Furthermore, comparing the results with those
of different studies indicated that the GMDH method could accu-
rately predict the moment capacity of ferrocement members. The
volume fraction of wire mesh importance plays a significant role
in the moment capacity of ferrocement members. It is also worth
to mention that the width of specimens and total depth specimens
generally affect the resulting properties of the moment capacity of
ferrocement members.
References
[1] ACI 549.1R-93: Guide for the Design, Construction Repair of Ferrocement,
n.d.
[2] M.T. Audu, O.W. Oseni, Comparative studies of engineering properties of
ferrocement and fibrecement material, Int. J. Curr. Res. Acad. Rev. 3 (2015)
182–189.
[3] M.A. Mashrei, N. Abdulrazzaq, T.Y. Abdalla, M.S. Rahman, Neural networks
model and adaptive neuro-fuzzy inference system for predicting the moment
capacity of ferrocement members, Eng. Struct. 32 (2010) 1723–1734, https://
doi.org/10.1016/j.engstruct.2010.02.024.
[4] J.C. Adajar, T. Hogue, C. Jordan, Ferrocement for hurricane–prone state of
Florida, in: Struct Faults + Repair – 2006, Edinburgh, Scotland, UK, 2006.
[5] H. Eskandari, A. Madadi, Investigation of ferrocement channels using
experimental and finite element analysis, Eng. Sci. Technol. Int. J. 18 (2015)
769–775, https://doi.org/10.1016/j.jestch.2015.05.008.
[6] A.M. Waliuddin, S.F.A. Rafeeqi, Study of the behavior of plain concrete confined
with ferrocement, J. Ferrocem. 24 (1994) 139–151.
[7] S.F.A. Rafeeqi, T. Ayub, Investigation of versatility of theoretical prediction
models for plain concrete confined with Ferrocement, Asian J. Civ. Eng. 12
(2011) 337–352.
[8] P.P. Bansal, B.E. Civil, C. Structures, Effect of wire mesh orientation on strength
of beams retrofitted using ferrocement Jackets, Int. J. 5 (2014) 8–19.
[9] E.H. Fahmy, Y.B.I. Shaheen, Y.S. Korany, Repairing reinforced concrete beams
by ferrocement, J. Ferrocem. 27 (1997) 19–32.
[10] T. Onet, C. Magureanu, V. Vescan, Aspects concerning the behaviour of
ferrocement in flexure, J. Ferrocem. 22 (1992) 1–16.
[11] K.C.G. Ong, P. Paramasivam, C.T.E. Lim, Flexural strengthening of reinforced
concrete beams using ferrocement laminates, J. Ferrocem. 22 (1992) 331.
[12] D.R. Seshu, Flexural strength assessment of ferrocement confined reinforced
concrete (FCRC) beams, J. Ferrocem. 31 (2001) 53–63.
[13] A.B.M.A. Kaish, M. Jamil, S.N. Raman, M.F.M. Zain, L. Nahar, Ferrocement
composites for strengthening of concrete columns: a review, Constr. Build.
Mater. 160 (2018) 326–340, https://doi.org/10.1016/j.conbuildmat.2017.11.054.
[14] S.U. Khan, S.F.A. Rafeeqi, T. Ayub, Strengthening of rc beams in flexure using
ferrocement, Iran. J. Sci. Technol. – Trans. Civ. Eng. 37 (2013) 353–365.
[15] A.E. Naaman, Ferrocement and Laminated Cementitious Composites, Techno
press, MI, 2000.
[16] S. Quek, S. On, Uncertainty in flexural capacity prediction of ferrocement
elements, J. Mater. Civ. Eng. 3 (1991) 263–277, https://doi.org/10.1061/(ASCE)
0899-1561(1991) 3:4(263).
[17] P. Paramasivam, R. Ravindrarajah, Effect of arrangements of reinforcements on
mechanical properties of ferrocement, Struct. J. 85 (1988) 3–11 (accessed April
5, 2018) https://www.concrete.org/publications/
internationalconcreteabstractsportal.aspx?m=detailsID=2944.
[18] M. Mansur, Ultimate strength design of ferrocement in flexure, J. Ferrocem. 18
(1988) 385–395.
[19] A.H. Gandomi, D.A. Roke, K. Sett, Genetic programming for moment capacity
modeling of ferrocement members, Eng. Struct. 57 (2013) 169–176, https://
doi.org/10.1016/j.engstruct.2013.09.022.
[20] K.J. Åström, P. Eykhoff, System identification – a survey, Automatica. 7 (1971)
123–162, https://doi.org/10.1016/0005-1098(71)90059-8.
[21] A.M. Hanna, D. Ural, G. Saygili, Evaluation of liquefaction potential of soil
deposits using artificial neural networks, Eng. Comput. 24 (2007) 5–16,
https://doi.org/10.1108/02644400710718547.
[22] D. Padmini, K. Ilamparuthi, K.P. Sudheer, Ultimate bearing capacity prediction
of shallow foundations on cohesionless soils using neurofuzzy models,
Comput. Geotech. 35 (2008) 33–46, https://doi.org/10.1016/J.
COMPGEO.2007.03.001.
[23] D. Rezazadeh Eidgahee, F. Fasihi, H. Naderpour, Optimized artificial neural
network for analyzing soil-waste rubber shred mixtures, Sharif. J. Civ. Eng. 31
(2) (2015) 105–111.
[24] E. Sanchez, T. Shibata, L.A. Zadeh, Genetic algorithms and fuzzy logic systems:
soft computing perspectives (advances in fuzzy systems-applications and
theory, Soft Comput. Perspect. 7 (1997).
[25] R. Madandoust, R. Ghavidel, N. Nariman-zadeh, Evolutionary design of
generalized GMDH-type neural network for prediction of concrete
compressive strength using UPV, Comput. Mater. Sci. 49 (2010) 556–567,
https://doi.org/10.1016/j.commatsci.2010.05.050.
[26] H. Adeli, Neural networks in civil engineering: 1989–2000, Comput. Civ.
Infrastruct. Eng. 16 (2001) 126–142, https://doi.org/10.1111/0885-
9507.00219.
[27] X. Wu, J. Ghaboussi, J.H. Garrett, Use of neural networks in detection of
structural damage, Comput. Struct. 42 (1992) 649–659, https://doi.org/
10.1016/0045-7949(92)90132-J.
[28] H. Naderpour, P. Fakharian, A synthesis of peak picking method and wavelet
packet transform for structural modal identification, KSCE J. Civ. Eng. 20
(2016) 2859–2867, https://doi.org/10.1007/s12205-016-0523-4.
[29] H. Adeli, H.S. Park, A neural dynamics model for structural optimization –
theory, Comput. Struct. 57 (1995) 383–390, https://doi.org/10.1016/0045-
7949(95)00048-L.
[30] H.M. Chen, K.H. Tsai, G.Z. Qi, J.C.S. Yang, F. Amini, Neural network for structure
control, J. Comput. Civ. Eng. 9 (1995) 168–176, https://doi.org/10.1061/(ASCE)
0887-3801(1995) 9:2(168).
[31] J. Ghaboussi, J.H. Garrett, X. Wu, Knowledge-Based modeling of material
behavior with neural networks, J. Eng. Mech. 117 (1991) 132–153, https://doi.
org/10.1061/(ASCE)0733-9399(1991) 117:1(132).
[32] H. Naderpour, A. Kheyroddin, G.G. Amiri, Prediction of FRP-confined
compressive strength of concrete using artificial neural networks, Compos.
Struct. 92 (2010) 2817–2829, https://doi.org/10.1016/
j.compstruct.2010.04.008.
93.13% 97.26%
2.66% 2.99%
172.29%
100.00%
0.00%
0%
50%
100%
150%
200%
b (mm) d (mm) fcu (MPa) ful (MPa) vf (%) Rall R0
Impact
on
RMSE
(%)
Variable
Fig. 6. Importance of input parameters by considering the influence on RMSE.
390 H. Naderpour et al. / Engineering Science and Technology, an International Journal 23 (2020) 382–391

10. [33] D. Rezazadeh Eidgahee, A.H. Rafiean, A. Haddad, A novel formulation for the
compressive strength of ibp-based geopolymer stabilized clayey soils using
ANN and GMDH-NN approaches, Iran J. Sci. Technol. Trans. Civ. Eng. (2019),
https://doi.org/10.1007/s40996-019-00263-1.
[34] C.M. Tam, C.F. Fang, Comparative cost analysis of using high-performance
concrete in tall building construction by artificial neural networks, ACI Struct.
J. 96 (1999), https://doi.org/10.14359/767.
[35] K.C. Sarma, H. Adeli, Life-cycle cost optimization of steel structures, Int. J.
Numer. Meth. Eng. 55 (2002) 1451–1462, https://doi.org/10.1002/nme.549.
[36] Ju-Won Oh, In-Won Lee, Ju-Tae Kim, Gyu-Won Lee, Application of neural
networks for proportioning of concrete mixes, ACI Mater. J. 96 (1999), https://
doi.org/10.14359/429.
[37] A.F. Bingöl, A. Tortum, R. Gül, Neural networks analysis of compressive
strength of lightweight concrete after high temperatures, Mater. Des. 52
(2013) 258–264, https://doi.org/10.1016/j.matdes.2013.05.022.
[38] M. Sarıdemir, _
I.B. Topçu, F. Özcan, M.H. Severcan, Prediction of long-term
effects of GGBFS on compressive strength of concrete by artificial neural
networks and fuzzy logic, Constr. Build. Mater. 23 (2009) 1279–1286, https://
doi.org/10.1016/j.conbuildmat.2008.07.021.
[39] F. Demir, Prediction of elastic modulus of normal and high strength concrete
by artificial neural networks, Constr. Build. Mater. 22 (2008) 1428–1435,
https://doi.org/10.1016/j.conbuildmat.2007.04.004.
[40] A. Öztas
ß, M. Pala, E. Özbay, E. Kanca, N. Çag
lar, M.A. Bhatti, Predicting the
compressive strength and slump of high strength concrete using neural
network, Constr. Build. Mater. 20 (2006) 769–775, https://doi.org/10.1016/
j.conbuildmat.2005.01.054.
[41] H. Naderpour, A.H. Rafiean, P. Fakharian, Compressive strength prediction of
environmentally friendly concrete using artificial neural networks, J. Build.
Eng. 16 (2018), https://doi.org/10.1016/j.jobe.2018.01.007.
[42] R.M. Haj-Ali, K.E. Kurtis, A.R. Sthapit, Neural network modeling of concrete
expansion during long-term sulfate exposure, ACI Mater. J. 98 (2001), https://
doi.org/10.14359/10158.
[43] S. Maru, A.K. Nagpal, Neural network for creep and shrinkage deflections in
reinforced concrete frames, J. Comput. Civ. Eng. 18 (2004) 350–359, https://
doi.org/10.1061/(ASCE)0887-3801(2004) 18:4(350).
[44] B. Wang, T. Man, H. Jin, Prediction of expansion behavior of self-stressing
concrete by artificial neural networks and fuzzy inference systems, Constr.
Build. Mater. 84 (2015) 184–191, https://doi.org/10.1016/
j.conbuildmat.2015.03.059.
[45] H. Naderpour, P. Fakharian, Predicting the torsional strength of reinforced
concrete beams strengthened with FRP sheets in terms of artificial neural
networks, J. Struct. Constr. Eng. (2017), https://doi.org/10.22065/
jsce.2017.70668.1023.
[46] X. Zhang, D.W. Pei, J. Dai, Fuzzy mathematics and rough set theory (2013).
[47] M. Battaini, F. Casciati, L. Faravelli, Fuzzy control of structural vibration. An
active mass system driven by a fuzzy controller, Earthq. Eng. Struct. Dyn. 27
(1998) 1267–1276, https://doi.org/10.1002/(SICI)1096-9845(1998110)
27:111267::AID-EQE7823.0.CO;2-D.
[48] S. Pourzeynali, H.H. Lavasani, A.H. Modarayi, Active control of high rise
building structures using fuzzy logic and genetic algorithms, Eng. Struct. 29
(2007) 346–357, https://doi.org/10.1016/j.engstruct.2006.04.015.
[49] S. Rajasekaran, M.F. Febin, J.V. Ramasamy, Artificial fuzzy neural networks in
civil engineering, Comput. Struct. 61 (1996) 291–302, https://doi.org/10.1016/
0045-7949(96)00044-2.
[50] H. Torkian, Z. Keshavarz, Determining the drift in reinforced concrete building
using ANFIS soft computing modeling, Pouyan Press, 2018, pp. 1–11, https://
doi.org/10.22115/CEPM.2018.53677.
[51] Z. Yuan, L.-N. Wang, X. Ji, Prediction of concrete compressive strength:
research on hybrid models genetic based algorithms and ANFIS, Adv. Eng.
Softw. 67 (2014) 156–163, https://doi.org/10.1016/J.
ADVENGSOFT.2013.09.004.
[52] F. Özcan, C.D. Atis
ß, O. Karahan, E. Uncuoğlu, H. Tanyildizi, Comparison of
artificial neural network and fuzzy logic models for prediction of long-term
compressive strength of silica fume concrete, Adv. Eng. Softw. 40 (2009) 856–
863, https://doi.org/10.1016/j.advengsoft.2009.01.005.
[53] _
I.B. Topçu, M. Sarıdemir, Prediction of compressive strength of concrete
containing fly ash using artificial neural networks and fuzzy logic, Comput.
Mater. Sci. 41 (2008) 305–311, https://doi.org/10.1016/
j.commatsci.2007.04.009.
[54] _
I.B. Topçu, M. Sarıdemir, Prediction of rubberized concrete properties using
artificial neural network and fuzzy logic, Constr. Build. Mater. 22 (2008) 532–
540, https://doi.org/10.1016/j.conbuildmat.2006.11.007.
[55] _
I.B. Topçu, M. Sarıdemir, Prediction of mechanical properties of recycled
aggregate concretes containing silica fume using artificial neural networks and
fuzzy logic, Comput. Mater. Sci. 42 (2008) 74–82, https://doi.org/10.1016/
j.commatsci.2007.06.011.
[56] H. Naderpour, S.A. Alavi, A proposed model to estimate shear contribution of
FRP in strengthened RC beams in terms of adaptive neuro-fuzzy inference
system, Compos. Struct. 170 (2017) 215–227, https://doi.org/10.1016/
j.compstruct.2017.03.028.
[57] K. Behfarnia, F. Khademi, A comprehensive study on the concrete compressive
strength estimation using artificial neural network and adaptive neuro-fuzzy
inference system, Iran Univ. Sci. Technol. 7 (2017) 71–80 (accessed April 7,
2018) http://ijoce.iust.ac.ir/browse.php?a_id=284sid=1slc_lang=en.
[58] C. Ferreira, Gene expression programming: a new adaptive algorithm, 6th
Online World Conf Soft Comput Ind Appl Complex Syst., 2001, pp. 87–129.
[59] H.Md. Azamathulla, Gene-expression programming to predict friction factor
for Southern Italian rivers, Neural. Comput. Appl. 23 (2013) 1421–1426,
https://doi.org/10.1007/s00521-012-1091-2.
[60] A. Guven, H.M. Azamathulla, Gene-expression programming for flip-bucket
spillway scour, Water Sci. Technol. 65 (2012) 1982, https://doi.org/10.2166/
wst.2012.100.
[61] H.M. Azamathulla, Gene expression programming for prediction of scour
depth downstream of sills, J. Hydrol. 460–461 (2012) 156–159, https://doi.org/
10.1016/J.JHYDROL.2012.06.034.
[62] P. Fakharian, H. Naderpour, A. Haddad, A.H. Rafiean, E.D. Rezazadeh, A
proposed model for compressive strength prediction of FRP-confined
rectangular columns in terms of genetic expression programming (GEP),
Concr. Res. (2018), https://doi.org/10.22124/jcr.2018.7162.1191.
[63] A.H. Gandomi, A.H. Alavi, Expression Programming Techniques for
Formulation of Structural Engineering Systems, in: Metaheuristic Appl Struct
Infrastructures, Elsevier, 2013, pp. 439–455, https://doi.org/10.1016/B978-0-
12-398364-0.00018-8.
[64] H. Naderpour, K. Nagai, P. Fakharian, M. Haji, Innovative models for prediction
of compressive strength of FRP-confined circular reinforced concrete columns
using soft computing methods, Compos. Struct. 215 (2019) 69–84, https://doi.
org/10.1016/j.compstruct.2019.02.048.
[65] M. Mansur, P. Paramasivam, Cracking behavior and ultimate strength of
ferrocement in flexure, J. Ferrocem. 16 (1986) 405–415 (accessed April 5,
2018) https://www.scopus.com/inward/record.uri?eid=2-s2.0-0022791694
partnerID=40md5=70132964822668b286a3274cdd797866.
[66] A. Naaman, Flexural design of ferrocement: computerized evaluation and
design aids, J. Ferrocem. 16 (1986) 101–116 (accessed April 5, 2018) https://ci.
nii.ac.jp/naid/10011872240/.
[67] H.R. Madala, A.G. Ivakhnenko, Inductive Learning Algorithms for Complex
Systems Modeling, CRC Press, Boca Raton, 1994.
[68] A. Azimi, GMDH-network to estimate the punching capacity of FRP-RC slabs,
Soft Comput. Civ. Eng. 1 (2017) 86–92. http://www.jsoftcivil.com/article_
48352.html.
[69] A.G. Ivakhnenko, Polynomial theory of complex systems, IEEE Trans. Syst. Man.
Cybern. 1 (1971) 364–378.
[70] S.J. Farlow, Self-organizing Methods in Modeling: GMDH Type Algorithms, CrC
Press, 1984.
[71] A.G. Ivakhnenko, E.A. Savchenko, G.A. Ivakhnenko, GMDH algorithm for
optimal model choice by the external error criterion with the extension of
definition by model bias and its applications to committees and neural
networks, Pattern Recognit Image Anal C/C Raspoznavaniye Obraz I Anal Izobr.
12 (2002) 347–353.
[72] D. Rezazadeh Eidgahee, A. Haddad, H. Naderpour, Evaluation of shear strength
parameters of granulated waste rubber using artificial neural networks and
group method of data handling, Sci. Iran. (2018), https://doi.org/10.24200/
sci.2018.5663.1408.
[73] D. Logan, S.P. Shah, Moment capacity and cracking behavior of ferrocement in
flexure, J. Am. Concr. Inst. 70 (1973) 799–804. https://www.scopus.com/
inward/record.uri?eid=2-s2.0-0015763310partnerID=40md5=
22c2cd789bf3a8386fa97fa2e5cddc0c.
[74] A.S. Alwash, Flexural Characteristics of Ferrocement Ms. c thesis, University of
Baghdad, Iraq, 1974.
[75] P.N. Balaguru, A.E. Naaman, S.P. Shah, Analysis and behavior of ferrocement in
flexure, ASCE J. Struct. Div. 103 (1977) 1937–1951. https://www.scopus.com/
inward/record.uri?eid=2-s2.0-0017549387partnerID=40md5=
8f2f8306360058b28ed6a186440d5e4a.
[76] P. Paramasivam, M.A. Mansur, K.C. Ong, Flexural behaviour of lightweight
ferrocement slabs, J. Ferrocem. 15 (1985) 25–33. https://www.scopus.com/
inward/record.uri?eid=2-s2.0-0021786520partnerID=40md5=
695902d4f91cb5b6f05ba2c8d2e2bfd4.
[77] P. Desayi, V. Reddy, Strength of ligthweight ferrocement in flexure, Cem. Concr.
Compos. 13 (1991) 13–20, https://doi.org/10.1016/0958-9465(91)90042-G.
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