Construction and Building Materials 280 (2021) 122523
Contents lists available at ScienceDirect
Construction and Building Materials
journal homepage: www.elsevier.com/locate/conbuildmat
Genetic programming based symbolic regression for shear capacity
prediction of SFRC beams
Wassim Ben Chaabene, Moncef L. Nehdi ⇑
Department of Civil and Environmental Engineering, Western University, London, ON, Canada
h i g h l i g h t s
TGAN generated synthetic data used for first time in SFRC RC beam shear model training.
‘‘Train on synthetic – test on real” modeling philosophy proved highly effective.
Genetic programming symbolic regression model outperformed 11 existing models.
Model feature importance analysis indicated most influential parameters on shear strength.
Proposed equation demonstrated consistency with experimental findings.
a r t i c l e
i n f o
Article history:
Received 18 September 2020
Received in revised form 21 January 2021
Accepted 24 January 2021
Available online 6 February 2021
Keywords:
Steel fiber
Concrete
Beam
Shear strength
Symbolic regression
Genetic programming
Generative adversarial network
Synthetic data
a b s t r a c t
The complexity of shear transfer mechanisms in steel fiber-reinforced concrete (SFRC) has motivated
researchers to develop diverse empirical and soft-computing models for predicting the shear capacity
of SFRC beams. Yet, such existing methods have been developed based on limited experimental databases, which makes their generalization capability uncertain. To account for the limited experimental
data available, this study pioneers a novel approach based on tabular generative adversarial networks
(TGAN) to generate 2000 synthetic data examples. A ‘‘train on synthetic - test on real” philosophy was
adopted. Accordingly, the entire 2000 synthetic data were used for training a genetic programmingbased symbolic regression (GP-SR) model to develop a shear strength equation for SFRC beams without
stirrups. The model accuracy was then tested on the entire set of 309 real experimental data examples,
which thus far are unknown to the model. Results show that the novel GP-SR model achieved superior
predictive accuracy, outperforming eleven existing equations. Sensitivity analysis revealed that the
shear-span-to-depth ratio was the most influential parameter in the proposed equation. The present
study provides an enhanced predictive model for the shear capacity of SFRC beams, which should motivate further research to effectively train evolutionary algorithms using synthetic data when acquiring
large and comprehensive experimental datasets is not feasible.
Ó 2021 Elsevier Ltd. All rights reserved.
1. Introduction
Shear failure of reinforced concrete (RC) beams is a major concern due to its brittle and sudden nature [1]. The use of conventional steel stirrups as shear reinforcement has proven to
effectively increase the shear capacity and avoid the sudden failure
of concrete. Yet, integrating stirrups can be challenging in the case
of thin, irregular, or congested sections. Placing concrete can
become a concern when the spacing between stirrups is small,
leading to voids in the concrete [2]. Moreover, using conventional
stirrups requires significant labor input and consequently higher
⇑ Corresponding author.
E-mail address: mnehdi@uwo.ca (M.L. Nehdi).
https://doi.org/10.1016/j.conbuildmat.2021.122523
0950-0618/Ó 2021 Elsevier Ltd. All rights reserved.
construction costs. The use of steel fibers has gained great momentum in recent years owing to its ability in replacing the minimum
shear reinforcement when used in sufficient volume fraction [3,4].
Several advantages have been reported as possible merits of adopting SFRC in the construction industry. For instance, Dinh et al. [3]
stated that fibers can increase the shear resistance through crack
width reduction and transfer of tensile stresses across diagonal
cracks. Fibers can also improve the post-cracking toughness and
shrinkage behavior of concrete [5,6]. Moreover, the dowel resistance is enhanced by fiber inclusion, which is caused by improvement of concrete tensile strength in the splitting plane along the
reinforcements [7,8].
Naryanan and Darwish [7] reported that the transition from
catastrophic shear failure to controlled ductile failure can be
W. Ben Chaabene and M.L. Nehdi
Construction and Building Materials 280 (2021) 122523
oped formula and the synthetic data. Sensitivity analysis is finally
conducted to reveal the most influential features of the model.
Deploying TGAN for training a GP-SR and using real experimental
test data has never been done for modeling the shear capacity of
SFRC beams in the open literature. The new predictive equation
established this novel approach should expand the accuracy of pertinent design codes, while mitigating the ‘‘black box” limitations of
existing soft computing models.
achieved with a fiber volume fraction of one percent. The aforementioned benefits of steel fibers encouraged the ACI building
code to allow the usage of steel fibers as a replacement for minimum stirrups [9]. Hence, accurate estimation of the shear capacity
of SFRC is of paramount importance. Providing accurate predictions is, however, not straightforward because of the complex
shear transfer mechanisms in SFRC beams and the fiber orientation
at the crack interface [6]. Existing models for assessing the shear
strength of SFRC beams have largely relied on empirical
approaches consisting of statistical analysis of experimental data.
For instance, Sharma [10] proposed an equation that involves the
tensile strength of concrete along with the shear span-to-depth
ratio and linked the compressive and tensile strengths of concrete
with Wright’s formula [11]. Kwak et al. [12] conducted twelve tests
on SFRC beams to develop a new formula that accounts for the
fiber effect based on Zsutty’s equation for RC beams [13].
One major drawback of such empirical models is that their
range of validity is limited. They are developed based on a few data
examples and their accuracy over new instances that fall outside
their range of validity is questionable. Recently, machine learning
(ML) models have emerged as a strong contender for predicting
the mechanical strength of concrete and to overcome the shortcomings of empirical models [14]. ML models are developed
through extensive databases that englobe features with a wide
range of values, which can impart to such models a strong generalization ability and leads to higher accuracy when assessing the
strength of concrete made with different mixture proportions.
For instance, artificial neural networks and support vector machines have been extensively used for predicting the mechanical
strength of SFRC [5,15–20]. Such models are either employed in
standalone form or combined with an optimization algorithm to
form a hybrid model. Performance assessment of these algorithms
revealed their superior predictive capacity. However, these models
are often referred to as ‘‘black box” models because they cannot
generate an explicit mathematical equation to describe the functional relationship between the input parameters and the shear
capacity.
Unlike ‘‘black box” models, several evolutionary algorithms
offer the advantage of modeling the relationship between inputs
and the corresponding output with transparent formulas [21].
Analysis of the evolved constitutive equations can generally
prompt human insights into underlaying mechanical and physical
phenomena characterizing the problem, which helps instill confidence in the developed equations. Several researchers employed
evolutionary algorithms to model the shear capacity of SFRC
beams. For instance, Sarveghadi et al. [22] developed two models
using multi-expression programming. One of the models provides
two different equations based on the compressive strength of concrete. Greenough and Nehdi [23] modified the RC beams shear
equation developed with genetic algorithms to account for the
improvement in dowel action caused by fiber inclusion [24]. Shahnewaz and Alam [2] adopted a meta-model of optimal prognosis to
identify the most important parameters to be used in developing a
genetic algorithm-based equation for SFRC beams.
Despite the satisfactory accuracy reported in the abovementioned studies, the databases used for evolving the different models were relatively small. Thus, the generalization capability of the
developed equations is lower than that of models developed with
significantly larger databases. The research presented herein uses
a novel approach to account for the limitation imposed by small
experimental databases using a tabular generative adversarial network (TGAN) to generate synthetic data for training a genetic
programming-based symbolic regression model (GP-SR). A philosophy of ‘‘train on synthetic data and test on real data” was adopted.
The accuracy of the GP-SR developed equation is assessed over the
experimental database to check the reliability of both the devel-
2. Database description
The experimental database used in the present study comprises
results obtained via three-point and four-point shear testing methods applied on 309 SFRC beams without stirrups. Only specimens
that exhibited pure shear failure mode were included in the database. The dataset englobes eight features including the beam width
ðbw Þ, effective depth of the beam ðdÞ, shear span-to-depth ratio
ða=dÞ, longitudinal steel ratio ðqÞ, compressive strength of concrete
0
f c , steel fiber volume fraction V f , fiber aspect ratio ðlf =df Þ, and
fiber type. Table 1 entails the range of features along with the references from which test results have been retrieved. It was
reported that the fiber volume fraction had a noteworthy effect
on the shear strength of SFRC beams. However, its combination
with the fiber aspect ratio led to a more significant effect [25]. This
combination is commonly known as the fiber factor ðFÞ, which also
involves the bond factor df determined by the type of steel fibers.
The fiber factor is expressed as follows:
F ¼ Vf lf
df
df
ð1Þ
The bond factor is equal to 0.5, 0.75, and 1 for straight, crimped,
and hooked fibers, respectively. Owing to its important influence,
the fiber factor was used herein as an input parameter that
accounts for the fiber type. The descriptive statistics of the various
input variables and the corresponding output of the final database
used in subsequent sections are provided in Table 2.
3. Feature selection
Irrelevant features with relatively low consequence on the target variable may adversely affect the model performance and
increase computation time. Thus, identifying informative features
with feature selection methods is key to decreasing the dimensionality of data, removing irrelevant data, simplifying the generated
model, and speeding up the learning mechanism [67]. One way
to select the most relevant features is through their importance
scores. Importance scores reflect the significance of the input
parameter to the target variable [68]. Decision tree models
included in the scikit-learn library of Python offer a simple way
to extract such importance scores using the ‘‘feature importance”
attribute. Hence, three models, namely classification and regression trees (CART), random forest (RF), and stochastic gradient
boosting (XGBoost) have been deployed to assess feature relevance. For each algorithm, five different values of ‘‘random state”
were specified to randomly select 70% of the experimental database used for training the model. Results are depicted in Fig. 1.
The results correspond to the mean values obtained from the
five simulations, and the error bars represent the standard devia0
tion. It can be observed that a=d, q, and f c had the most significant
effect on the shear capacity. The beam width presented the lowest
importance score except for CART, where its standard deviation
was significant. The effective depth of the beam had lower effect
compared to that of the fiber volume fraction. The fiber factor
had superior influence over V f and lf =df , affirming the aforemen2
Construction and Building Materials 280 (2021) 122523
W. Ben Chaabene and M.L. Nehdi
Table 1
Database of SFRC beams without stirrups.
No. of specimens
32
3
6
8
3
2
3
28
4
2
2
21
1
4
6
1
1
6
9
8
13
6
5
6
6
3
2
19
5
34
6
3
4
7
2
1
5
8
6
2
2
1
2
2
7
2
Geometric properties of the beam
Steel
properties
Concrete
properties
Fiber properties
Shear capacity
bw ðmmÞ
dðmmÞ
a=d
qð%Þ
f 0 c ðMPaÞ
Vf ð%Þ
min
max
min
max
min
max
min
max
min
max
min
max
min
max
150
150
140
150
150
300
100
85
150
125
100
152
125
150
200
125
125
152
150
152
200
200
55
150
120
200
200
200
175
101
100
200
152
100
150
100
125
100
100
200
80
300
125
310
300
200
150
150
140
150
150
300
100
85
150
125
100
205
125
300
200
125
125
152
150
610
300
200
55
150
120
200
200
200
175
101
100
200
152
100
150
100
125
100
100
200
80
300
125
310
300
200
251
261
175
200
217
622
135
126
219
212
130
381
225
202
435
210
225
221
197
254
180
314
265
560
168
265
285
260
210
127
175
300
283
159
219
275
210
140
85
273
165
420
222
240
523
300
251
261
175
200
217
622
135
130
219
212
130
610
225
437
910
210
225
221
197
813
570
314
265
560
168
265
310
305
210
127
175
300
283
166
219
275
212
245
85
273
165
420
222
258
923
300
3.49
3.45
1.50
2.50
1.59
2.81
2.22
2.02
2.00
2.00
3.08
3.44
2.89
2.97
2.50
4.00
2.89
1.50
2.00
3.45
2.77
3.50
2.00
1.63
1.43
3.02
2.55
1.54
4.50
1.20
2.00
2.50
2.50
3.02
2.00
2.00
3.77
0.90
3.52
2.75
2.99
3.21
1.80
3.00
3.00
2.00
3.49
3.45
2.50
4.50
2.95
2.81
2.22
3.52
2.80
3.00
3.08
3.50
2.89
3.09
2.51
4.00
2.89
3.50
3.60
3.61
3.33
3.50
4.91
1.63
1.43
3.02
2.77
4.04
4.50
5.00
4.50
4.50
2.50
3.14
2.80
2.00
3.81
2.50
3.52
2.75
2.99
3.21
1.80
3.00
3.00
3.50
2.67
1.95
1.28
1.34
1.85
1.98
1.16
2.05
1.91
1.52
3.09
1.52
3.49
1.17
0.99
1.53
3.49
1.20
1.36
2.47
2.87
3.50
2.76
2.14
1.32
1.78
1.13
1.03
3.10
3.09
3.59
3.08
1.99
3.43
1.91
0.55
1.52
0.64
1.66
3.48
1.71
3.22
1.45
2.50
1.44
3.60
2.67
1.95
1.28
1.34
1.85
1.98
1.16
5.72
1.91
1.52
3.09
2.63
3.49
1.50
1.04
1.53
3.49
2.39
2.04
2.86
4.47
3.50
4.31
2.14
2.82
1.78
3.33
3.55
4.01
3.09
3.59
3.08
1.99
4.78
1.91
0.55
2.28
1.12
1.66
3.48
1.71
3.22
1.45
4.03
2.55
3.60
24.90
23.80
82.00
9.77
35.00
34.00
64.20
30.60
40.85
30.80
38.69
28.70
90.00
19.60
24.40
44.60
90.00
34.00
20.60
29.00
60.20
132.00
33.95
40.80
25.70
38.00
39.80
26.50
36.41
33.22
80.00
110.00
33.03
35.50
80.04
28.40
49.60
36.08
49.30
109.20
39.87
62.30
30.00
23.00
23.00
199.00
64.60
32.90
83.80
33.68
35.00
36.00
64.20
53.55
43.23
30.80
42.40
50.80
90.00
21.30
55.00
44.60
90.00
34.00
33.40
50.00
93.30
154.00
41.90
56.50
86.10
47.90
39.80
47.60
40.84
40.21
80.00
111.50
34.38
88.00
80.04
28.40
59.40
36.90
54.80
110.90
41.23
62.30
30.00
41.00
80.00
215.00
0.50
0.75
0.50
1.00
0.75
0.32
0.50
0.25
1.00
0.50
1.00
0.75
1.25
0.50
0.25
0.50
1.25
0.50
0.50
0.75
0.50
1.00
1.00
0.40
0.50
0.50
0.38
0.25
0.40
0.22
0.50
0.75
1.00
0.50
1.00
0.50
0.50
0.50
0.50
0.75
1.00
0.75
0.50
1.00
1.00
2.00
1.50
1.25
1.50
3.00
0.75
0.69
0.50
2.00
2.00
0.50
2.00
1.50
1.25
1.00
0.38
0.50
1.25
1.00
0.75
0.75
1.00
2.00
1.00
1.50
1.50
1.00
0.38
0.75
1.20
1.76
1.00
0.75
2.00
2.00
1.00
0.50
1.00
0.75
2.00
0.75
1.50
0.75
0.50
1.00
1.00
2.00
50
80
80
55
80
65
65
100
60
63
60
55
60
55
50
63
60
60
60
67
40
75
100
60
60
50
80
45
100
62
100
75
100
60
55
75
55
63
127
64
50
65
80
55
55
55
85
80
80
55
80
65
65
133
60
63
60
80
60
55
50
63
60
60
60
67
86
75
100
60
60
50
80
80
100
102
100
75
100
60
55
75
80
63
191
67
50
65
80
55
55
55
lf =df
Ref.
vu ðMPaÞ
Fiber type
hooked, crimped
hooked
hooked
hooked
hooked
hooked
hooked
crimped
hooked
hooked
straight
hooked
hooked
hooked
hooked
hooked
hooked
hooked
hooked
hooked
hooked, straight
straight
crimped
hooked
hooked
hooked
hooked
hooked
crimped
straight, crimped
straight
hooked
hooked
hooked
hooked
hooked
hooked
hooked
crimped
hooked
hooked
hooked
hooked
hooked
hooked
hooked
min
max
1.726
2.376
2.531
1.067
2.581
1.527
3.037
1.901
2.922
2.528
4.462
1.815
5.582
1.220
1.368
1.333
4.907
1.459
1.523
2.477
2.673
3.201
2.882
2.441
2.985
1.717
2.145
1.584
1.905
1.715
2.743
3.517
3.088
1.813
3.470
1.527
1.623
1.235
1.994
3.681
2.424
3.302
2.811
2.638
1.555
6.217
5.179
2.912
7.592
2.133
4.547
1.903
3.185
7.096
3.531
4.038
5.692
3.782
5.582
1.848
1.857
1.333
4.907
4.376
2.910
3.506
8.306
5.717
5.489
3.869
9.254
2.811
3.895
5.789
3.238
11.226
7.371
4.767
3.367
5.094
4.292
1.527
2.248
6.143
2.581
3.846
3.030
3.302
3.063
3.777
2.843
9.767
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[7]
[33]
[12]
[34]
[3]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[23]
[48]
[49]
[8]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
Table 2
Descriptive statistics of the database.
l
r
Minimum
25%
50%
75%
Maximum
Skewness
Kurtosis
bw ðmmÞ
d(mm)
a=d
qð%Þ
f c ðMPaÞ
Vf ð%Þ
lf =df
F
vu ðMPaÞ
150.889
60.332
55.000
101.000
150.000
200.000
610.000
2.066
13.450
263.582
161.975
85.250
150.000
221.000
275.000
923.000
2.008
7.469
3.043
0.812
0.900
2.500
3.080
3.500
5.000
0.229
2.883
2.505
1.034
0.550
1.810
2.540
3.090
5.720
0.857
4.008
47.922
26.365
9.770
33.220
40.210
53.200
215.000
2.704
13.296
0.833
0.470
0.220
0.500
0.750
1.000
3.000
1.356
5.722
74.803
25.700
40.000
60.000
65.000
80.000
191.000
1.995
8.404
0.538
0.346
0.102
0.315
0.499
0.645
2.865
2.043
11.040
3.252
1.642
1.067
2.183
2.882
3.598
11.226
1.927
7.605
0
3
W. Ben Chaabene and M.L. Nehdi
Construction and Building Materials 280 (2021) 122523
Fig. 1. Feature importance scores.
tions such as addition, multiplication, and subtraction, with input
parameters and random constants. Each developed expression is
evaluated via the mean squared error (MSE), which represents
the selected fitness function. Expressions with the best fitness
value are prone to a probabilistic selection and recombination via
two genetic operations named crossover and mutation [72]. As
illustrated in Fig. 3, crossover corresponds to an exchange of subtrees between a pair of expressions that recorded high fitness
values.
This exchange leads to new offspring formulations. Regarding
mutation, a randomly selected node of the previously formed tree
is modified as shown in Fig. 4, to stimulate diversity within the
tree-structured populations and expand the exploration of better
data fitting models. The newly generated expressions replace those
with lower fitness value in the tree population. The iterative process of evaluation, selection, crossover, and mutation defines one
generation of the analysis. The process is repeated until the maximum number of generations is achieved.
As entailed in Table 3, different parameters have been selected
for model deployment in python. Ramped half-and-half is an initialization approach that combines the ‘‘full” and ‘‘grow” methods,
leading to a population of trees with different shapes and depths
that range from the selected initial tree depth to the maximum tree
depth. Tournament size refers to the number of individuals that
will vie to produce the next generation. Parsimony coefficient is
applied to penalize large programs by making their fitness less
favorable and ensure that trees have reasonable length and enough
transparency.
tioned assumptions. Furthermore, the correlation matrix illustrated in Fig. 2 reflects a strong linear dependency between F
and V f . All of these factors can help disregard bw , d, V f , and lf =df
due to their little impact and/or linear dependency with other
0
more influential parameters. Therefore, only a=d, q, f c , and F have
been selected for training TGAN in the next sections.
4. Genetic programming based symbolic regression
Genetic programming (GP) is an evolutionary algorithm that
develops a population of computer programs for solving specific
problems [69]. GP is based on Darwinian principles of natural
selection and genetic spread of features adopted by biologically
developing species [70,71]. Symbolic regression (SR) is a particular
application of GP in which GP evolves populations of symbolic tree
expressions to generate the mathematical formula that provides
the best fitness value. Accordingly, GP based symbolic regression
(GP-SR) aims to optimize the expression and the corresponding
parameters of a linear or nonlinear function expressed as follows:
y ¼ f ðx; pÞ
ð2Þ
T
where x ¼ ðx1 ; ; xn Þ is an n-dimensional vector of model inputs,
y represents the model output and p ¼ ðp1 ; ; pm ÞT denotes the
m-dimensional vector of the model parameters.
Before the start of the analysis, the tree depth and size are initialized. Then, the algorithm generates a random population of
tree-structured symbolic expressions by combining several func4
Construction and Building Materials 280 (2021) 122523
f'c
Vf
lf/df
F
-0.18
0.11
0.04
-0.38
-0.06
-0.05
-0.22
0.05
0.03
-0.34
-0.04
-0.05
1.00
0.30
-0.07
-0.10
0.13
-0.09
-0.18
-0.22
0.30
1.00
0.30
0.14
0.21
0.08
f'c
0.11
0.05
-0.07
0.30
1.00
0.22
0.01
0.09
V
f
W. Ben Chaabene and M.L. Nehdi
bw
d
a/d
bw
1.00
0.65
0.01
d
0.65
1.00
a/d
0.01
0.04
0.03
-0.10
0.14
0.22
1.00
-0.06
0.83
lf/df
-0.38
-0.34
0.13
0.21
0.01
-0.06
1.00
0.34
F
-0.06
-0.04
-0.09
0.08
0.09
0.83
0.34
1.00
Fig. 2. Correlation matrix of input features.
process of developing GP-SR model is schematically represented
in Fig. 6. It is to be understood that the ‘‘fake” synthetic data is only
used for training the model. The robustness and predictive accuracy of the GP-SR model is exclusively validated on real experimental data that is unknown to the model. and thus far never
presented to it.
5. Model training through TGAN
Training machine learning models with limited databases can
create overfitting issues. Models trained with further clean data
acquire greater generalization capability compared to those developed with few data examples. Nevertheless, gathering large datasets might be challenging for certain problems due to the
restricted number of experimental tests. Recently, tabular generative adversarial networks (TGAN) has been introduced as a new
approach to expand existing databases with synthetic data. TGAN
consists of building a generative model that can produce synthetic
data with similar characteristics to the actual data [73]. The
approach comprises two key components called the generator
and the discriminator. As shown in Fig. 5, the generator receives
a random noise as input and generates synthetic data. Conversely,
the discriminator learns to distinguish between real examples and
‘‘fake” examples and assigns a specific score to indicate whether
data is actual or synthetic. The key objective of the generator is
to fool the discriminator by enhancing the quality of synthetic data
and make the distinction process harder.
Generally, long short-term memory (LSTM) network and multilayer perceptron are used as generator and discriminator, respectively. For the present study, the collected experimental database
of 309 samples was used to train TGAN. The generative model produced 2000 synthetic samples that will be involved in the training
GP-SR model. The trained GP-SR model is tested afterwards over
real experimental data to verify the generalization capability of
the model as well as the reliability of synthetic data. The entire
6. Results and discussion
6.1. Statistical metrics
Statistical metrics are commonly used for assessing the predictive accuracy of ML models as well as comparing the performance
of various algorithms. The metrics used in this study are the correlation coefficient (R), root mean squared error (RMSE), mean absolute error (MAE), and three other parameters characterizing the
ratio of the predicted value v p and tested value ðv t Þ including
the mean ðlÞ, standard deviation ðrÞ, and coefficient of variation
ðCOV Þ. The expression of each metric is indicated below.
P
P
P
n ni¼1 v pi v ti ð ni¼1 v pi Þð ni¼1 v ti Þ
ffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
n
n
n
P
P
P
P
nð v 2pi Þ ð v pi Þ2 nð v 2ti Þ ð v ti Þ2
i¼1
RMSE ¼
5
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uP
un u
v v ti 2
ti¼1 pi
n
i¼1
ð3Þ
i¼1
ð4Þ
W. Ben Chaabene and M.L. Nehdi
Construction and Building Materials 280 (2021) 122523
Fig. 3. Crossover in GP-SR model.
×
×
×
+
-
w
y
z
√
/
+
y
-
z
y
w
√
y
z
z
Fig. 4. Mutation in GP-SR model.
MAE ¼
n 1X
v p v t i
i
n i¼1
n v pi
1X
l¼
n i¼1 v ti
ð5Þ
r¼
ð6Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uP
2
u n v pi u
l
v
ti
ti¼1
COV ð%Þ ¼
6
n
r
100
l
ð7Þ
ð8Þ
Construction and Building Materials 280 (2021) 122523
W. Ben Chaabene and M.L. Nehdi
Table 3
Set of parameters used to develop GP-SR.
Parameter
Values
Population size
Crossover rate (%)
Mutation rate (%)
Tournament size
Parsimony coefficient
Function set
Methods used
1000, 2000, 3000, 4000, 5000
70, 80, 90
30, 20, 10
5, 10, 15, 20, 25, 30
0.1, 0.01, 0.001, 0.0001
p
+, -, , /, , ln, exp
Initialization: Ramped half-andhalfSelection: Tournament
selectionCrossover: Subtree
exchangeMutation: Subtree replacement
2
12
Initial tree depth
Maximum tree depth
Fig. 5. Simplified process of TGAN.
Higher values of RMSE and MAE reflect greater error between
tested and predicted values, while lower COV indicates a lower
level of dispersion of the different data points around the mean.
action. The arch action tends to resist the applied shear, leading
to higher capacity [74]. Moreover, an increase in the value of the
fiber factor engenders higher shear strength. The increase of shear
capacity is attributed to the improvement of both the arch action
and dowel action caused by fiber inclusion [7]. The shear capacity
is also improved by the increase of the compressive strength of
concrete. The rate of increase diminishes with higher compressive
strength, contrarily to some studies that reported exponential relationships [1]. Similarly, the rate of increase in the shear capacity is
reduced when a higher reinforcement ratio is included. This is
reflected by the logarithmic function and the square root function
applied to the reinforcement ratio. This was previously confirmed
by Swamy and Bahia [8]. The decrease of the shear capacity
improvement rate for the longitudinal reinforcement ratio was
linked to the influence of the steel area and net concrete width
at the steel level [75,76]. These two factors exhibited an important
influence on the dowel resistance and consequently on the shear
capacity.
The other important aspect that needs further investigation is
the size effect. Several recent studies reported a significant influence of this factor on shear strength [64,77,78]. The abovementioned equation determines the shear stress, which represents
the shear force per unit area. When the original database was collected, the shear stress representing the shear force divided by the
product of beam width and effective depth of beam was considered
as output variable. This dependency explains the low importance
scores attributed to these two input features. The resultant shear
force is therefore expressed as follows:
6.2. Proposed new shear equation
As indicated earlier, the proposed equation for predicting the
shear capacity of SFRC beams without stirrups is a function of four
parameters that have the highest influence on its value. It is worth
mentioning that the suggested equation has been selected based
on two criteria. The first criterion is the program length, which represents the total number of nodes. Extremely complex equations
were discounted and only those with a length of less than 30 were
considered to ensure enough model transparency. The second criterion is the fitness value described above. The equation that provides the best fitness value will be considered as the best solution.
The final equation obtained from the GP-SR model is expressed as
follows:
0:694 ln ð1:786F þ 1:091qÞ a
v u ðMPaÞ ¼ 0:921 þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
0
0:787f c þ
0
4:863q
0:798f c
ðÞ
a 3
d
d
ð9Þ
Several observations can be extracted from the proposed formula by altering one variable and keeping the others constant at
their mean values. Fig. 7 shows that an increase shear span-todepth ratio leads to lower shear capacity. This can be explained
by the arch action effect, which represents the compressive force
generated along the loading points and beam supports. In the
region of short shear spans, loads are carried in part via the arch
Fig. 6. Process of training and testing GP-SR.
7
W. Ben Chaabene and M.L. Nehdi
Construction and Building Materials 280 (2021) 122523
Fig. 7. Effect of features on shear capacity.
Table 4
Existing equations for SFRC shear strength.
Reference
Sharma [10]
Suggested model
8
< 1 if f t is obtained by direct tension test
2=3 if f t is obtained by indirect tension test
:
4=9 if f t is obtained via modulus of rupture
pffiffiffi
1
ifða=dÞ 2:8
vu ¼ e 0:24f spfc þ 80qðd=aÞ þ vb vb ¼ 0:41sF; F ¼ ðlf =df ÞVf qf ; s ¼ 4:15MPae ¼
f
¼ f cufpffiffi þ 0:7 þ F
2:8ðd=aÞ ifða=dÞ < 2:8 spfc ð20 FÞ
8
qffiffiffiffi
1
0
3
>
>
qffiffiffiffi
ifða=dÞ 2:5
2:11 f c þ 7F ðqðd=aÞÞ3
<
0
qffiffiffiffi
vu ¼ 0:7 f c þ 7F ðd=aÞ þ 17:2qðd=aÞ
vu ¼ 1
0
>
3
>
: 2:11 f c þ 7F ðqðd=aÞÞ3 ð5d=2aÞ þ vb ð2:5 a=dÞ ifða=dÞ < 2:5
qffiffiffiffi
0
1
ifða=dÞ 2:5
vu ¼ ð0:167a þ 0:25FÞ f c a ¼
2:5ðd=aÞ ifða=dÞ < 2:5
1
vu ¼ kf t ðd=aÞ4 k ¼
Narayanan & Darwish [7]
Ashour et al. [81]
Khuntia et al. [1]
ifða=dÞ > 3:4
ifða=dÞ 3:4
Kwak et al. [12]
vu ¼ 3:7ef 3spfc ðqðd=aÞÞ3 þ 0:8vb e ¼
Shahnewaz & Alam [79]
vu ¼ 0:2 þ 0:034f c þ 19q0:087 5:8ða=dÞ2 þ 3:4V0:4
800ðlf =df Þ
f
2
1
0
197ðða=dÞðlf =df ÞÞ
0:07
Shahnewaz & Alam [2]
Sarveghadi et al. [22]
Arslan [83]
RILEM [84]
þ 105ðVf ðlf =df ÞÞ
2:12
2
þ 13:5V0:07
þ 450ðlf =df Þ
f
2:69
Gandomi et al. [82]
1:6
1
1:4
24ða=dÞ
1
3:4ðd=aÞ
þ1181ðVf ðlf =df ÞÞ
0
d
vu ¼ 2d
a qf c þ vb þ 2a
0:0002ðða=dÞVf Þ
21:89ðða=dÞVf Þðlf =df Þ
q
0:05
12ðða=dÞVf Þ
0 0:85
ifða=dÞ 2:5vu ¼ 0:2 þ 0:072 f c
þ 12:5q0:084
0:9
3:9
27:69ðða=dÞðlf =df ÞÞ
0:84
ifða=dÞ > 2:5
þ2
0
vu ¼ 3:2 þ 0:072f c þ qVf 1:26 0:25 da da 1:92 þ 0:017f c 0:38 da
8
0
0
f t þvb
>
>
iff c < 41:4MPa
þ vb
qffiffiffiffi
<
3q
a
qþv ðvb þ2þdaf t þ4qf t Þ
0
0
d
b
vu ¼
f t ¼ 0:79 f c
a
0
0
>
d
d
>
: a 2f t þ qþqð4þv Þ aþq 1a 2 þ vb iff c > 41:4MPa
Þð
b ðd
dÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2
2
0 3 0
3
vu ¼ 0:2f c dc þ qð1 þ 4FÞf c 3a dc isasolutionof dc þ 6000 q dc 6000 q ¼ 0
fc
fc
d
qffiffiffiffiffiffi
1=3
vRd ¼ vcd þ vfd vcd ¼ 0:12kð100qf 0 c Þ ; k ¼ 1 þ 200
2;
q
2%v
fd
d
qffiffiffiffiffiffi
¼ kf kl sfd ; kl ¼ 1 þ 200
2; kf ¼ 1ðrectangular sectionÞ; sfd ¼ 0:12f Rk;4 ; f Rk;4 ¼ 1MPaðassuming sufficient fiber dosageÞ
d
0
ð288q11Þ4
8
Construction and Building Materials 280 (2021) 122523
W. Ben Chaabene and M.L. Nehdi
2
0:694 ln ð1:786F þ 1:091qÞ 6
6
a v u ðMN Þ ¼ 660:921 þ
4
d
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi3
0
0:787f c þ
Still, the applicability of the abovementioned equation to very
large-scale beams needs to be verified since their behavior might
differ significantly due to size effects. Thus, it is important at this
stage to ensure that the value of d is within the specified range of
the database to obtain realistic results.
0
4:863q
0:798f c
ðadÞ
3
7
7
7 bw d
7
5
ð10Þ
where 85:250 dðmmÞ 923:000. It was reported that larger
beams would generally fail at lower stress because of the size effect.
In the resultant shear force equation, the shear span and effective
depth of the beam are both related to the size effect. Chao [78]
reported that the shear span is a significant parameter that impacts
the size effect. When the depth of the beam is increased, the shear
span also increases to maintain specific ratios and ensure realistic
specimen configuration. Thus, the compression zone for larger
beams typically has lower contribution to shear resistance, which
leads to lower shear capacity [78]. The fiber factor in the equation
also influences the size effect. Minelli et al. [77] posited that incorporating steel fibers can mitigate the size effect in shear, which
makes the results for plain concrete not applicable to SFRC beams.
6.3. Performance assessment of proposed equation
Evaluation of the proposed model has been performed through
the statistical metrics described earlier. Also, the model was
benchmarked against several existing equations outlined in Table 4.
Some of the equations are based on an empirical approach, while
others were developed using soft computing techniques
[2,22,79]. The performance of each model over the experimental
database is captured in Table 5. It can be observed that the proposed equation outperformed all the other models with
R ¼ 0:8878, RMSE ¼ 0:8421, and MAE ¼ 0:6099. This indicates that
the model generates lower error as well as stronger uphill linear
Table 5
Performance evaluation of existing models.
Reference
R
RMSE
MAE
Sharma [10]
Narayanan and Darwish [7]
Ashour et al. [81]
Khuntia et al. [1]
Kwak et al. [12]
Shahnewaz and Alam [79]
Gandomi et al. [82]
Shahnewaz and Alam [2]
Sarveghadi et al. [22]
Arslan [83]
RILEM [84]
Suggested Model
0.6622
0.8038
0.7989
0.6489
0.8086
0.7464
0.8133
0.8172
0.6156
0.765
0.6072
0.8878
1.3933
1.2902
1.1665
1.6794
0.9811
1.9003
1.0438
0.9668
1.9592
1.1011
2.0013
0.8421
0.8792
0.9926
0.8646
1.2247
0.6761
1.5764
0.7749
0.6712
1.5749
0.6716
1.8154
0.6099
vp =vt
l
r
COVð%Þ
0.9438
0.7571
0.8486
0.716
1.0142
1.0855
1.2177
1.0376
0.7881
0.9767
1.6841
0.9489
0.2987
0.2842
0.3009
0.2657
0.3557
0.4591
0.3581
0.3035
0.6065
0.2374
0.5455
0.2242
31.6513
36.1127
35.4641
37.1101
35.0734
42.2946
29.4112
29.2529
76.9602
24.3084
32.3902
23.6299
Table 6
Validation of proposed new equation.
Condition number
1
2
Formula
Calculated values
Pn
vt vp
0:85 k ¼ Pi¼1n i 2 i 1:15
vp
Pni¼1 i
0
v v
i¼1 ti pi
1:15
0:85 k ¼ P
n
2
k ¼ 0:9998
0
k ¼ 0:9573
v
i¼1 t
Pn
Pn
2 2
2 02
2
2
ðvpi kvpi Þ 0 2
ðvti k0 vti Þ
R R0 R R i¼1
R2 < 0:1 R2 0 < 0:1 with R20 ¼ 1 Pi¼1
n
2 R 0 ¼ 1 Pn
2
ðvpi vp Þ
ðvti vt Þ
i¼1
i¼1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rm ¼ R2 ð1 R2 R20 > 0:5
i
3 and 4
5
R2 R20
R2
R2 R0 20
R2
¼ 0:0612
¼ 0:0254
Rm ¼ 0:5080
Table 7
Ranking of the different shear equations.
Reference
Proposed Model (this study)
Shahnewaz and Alam [2]
Arslan [83]
Gandomi et al. [82]
Kwak et al. [12]
Ashour et al. [81]
Sharma [10]
Narayanan and Darwish [7]
Khuntia et al. [1]
Shahnewaz and Alam [79]
Sarveghadi et al. [22]
RILEM [84]
Ranking
Overall Ranking
R
RMSE
MAE
COV
Average
1
2
7
3
4
6
9
5
10
8
11
12
1
2
5
4
3
6
8
7
9
10
11
12
1
2
3
5
4
6
7
8
9
11
10
12
1
3
2
4
7
8
5
9
10
11
12
6
1
2
4
4
4
6
7.5
7.5
9.5
10.5
11
12
9
1
2
3
3
3
6
7
7
9
10
11
12
W. Ben Chaabene and M.L. Nehdi
Construction and Building Materials 280 (2021) 122523
model. The system uses the average scores based on the values
of statistical metrics. Higher scores are assigned with higher R values, and lower scores are attributed to higher RMSE; MAEandCOV.
The other existing equations demonstrated lower accuracy compared to the proposed model. For example, the formula suggested
by Shahnewaz and Alam [2] achieved the second best predictive
accuracy in terms of R, RMSE, and MAE, which is also reflected in
the ranking system. However, this equation is not consistent with
real-world findings. The longitudinal steel reinforcement ratio has
a linear relationship with the output, which means that the rate of
increase of the shear capacity remains constant for higher reinforcement ratios. Similarly, Narayanan and Darwish’s equation
[7] assumes a linear relationship between both variables. The
low accuracy of existing formulas can also be attributed to the lack
of important parameters involved in it. For instance, Khuntia et al.
[1] disregarded the influence of q. Sharma’s formula [10] did neither consider the effect of fibers nor that of q. For these reasons,
both equations achieved relatively low accuracy with a correlation
coefficient less than 0.7.
The RILEM equation revealed the worst precision amongst the
different equations with R ¼ 0:6072, RMSE ¼ 2:0013, and
MAE ¼ 1:8154. The low accuracy presented by RILEM formula is
mainly linked to safety factors included in the equation for design
precautions, which leads to inaccurate results when predicting the
shear capacity measured by laboratory tests. The Taylor diagram
illustrated in Fig. 8 is a visual metric that can further depict the
performance of the various equations. Taylor diagram considers
the correlation coefficient, the standard deviation, and the root
pattern between predicted and experimental values. Smith [80]
argued that if a positive correlation coefficient is higher than 0.8
then there is a strong correlation between actual and predicted
values.
To better investigate the model validity, Golbraikh and Tropsha
[85] introduced several validation criteria. As outlined in Table 6, k
0
and k that denote the slope of regression lines through the origin
should be close to 1, and the correlation coefficients through the
origin (R20 and R0 20 ) should be close to R. Moreover, Roy and Roy
[86] suggested another validation criterion ðRm Þ that evaluates
the external predictability of a model, and stated that for a good
model, Rm should be greater than 0.5. It was found that the proposed equation satisfied all the validation criteria, indicating a
strong predictive ability for new unseen data.
In addition, the model exhibited the lowest COV amongst the
presented equations, which means that the obtained values had
lower level of dispersion around the mean. The main reason behind
the superior accuracy of the proposed model is the extensive database used in its training. Even though the data used for training
was synthetic and generated from TGAN, the model exhibited
strong generalization capability, which indicates that TGAN can
produce reliable data examples for training. It is worth mentioning
that this is the biggest database that has ever been used for training a soft computing model to predict he shear strength of SFRC
beams. Moreover, the model has shown compliance with previous
research findings as described in the previous section, which also
explains its reliability. The ranking system presented in Table 7
can further confirm the superior performance of the developed
Fig. 8. Taylor diagram for SFRC shear strength prediction.
10
Construction and Building Materials 280 (2021) 122523
W. Ben Chaabene and M.L. Nehdi
mean-square-centered difference to illustrate the predictive accuracy of each model. Based on the diagram, the closest point to the
point representing observed data is that of the proposed equation.
This means that the suggested formula has the highest accuracy,
confirming the abovementioned results. Conversely, The equation
proposed by Saverghadi et al. [22] showed the lowest accuracy,
which was indicated by the relatively high root mean-squarecentered difference. Thereupon, this model has a rather low accuracy compared to its counterparts.
The model proposed in the present study proved to have superior predictive accuracy. Yet, it is crucial to understand the degree
to which each parameter of the equation affects the shear capacity.
Thus, sensitivity analysis can provide a better understanding of the
equation by revealing the most influential parameters. The process
adopted for sensitivity analysis is discussed below.
Fig. 9. Sobol indices of different variables.
6.4. Sensitivity analysis
Sensitivity analysis (SA) aims to quantify the uncertainty of a
model output linked to various sources of uncertainty in the different inputs [87,88]. For complex nonlinear problems, global sensitivity analysis models such as variance-based sensitivity analysis
are widely adopted owing to their ability in considering the interactions of the input with the other variables when quantifying the
output uncertainty [89]. Variance-based methods present a specific methodology to determine first-order and total-order sensitivity indices for each input variable of the developed equation. For
a model having the form Y ¼ f ðX 1 ; X 2 ; ; X k Þ, the variance-based
approach employs a variance ratio to assess the relevance of
parameters through variance decomposition that can be expressed
as follows:
V¼
k
X
Vi þ
i¼1
k X
k
X
i¼1
V ij þ þ V 1;2;;k
ferent random state. Consequently, different variable bounds are
specified in each analysis. Results are illustrated in Fig. 9. It can
be observed that shear span-to-depth ratio demonstrated the
greatest effect on the model output. The second most influential
parameter was the longitudinal steel reinforcement ratio, followed
by the compressive strength of concrete. The lowest influence was
recorded for the fiber factor. Moreover, total-order sensitivity
index was greater than the first-order index for each parameter
due to the interaction of the feature with the other variables. The
presented results of sensitivity analysis are also consistent with
previous studies in the open literature [2]. However, the results
are not in compliance with studies which reported that the fiber
factor had the most significant effect. It is worth mentioning that
those studies were developed based on a limited number of data
examples, as well as a specific types of fibers. The type of fiber
has a significant effect on the experimental results. Thus, equations
developed from data using certain types of fibers are more sensitive to the fiber factor than others developed with a different type
of fiber. For example, some studies mentioned that hooked fibers
have higher efficiency in resisting pull out forces compared to
straight fibers [91], which also affects the shear capacity. This contrast engenders different results when the model is developed with
extensive databases because the model will search for a solution
that can be applied to different fiber types.
ð11Þ
j>i
where V denotes the variance of the model output, V i is the firstorder variance for the input X i , and V ij to V 1;2;;k represent the variance of the interaction of the k parameters. V i and V ij , which indicates the significance of the input to the variance of the output,
depends on the variance of the conditional expectation as shown
in Eq. 12 and Eq. 13.
V i ¼ V Xi EX i ðYjX i Þ
ð12Þ
h
i
V ij ¼ V Xi Xj EX ij YjX i ; X j V i V j
ð13Þ
7. Verification of model applicability to reinforced concrete
beams
with X i notation corresponds to the set of all variables
excluding X i . The first-order sensitivity index ðSi Þ for an input X i
is therefore given by:
Si ¼
Vi
V ðY Þ
Verification of model applicability to plain reinforced concrete
(RC) beams without steel fibers consists of evaluating the predictive accuracy of the equation when the fiber factor approaches
zero. Therefore, the equation for RC without fiber reinforcement
will have the following expression:
ð14Þ
v u ðMPaÞ ¼ 0:921
Conversely, the total effect of the input factor X i , which comprises the first-order effect along with effects coming from the
interaction with other features, is given by the following formula
[89]:
STi ¼
EX
i
V Xi ðYjX i Þ
V X i EX i ðYjX i Þ
¼1
VðYÞ
VðYÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi0ffi
0
4:863q
0:798f c
0:694 ln ð1:091qÞ 0:787f c þ
3
ðadÞ
a
þ
d
ð15Þ
ð16Þ
The approach suggested by Saltelli et al. [90] for calculating
first- and total-order sensitivity indices has been used in the current study. The number, names, and bounds of the variables were
first specified. Data sampling was then conducted via Saltelli’s
extension of the Sobol sequence to generate 10,000 samples. The
analysis was repeated 5 times, each time a random subset that
consists of 70% of the entire database was generated by using a dif-
Such verification is of paramount importance since it can
extend the range of validity of the proposed equation to RC beams
without steel fibers. For this purpose, 20 different RC sample
beams were used for assessing the model accuracy. The specimens
were gathered from different references in the open literature to
ensure sufficient data diversity and adequate model generalization
capability [92–101]. The results are shown in Fig. 10.
11
W. Ben Chaabene and M.L. Nehdi
Construction and Building Materials 280 (2021) 122523
The feature importance analysis indicated that the shear spanto-depth ratio, longitudinal reinforcement ratio, compressive
strength of concrete, and fiber factor had the most significant
effect on SFRC beam shear capacity. Further analysis with a correlation matrix revealed a strong linear dependency between V f
and F.
Validation of the proposed equation reflected its superior predictive accuracy on new data unknown to the model, which
was indicated by the values of k, k0 , and Rm .
The proposed new model exhibited the highest predictive accuracy with an R value of 0.8878 and RMSE and MAE of 0.8421 and
0.6099, respectively. This reflects the reliability of the model in
providing accurate estimations as well as the effectiveness of
synthetic data generated by TGAN in properly training the GPSR model.
Analysis of the proposed equation showed consistency with
experimental findings. Higher a=d values led to decreased shear
0
capacity, while greater values of q, F, and f c led to enhanced
shear capacity. This is attributed to the effect of the arch action,
which increases with lower a=d, in addition the influence of the
dowel action which is enhanced with greater q and F.
Sensitivity analysis showed that a=d had the greatest influence
on the shear capacity, while the lowest effect was linked to F.
Based on the results, the ability of TGAN in generating reliable
synthetic data examples can be explored to overcome the lack
of data examples for other concrete types and predicting diverse
engineering properties.
The new proposed model can accurately predict the shear
capacity of SFRC beams. However, this model does not reflect
the behavior of the entire structure. Designers need to understand the load distribution in the structure, which can be
achieved by alternative techniques such as finite element
methods.
Fig. 10. Correlation between predicted and experimental RC shear strength.
The obtained coefficient of determination is R2 ¼ 0:9216, which
indicates that there is a strong linear relationship between the predicted and experimental values of RC beam shear strength. Therefore, the model validity can be extended to RC beam specimens
without steel fiber reinforcement.
8. Model limitations
Even though the predictive equation developed in the present
study achieved superior accuracy in predicting the shear strength
of SFRC beams, there are some rational limitations associated with
its implementation. Sensitivity analysis reflected la rather low
effect of the fiber factor, which is not in compliance with some previous studies. This is probably due to the variable influence of the
different fiber types in the database on the shear strength, which
makes the model incapable of capturing the noteworthy influence
of fibers in some cases because it rather captures the average effect
of such feature over the entire dataset. Such a limitation is worth
further investigation in future studies. While this could be
achieved, for instance through using the fiber characteristics as
input variables, there is need for developing robust and comprehensive data that can effectively capture the complex effects of
such features in both model training and in its testing stage. Moreover, it is important to point out that the applicability of the proposed equation is limited to the specific ranges of variables
entailed in Table 2. This implies that generalizing the pertinence
of the equation to very large-scale beams that can have different
behavior caused by size effects needs to be verified. The model
can be extended to large-scale beams if sufficient robust and comprehensive data on such beams becomes available. This aspect
warrants further future investigation.
CRediT authorship contribution statement
Wassim Ben Chaabene: Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing - original
draft. Moncef L. Nehdi: Conceptualization, Funding acquisition,
Methodology, Project administration, Resources, Supervision, Validation, Writing - review & editing.
Declaration of Competing Interest
This research work abides by the highest standards of ethics,
professionalism and collegiality. All authors have no explicit or
implicit conflict of interest of any kind related to this manuscript.
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