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Applied Soft Computing Journal 105 (2021) 107281
Contents lists available at ScienceDirect
Applied Soft Computing Journal
journal homepage: www.elsevier.com/locate/asoc
Bayesian optimization algorithm based support vector regression
analysis for estimation of shear capacity of FRP reinforced concrete
members
Md Shah Alam a,∗, N. Sultana b, S.M. Zakir Hossain c
aDepartment of Civil Engineering, College of Engineering, University of Bahrain, Zallaq, Bahrain
bDepartment of Computer Science, College of Computer Science and Information Technology, Imam Abdulrahman Bin Faisal
University, Dammam, Saudi Arabia
cDepartment of Chemical Engineering, College of Engineering, University of Bahrain, Zallaq, Bahrain
article info
Article history:
Received 11 October 2020
Received in revised form 25 February 2021
Accepted 4 March 2021
Available online 13 March 2021
Keywords:
Shear resistance
FRP reinforcement
Reinforced concrete
Bayesian optimization algorithm
SVR model
abstract
The use of fiber-reinforced polymer (FRP) rebars in lieu of steel rebars has led to some deviations
in the shear behavior of concrete members. Several methods have been proposed to forecast the
shear capacity of such members. Nonetheless, there are differences in the methods of considering
the various parameters affecting shear capacity, and some of them provide widely scattered and
conservative results. This paper presents a hybrid of the Bayesian optimization algorithm (BOA) and
support vector regression (SVR) as a novel modeling tool for the prediction of the shear capacity of
FRP-reinforced members with no stirrups. For this purpose, a large dataset of simply supported beams
and unidirectional slabs reinforced with FRP were utilized. The model performance was assessed using
several statistical performance indicators and compared with the Japan Society of Civil Engineers (JSCE),
British Institution of Structural Engineers (BISE), Canadian Standard Association (CSA), and American
Concrete Institute (ACI) design codes and guidelines, as well as some other artificial intelligence (AI)
models. For development of the model, all the hyperparameters, i.e., kernel function type, epsilon,
box constraint, and kernel scale, were optimized using the BOA technique. The k-fold cross validation
approach was utilized to avoid overfitting of the model. It was found that the mean, median, standard
deviation, minimum, maximum, and interquartile range of the developed hybrid model predictions
are very close to the experimental results. The predicted results overlap the experimental data with a
coefficient of determination of 95.5%. The plot of relative deviations and residual plots are scattered
around the zero reference line with low deviation, which indicates that the model is reliable and
valid. The error terms (e.g., mean absolute error, root mean square error) obtained for all specimens
were 4.85 and 11.03, which are very low values. The correlation coefficient (R) and fractional bias
(FB) were found to be 0.977 and 0.0033, which are very close to 1 and 0, respectively, thus implying
a reliable prediction. The comparative investigations with other codes and guidelines show that the
hybrid BOA–SVR model predictions are more accurate and robust than those of the other models.
©2021 Elsevier B.V. All rights reserved.
1. Introduction
The use of fiber-reinforced polymer (FRP) rebars in lieu of steel
rebars has increased widely due to the non-corrosive and other
favorable properties of FRP. However, if these rebars are used as
the main flexural reinforcement, some of the shear transfer mech-
anisms change [1–3]. This is partly due to the fact that the shear
strength of reinforced concrete (RC) members without stirrups
depends on the shear span-to-depth ratio (a/d), reinforcement
ratio (ρ), depth of beam (d), and concrete strength (f′
c) [4,5].
∗Corresponding author.
E-mail address: malam@uob.edu.bh (M.S. Alam).
The relative contribution of shear resistance from these param-
eters will be different for FRP-reinforced members. Accordingly,
several design codes and guidelines have been proposed from
the modification of available codes for steel-reinforced members
or from the empirical or semi-empirical models based on the
limited set of available test data [6–9]. These design methods
have various forms and lack of general consensus among the
different codes on the parameters affecting the shear strength.
These design methods are continually changing and still open for
further investigation.
Recently, artificial intelligence (AI) methods (e.g., genetic al-
gorithms, adaptive regressions, fuzzy logics, artificial neural net-
works) in lieu of classical or conventional methods are becoming
popular techniques among researchers. These techniques have
https://doi.org/10.1016/j.asoc.2021.107281
1568-4946/©2021 Elsevier B.V. All rights reserved.
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
been used successfully in engineering research including that
of FRP-reinforced members [10–18]. They do not rely on con-
ventional structural mechanisms; rather, they use high-precision
fit to match results as closely to true values as possible. Re-
cently, this technique has been used for experimental design and
prediction of the mechanical properties of jute fiber-reinforced
concrete [19,20]. Nehdi et al. [11], Kara [12], and Shahnewaz
et al. [17] suggested shear design formulas for FRP-reinforced
members with no stirrups using genetic algorithm method. The
authors used datasets of 50, 104, and 117 specimens, respec-
tively, to develop their models for training and testing. The model
proposed by Nehdi et al. [11] for members without stirrups has
two forms: one for a shear span-to-depth ratio (a/d) greater than
2.5 and another for an a/dless than 2.5. The average absolute
error (AAE) and coefficient of variations (CoV) of the model pre-
dictions are 22.42% and 33%, respectively. On the other hand,
the predictions using the model proposed by Kara [12] are in
better agreement with the experimental results. However, the
author utilized a dataset of only 104 specimens with a maximum
effective depth (d) of 360 mm. It is known that the shear strength
decreases with an increase in dof the beam, which is defined as
the size effect. Therefore, this model needs further investigation
for large beam depths. Shahnewaz et al. [17] proposed modifica-
tions to some of the existing methods. Although the modifications
yield some improvement in the predictions, the CoV and the AAE
are still high for members with no shear reinforcement.
The artificial neural network (ANN) technique for modeling
the shear strength of FRP-reinforced members was utilized by
some authors [13–15,21–24]. Lee and Lee [14] utilized a dataset
of 106 specimens to develop the ANN model. The authors men-
tioned that more data were needed to investigate the size effect.
Nasrollahzadeh and Basiri [16] developed a method using the
fuzzy interface system (FIS) and using a dataset of 128 speci-
mens. However, the CoV and the mean absolute percentage error
(MAPE) of the model predictions were 36.6% and 32.8%, respec-
tively. The authors also mentioned that more experimental data
were needed to refine the model. Alam and Gazder [22] proposed
a model using the ANN technique which yielded a 0.38 standard
deviation with a coefficient of variation of 36% from a dataset
of 196 specimens. On the other hand, Naderpour et al. [23]
proposed a model using the computational intelligence technique
which showed superior results. However, the authors utilized a
dataset of only 110 specimens to develop the model. Golafshani
and Ashour [25] proposed a new model using biogeography-
based programming (BBP) for the prediction of the shear capacity
of FRR-reinforced members and showed the potentiality of us-
ing BBP utilizing a 138-specimen dataset. Although some of the
above-mentioned models could predict the shear strength of FRP-
reinforced members with reasonable accuracy, most of these
models were based on smaller datasets. Nevertheless, the size of
the dataset used to train and test the model plays an important
role in making the model successful and robust [15]. Also, the
above models did not use optimization of the parameters, which
may lead to overfitting or underfitting of the results. On the
other hand, support vector regression (SVR) is a familiar machine
learning algorithm (MLA) that is used for complex engineering
systems. It is a notable aspect that SVR can identify linear or
nonlinear patterns, make connections between input and output,
and link the descriptor with the target [26]. This technique has
been used to estimate the mechanical, physical, and hydrological
properties of permeable concrete [27] and was found to be very
promising. As discussed before, there are some advantages and
disadvantages of the available methods for determining the shear
capacity of FRP-reinforced members. Therefore, there are some
areas to explore for a better procedure for calculating the shear
capacity of such members, and SVR can be one of those.
Recently, machine learning techniques, including SVR with
multi-objective optimization technique, have been successfully
used to recognize the COVID-19 strains, to capture the non-linear
properties of crude oil time series, and to forecast financial time
series [28–30].
Therefore, the present study focuses on the development of a
novel hybrid model using support vector regression (SVR) with
the Bayesian optimization algorithm (BOA) to forecast the shear
capacity of FRP-reinforced members without stirrups. It is im-
portant to note that the proper selection of hyperparameters
such as epsilon (ε), box constraint (C), and kernel scale (γ) and
kernel function (i.e., linear, polynomial, gaussian, sigmoidal) are
important in SVR modeling. High or low values/types of these
parameters can lead to overfitting or underfitting. Thus, it is
crucial to tune all these parameters. The Bayesian optimization
algorithm (BOA) along with k-fold cross validation is integrated
with SVR to tune all these hyperparameters aromatically in order
to obtain an optimal hybrid BOA–SVR model. It is also worth
noting that k-fold cross validation protects the model against
overfitting. The proposed SVR–BOA model utilizes a dataset of
216 specimens of FRP-reinforced members having no stirrups.
The model’s predicted results are compared with the current JSCE
1997 [6], BISE 1999 [7], CSA S806-12 [8], and ACI 440.1R-15 [9]
design codes and guidelines and with some AI models proposed
by Nehdi et al. [11], Lee and Lee [14], and Jumaa and Yousif [15].
The paper is organized in the following sequence: Section 2
describes some shear resistance mechanisms in reinforced con-
crete members followed by a discussion of some available design
codes and guidelines and some related AI models in Section 3. The
details of the data used in the model development are shown in
Section 4. Section 5shows the details of the model development
followed by the results and discussion of the proposed model in
Section 6. Finally, some conclusions are presented in Section 7.
2. Shear resistance mechanisms
Stirrupless reinforced-concrete components resist shear
through five mechanisms. These are: (1) resistance of the un-
cracked concrete compression zone (Vcz ), (2) aggregate inter-
lock (Va), (3) dowel action of the main reinforcement (Vd), (4)
arching action, and (5) remaining tensile stresses in cracks (ft)
(Fig. 1) [31]. In conventional design methods, the contribution
of these mechanisms is usually combined together into one
term and described as the contribution of concrete to the shear
strength (Vc). The method of determination of this contribution
varies in different guidelines. Thus, the aim of this study is to
predict the contribution of shear strength (Vc) only. The probable
contributions from different mechanisms are explained below.
The resistance of the compression zone of uncracked concrete
relies on the depth of the uncracked zone and the concrete
strength. The greater the uncracked area or the concrete strength,
the greater the resistance. The contribution of the aggregate
interlock relies on the coarseness of the crack gravity, the width
of the crack, and the concrete strength. The severity of the crack
depends on the maximum aggregate size, and the width of the
crack depends on the stiffness of the reinforcement. Once the
crack occurs and begins to widen, the reinforcement restricts
the expansion of the crack. For narrow cracks, the added locking
mechanism can make a significant contribution [31]. The dowel
action is described as the ability of the longitudinal reinforce-
ment to transfer forces perpendicular to its axis. These forces
are effective when the crack surfaces slip and the longitudinal
bars counteract this slippage. This capacity depends mainly on
the transverse stiffness and the longitudinal bar strength. The
dowel effect can be important for elements with a high propor-
tion of longitudinal reinforcement, especially if the longitudinal
reinforcement is dispersed over several layers [31].
2
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
Fig. 1. Mechanisms of shear transfer in a beam with no stirrups after crack.
On the other hand, the prevention of the transmission of the
shear flow through an inclined crack that extends from the load
to the reaction is defined as an arch action. This is typically the
case for beams with a shear span-to-depth ratio of less than 2.5.
This type of beam is defined as a deep beam. In this regard,
the shear is transmitted through an arch rather than by beam
action. The tension reinforcement works as a link to form a
tied arch. The remaining tensile stress through the crack is the
amount of stress carried by small pieces of concrete that bridge
the crack. The cracks can transmit the tensile force for crack
width ranges from 0.05 to 0.15 mm [31]. The tooth model of
Reineck [32] showed that the residual tensile stress accounts for
much of the shear strength for very thin elements (depth of less
than 100 mm) with small widths of bending and diagonal cracks.
Although the basic mechanisms of shear transfer can be the same
for both conventional and FRP-reinforced elements, the relative
contribution of each of the mechanisms can be very different due
to the lower modulus of elasticity, the increased strength, and the
linear elastic behavior of FRP bars.
3. Details of some codes and guidelines and artificial intelli-
gence (AI) models
This section describes some of the current design codes and
guidelines that are used in different countries together with
some AI models. Among these, the Japan Society of Civil En-
gineers (JSCE) proposed an equation for the shear strength of
FRP-reinforced concrete members (JSCE 1997) [6] as given by
Eq. (1):
Vc=βdβρβnfvudbwd
γb
(1)
where βρ=3
100ρfEf
Es≤1.5, βd=100
d1/4≤1.5,
βn=1+Mo
Md≤2.0forN′
d≥0, βn=1+2Mo
Md≥0forN′
d<0,and
fvud =0.2f1/3
mcd ≤0.72, where fmcd is the design compressive
strength of concrete, γb=the reduction factor for strength,
usually 1.3, Mo=the decompression moment, Md=the bending
moment (design), N′
d=the axial compressive force (design),
βn=1.0 for sections with no axial force, Efis the elastic
modulus of FRP, Esis the elastic modulus of steel, dand bware
the effective depth and width of the member, respectively, and
ρfis the reinforcement ratio.
The interim guideline of the British Institution of Structural
Engineers (BISE) for the design of FRP-reinforced members in
shear [7] is based on the design equation of steel-reinforced
members and is given by Eq. (2):
Vc=0.79 100ρf
Ef
Es1/3400
d1/4fcu
25 1/3
bwd,
where fcu =1.25f′
c(2)
where fcu is the cubic strength of concrete and f′
cis the cylinder
strength of concrete.
As per the Canadian Standard Association (CSA) [8], the shear
capacity of FRP-reinforced members can be calculated using Eq.
(3):
Vc=0.05λφckmkrf′
c1/3bwdv,where dv=0.9d,
km=Vd
M,andkr=1+ρfEf1/3
0.11λφcf′
cbwd≤Vc≤0.2λφcf′
cbwd(3)
where φcis the concrete resistance factor, λis the concrete
density factor, and Vand Mare the shear and moment at a
section, respectively.
When the ratio of shear span to depth (a/d) is less than 2.5, Vc
should be multiplied by ka, where ka=2.5Vd/Mand1.0≤ka≤
2.5. For an effective depth more than 300, Vcmust be multiplied
by ks, where ks=750/(450 +d)≤1.0.
The proposed guideline of the American Concrete Institute
(ACI) [9] for shear design of members reinforced with FRP bars
is similar to that of the ACI 318-14 [33] code for steel-reinforced
members. The equation for calculating shear capacity is given in
Eq. (4):
Vc=0.4f′
cbwkd,where k=2ρfnf+ρfnf2−ρfnf,and
nf=Ef/Ec(4)
Here, the calculation of k(the ratio of neutral axis depth to the
effective depth, d) takes into account the effect of the axial stiff-
ness of the reinforcement, which depends on the reinforcement
ratio (ρf) and the modular ratio nf.
Among the AI models, Nehdi et al. [11] proposed the following
model for estimation of the shear capacity of FRP-reinforced
members:
Vc=2.1f′
cρfdEf
aEs0.3
bwdfor a/d>2.5 (5)
For a/d<2.5, Vcshould be multiplied by 2.5d/a, where ais the
shear span.
The model proposed by Lee and Lee [14] is given by the
following equation (Eq. (6)):
Vc=(A1d+A2)bw(6)
where A1andA2are the linear functions of Efρf.
On the other hand, the non-linear regression model proposed
by Jumaa and Yousif [15] is derived from the artificial neural
3
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
Table 1
Statistical breakdown of the database.
Parameters
a/d d (mm) f′
c(MPa) ρeff (%)
No. of Specimens 216 216 216 216
Mean 3.9 289 41.6 0.35
Median 3.5 225 36.3 0.32
Std. Deviation 1.2 171 16.1 0.23
Minimum 2.5 141 24.0 0.08
Maximum 6.5 938 88.3 1.48
Interquartile range 1.4 103 13.3 0.29
network (ANN) and gene expression programming (GEP) and is
given by Eq. (7):
Vc=0.32 1
d1/3Efρf
a/d2/3
f′
c1/5bwd(7)
4. Data collection
To examine the shear strength of members reinforced with
FRP bars and to develop a robust model, 216 beam and unidirec-
tional slab samples were taken from the literature [1–4,34–53].
Among the specimens, 120 of them were glass fiber-reinforced
polymer (GFRP), 94 of them were carbon fiber-reinforced polymer
(CFRP), and two of them were aramid fiber-reinforced polymer
(AFRP). The specimens were reinforced longitudinally without
stirrups and loaded through 3 or 4-point bending. The parameters
considered in this investigation were depth of beam (d), shear
span-to-depth ratio (a/d), concrete strength (f′
c), and effective
reinforcement ratio (ρeff ). Due to the different elastic modulus of
the bars, the effective reinforcement ratio was considered. The
ranges of these parameters were 2.5 to 6.5, 141 mm to 938 mm,
24 MPa to 88 MPa, and 0.08% to 1.48%, respectively. The detailed
statistical values of the parameters are shown in Table 1.
5. Details of the proposed model
The background of the proposed hybrid model (BOA–SVR) is
presented in this section. Brief explanations of the SVR intelli-
gence technique and BOA are given.
5.1. Support vector regression (SVR)
Support vector machine (SVM) algorithm is a general clas-
sification and regression method, which is based on statistical
learning theory [26]. This has been extended to solve regression
problems with the introduction of Vapnik’s ε-insensitive loss
function, which is known as support vector regression (SVR). SVR
utilizes the principle of structural risk minimization (SRM) to im-
prove its generalization strength, even if it is established utilizing
a limited number of learning data. The aim of this minimization is
to get a function that has at most εerror from the actual targets
for all training data. Since it is inevitable to avoid the errors, the
aim is to keep the error within a certain value range (ε). The
connection between input and output variables for non-linear
mapping can be expressed as Eq. Eq. (8) [54]
k(z)=⟨v.φ(z)⟩+c(8)
where z=(z1,z2,z3,..........zn) denotes the input value,
yirepresents the output value, and φ(z)denotes an irregular
function to assign input data to the high-dimensional domain.
Furthermore, v∈Rnrepresents the weight vector, which deter-
mines the orientation of a discriminating plane, c∈Rrepresents
the scalar threshold, which determines the offset of the discrim-
inating plane from the origin (i.e. bias term), and nis the size of
training data. The Vapnik’s ε-insensitive loss function is defined
as [26]:
|y−k(z)|ε=max {0,|y−k(z)|−ε}, ε > 0 (9)
The flatness of Eq. (8) depends on a smaller value of v. In practice,
it is not possible that a function will provide an error for all data
points that is less than ε. In order to allow some more errors, slack
variables ξi,ξ∗
iis introduced. Consequently, the optimization
function in SVR can be written as [26]:
Minimize: 1
2
v2
+C
n
i=1ξi−ξ∗
i
Subject to:
yi−{v.φ (zi)+c}≤ε+ξi
{v.φ (zi)}+c−yi≤ε+ξ∗
i
ξi, ξ ∗
i≥0
(10)
where Cis a regularization constant that is defined as the penalty
factor to indicate the trade-off between the empirical error and
the flatness of the model. The slack variables are zero for all data
points within the ε-insensitive zone and progressively increase
for data points outside the zone. Smola and Scholkopf [55] ex-
plained the solution of the optimization problem in Eq. (10) by
converting it into a dual formulation using Lagrange multipliers
ηi, η∗
i,αi, α∗
i. The final form of the solution can be written as:
Maximum: −1
2
n
i=1
n
j=1αi−α∗
iαj−α∗
jKzi,zj
−ε
n
i=1αi+α∗
i+
n
i=1
yiαi−α∗
i
Subjected to:
n
i=1αi−α∗
i=0andαi, α∗
i∈|0,C|(11)
where Kzi,zj=ϕ(zi).ϕ zjis defined as the kernel function.
After solving Eq. (11) for the values of αi, α∗
i, the final form of
Eq. (8) can be written as:
k(z)=
n
i=1αi−α∗
iKzi,zj+c(12)
In this optimization problem, the kernel function is calculated
instead of φ(z) in order to reduce the computational costs for
high-dimensional feature space treatment. The kernel functions
commonly used in regression include the linear kernel function,
the polynomial kernel function, the radial basis function (RBF),
and the sigmoid kernel function. In view of the infinitely dimen-
sional feature space corresponding to the RBF, the following RBF
is used in this investigation:
Kzi,zj=e−γ∥zi−zj∥2where γ=1/σ 2is the RBF parameter.
(13)
5.2. Principle of Bayesian optimization algorithm (BOA)
The BOA works based on Bayes’ rule, as shown in Eq. (14)
below [56]:
p(w|D)=p(D|w)p(w)
p(D)(14)
where wdenotes an unseen value, p(w) denotes the preceding
distribution, p(D|w) denotes the probability, and p(w|D)
indicates the posterior distribution.
Bayes’ rule covers prior knowledge to determine the posterior
possibility, which means that when choosing the values for the
4
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
Fig. 2. A general pseudocode of BOA.
next iteration, the results of previous iterations will be used.
Hence, it can approach the optimum point more efficiently than
arbitrary selection. The BOA can be used with two sub-models,
i.e., the substitute and the acquisition. The substitute model eval-
uates the objective function using the Gaussian process (GP),
which is a standard surrogate for modeling objective function.
This is a generalization of Gaussian distribution. Generally, GP
defines a prior over function and, after having observed some
functions values, it can be converted into a posterior over func-
tions. In this technique, the function f(z) is assumed to be a
realization of GP with the mean function m(z) and the covariance
function kzi,zjas shown below, where zis the function value
with any potential pair of zi,zjin the input domain. The detail
of this can be found elsewhere [57]:
f(z)∼GP m(z),kzi,zj (15)
Every input in this function is a variable that is related to the
other variables in the input domain, such as those defined by
the covariance function. The covariance function is a kernel that
controls the smoothness and amplitude of the Gaussian process
samples. On the other hand, the acquisition function of the BOA
relies on the preceding observations, and it is maximized over
repetitions. The acquisition model suggests the next point to
iterate using the results of the substitute model. Mathematically,
the hyperparameter optimization using BOA can be stated as
Eq. (16):
g⋆=arg min
g›G f(g) (16)
Here, f(g) is the objective score to reduce root mean square
error (RMSE), g⋆is the set of hyperparameters that generates
the lowest value of the score, and gis any value of space G.
In this study, the BOA was utilized since it is more effective
than other available optimization approaches (e.g., grid, random
search, manual, particle swarm optimization). It is also an orderly
procedure for global optimization of black box functions [58,59].
5.3. Hyperparameter optimization using BOA
The hyperparameters (i.e., ε,C, γ, kernel function) are opti-
mized by integrating the Bayesian optimization algorithm (BOA)
with the SVR algorithm. Prior to the application of the BOA, the
k-fold cross validation approach was utilized. Concisely, the data
are separated into ksubsets similarly. One dataset is selected as
a test subset, while the others are used as training subsets. This
approach is reiterated ktimes, and hence, each subset is applied
exactly once for the test. A general pseudocode of the BOA is
presented in Fig. 2.
5.4. Performance evaluation indicator
The performance of each model is assessed using several in-
dicators such as correlation coefficient (R), mean absolute error
(MAE), root mean square error (RMSE), and fractional bias (FB).
Ris utilized to evaluate the strength of the relationship between
predicted and experimental responses. The range of Rvaries from
−1 to +1. If the Rvalue is close to 1, the model can accurately
predict the experimental observations. MAE measures the av-
erage prediction errors without considering their direction. In
RMSE, the errors are squared before they are averaged, making
it more powerful when large errors are particularly undesirable.
The model with high Rand low error evaluations is always
desirable. Fractional bias (FB) is a degree of the shift between the
experimental and anticipated values. Mathematically, all these
indicators are defined as follows:
Pearson correlation coefficient,R=
1−N
i=1(VExp −VM)2
N
i=1(VExp −VExp)2
(17)
5
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
Fig. 3. The progress of Bayesian optimization for tuning hyperparameters of
SVR.
Mean absolute error,MAE =N
i=1VExp −VM
N(18)
Root mean square error,RMSE =N
i=1(VExp −VM)2
N(19)
Fractional bias,FB =2N
i=1(VExp −VM)
N
i=1(VExp +VM)(20)
where VExp denotes the experimental shear strength, VExp de-
notes the mean experimental shear capacity, VMis the model-
forecasted shear capacity, and Ndenotes the number of test data
specimens.
6. Results and discussion
6.1. Hybrid BOA-SVR model development
The BOA–SVR model is developed by tuning its hyperparam-
eters (i.e., kernel function, C,ε,γ) since the model performance
as well as computational time depend heavily on these param-
eters. Several popular approaches to tune the hyperparameters
are available in the literature, including the grid search algo-
rithm, random search algorithm, particle swarm optimization,
and Bayesian optimization algorithm (BOA). It is worth men-
tioning that both the grid and random search algorithms re-
quire many different trials and can be time consuming. Particle
swarm optimization is a well-known traditional approach and is
also time-consuming. Conversely, the BOA is a state-of-the-art
optimization framework that identifies the best parameters in
significantly less time than the other methods because it uses an
acquisition function that measures the next point to assess. The
BOA is used in the present research because it is also a systematic
process for tuning that does not require derivatives [59–61]. It
also provided better results compared to other methods. A 5-
fold cross-validation approach was chosen to prevent overfitting
because it showed a low RMSE with a lower computing time of
29.88 s The type of kernel function and the values of epsilon,
box constraint, and kernel scale were optimized using the BOA
technique, and the predictive accuracies of the models were
assessed. Fig. 3 shows the progress of the SVR hyperparameter
optimization, including the optimal point. The score for the min-
imum objective observed of 6.662 was observed at 47 iterations.
The level of precision was used to determine the optimal model
using the tuned parameters as given in Table 2.
Since the intelligence-based models are data dependent, the
applied BOA–SVR models were generated using real-life labora-
tory trials acquired from the literature. It is a noteworthy aspect
Table 2
The optimized parameters for SVR model.
Parameters Range/type Optimized parameters
Kernel function Linear, Gaussian, Polynomial Gaussian
Epsilon [10−4, 10] 0.37456
Box Constraint [10, 1000] 975.654074
Kernel Scale [10−4, 100] 12.783064
Table 3
Statistical summary of the experimental and the SVR model predictions.
Parameters Expt. SVR
Observation 216 216
Mean 58.85 58.63
Median 38.06 39.28
Std. Deviation 51.69 49.11
Minimum 8.76 9.27
Maximum 264.80 264.43
Interquartile range 48.94 50.28
Fig. 4. Box plot for experimental and anticipated shear strengths.
that the descriptive statistics and distribution of results were
achieved by both experiments and the SVR model. Table 3 dis-
plays the statistical analyses of the experimental and anticipated
shear strengths using the model.
On the other hand, a boxplot is a common technique of
demonstrating the distribution of data via 5-number summary,
i.e., median, minimum, and maximum scores, and first and third
quartiles. Fig. 4 indicates the boxplots of experimental and antici-
pated results for shear strength. The results demonstrate that the
experimental and anticipated output distributions were compa-
rable. In addition, right-skewed behaviors were detected for all
datasets. The difference between the datasets was not observed
since all the median lines fall within the boxes. The interquartile
ranges for both the experimental (48.94) and predicted (50.28)
strengths did not fluctuate much, which indicates that the results
were not broadly distributed. Besides, few outliers were detected
in both cases.
6.2. Evaluation of the hybrid BOA-SVR model
The SVR model developed in this study was used to antic-
ipate the shear capacity of FRP-reinforced members with no
stirrups. The comparison between anticipated and experimen-
tally determined shear strengths with respect to frequency is
shown in Fig. 5. Clearly, most of the experimental data over-
lap the predicted data, indicating that the developed model is
efficient. As mentioned earlier, the model utilizes automatic hy-
perparameter tuning which optimized the results nearest to the
experimental results, and k-fold cross validation was used to
prevent overfitting. Thus, better predictions were achieved.
6
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
Fig. 5. Comparison between estimated and experimentally measured shear
strength.
Fig. 6. Fitted line plot of experimental and predicted values.
Table 4
Performance indicator for training, testing and overall dataset.
Criterion Training (173) Testing (43) All (216)
R0.9814 0.9591 0.9773
MAE 4.0166 8.2114 4.8517
RMSE 10.1693 13.9400 11.0233
FB 0.0124 −0.0276 0.0033
Fig. 6 indicates the fitted line between the experimental and
model-forecasted shear strengths with the corresponding 45◦
line. It is evident that the fitted line (red in color) is very close
to the 45◦line. The coefficient of determination (R2) as well as
the adjusted coefficient of determination (adj R2) values of the
fitted curve are 95.5%, which indicates a proper fit. Hence, it can
be argued that the forecasted results are close to the real-life
laboratory data.
In addition, to assess the compatibility of the model, the
results of residual analyses (relative deviation vs. frequency and
residual vs. frequency) are shown in Figs. 7–8. It is worth men-
tioning that the residual plot is a scatter plot, and if relative
deviations or residual data points are scattered around the zero
line, then the model is acceptable. Interestingly, all of the plots
specify that the relative deviations in the case of training (Fig. 7a),
testing (Fig. 7b), and overall (Fig. 7c) are well distributed around
the zero line with little deviation. Similarly, Fig. 8 shows that all
data points of residuals are observed around the zero line, thus
supporting the reliability and validity of the applied model.
For further assessment of the performance of the SVR–BOA
model, the predicted results of training, testing, and overall
dataset are assessed using several performance-measuring in-
dicators such as R, MAE, RMSE, and FB. The results of all the
performance indices for the model are displayed in Table 4. High
values of Pearson correlation coefficient (>0.95) were observed
for training, testing, and overall, indicating that the predicted
and experimental results were superimposed. All the statistical
Fig. 7. Plot of relative deviation vs frequency for; (a) training, (b) testing, and
(c) all data.
Fig. 8. Residual vs. frequency plot for all data.
error parameters (MAE, RMSE) were observed to be low (Table 4).
Besides, the performance of a model is satisfactory if |FB| ≤
7
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
Fig. 9. Variation of experimental to the forecasted shear strength: (a) CSA [8], (b) ACI [9], (c) JSCE [6], (d) BISE [7], and (e) Proposed BOA–SVR Model.
Table 5
Comparison of nominal strengths.
Criterion Vexp/Vpred
JSCE [6] BISE [7] CSA 806-12 [8] ACI 440.1R-15 [9] Nehdi et al. [11] Lee and Lee [14] Jumaa and Yousif [15] BOA–SVR model
Mean 1.320 1.100 1.523 1.817 1.35 1.01 1.00 0.995
STDV 0.289 0.256 0.317 0.446 0.45 0.14 0.17 0.136
CoV (%) 21.925 23.257 20.801 24.562 33 14 17.32 13.648
R0.900 0.953 0.851 0.828 – – – 0.977
MAE 14.852 8.993 19.808 23.875 – – 7.3 4.852
RMSE 23.368 15.805 28.764 31.892 – – 10.7 11.023
FB 0.282 0.104 0.370 0.509 – – – 0.003
Note: ‘‘–’’ data not available.
0.5 [62]. The model-predicted bias is 0.0033. Hence, the results
shown in Table 4 indicate that the model used in this research
is strongly acceptable in forecasting the shear strengths. It is a
noteworthy aspect that the hybrid BOA–SVR model predictions
8
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
are reliable if the correlation coefficient and error values are close
to 1 and 0, respectively. Clearly, the SVR model satisfies this
criterion.
6.3. Evaluation of the results with codes and guidelines
The forecasted results using the applied BOA–SVR model are
compared with the existing JSCE-1997 [6], BISE-1999 [7], CSA
S806-12 [8], and ACI 440.1R-15 [9] methods for predicting the
shear capacity of FRP-reinforced members. In addition, the model
results are compared with some other artificial intelligence mod-
els proposed by Nehdi et al. [11], Lee and Lee [14], and Jumaa and
Yousif [15]. These models are selected based on the availability
of the results mentioned by the authors. The nominal strengths
are used for the comparison. The comparison is based on average,
standard deviation (STDV), and coefficient of variations (CoV) of
the ratio of the experimental (Vexp) to predicted shear strengths
(Vpred). Table 5 shows the detailed results of this comparison. The
mean, STDV, and coefficient of variation of the SVR model are
0.99, 0.14, and 13.6%, respectively. The mean value is close to
1.0 with lower STDV and coefficient of variation compared to the
other methods. This could imply that the SVR model predictions
are superior to those of the other four methods. Similarly, the
Pearson correlation coefficient of the model is the highest, and
MAE, RMSE, and fractional bias are the lowest compared to the
other models. In addition, the proposed model yields the lowest
values of the STDV and CoV of the ratio of experimental to
predicted shear resistances than the other artificial intelligence
models [11,14,15]. Also, the MAE and RMSE values are smaller
than the Jumaa and Yousif [15] model.
Fig. 9 shows the variations of the ratio of the experimental
(Vexp) to forecasted shear strengths (Vpred ) of the models with
respect to the number of specimens. The figures are plotted
against the number of specimens irrespective of their values of
shear span-to-depth ratio (a/d), effective depth (d), reinforcement
ratio (ρ), and concrete strength. Among the five methods, the
hybrid BOA–SVR model shows uniform and consistent predictions
with less scattering around the 1.0 line irrespective of the shear
span-to-depth ratio (a/d), effective depth (d), reinforcement ratio
(ρ), and concrete strength (f′
c).
7. Conclusions
FRP rebars are a feasible alternative to traditional steel re-
inforcement in concrete members. Nonetheless, the shear na-
ture of FRP-reinforced members is unlike that of steel-reinforced
members. This paper examines the shear behavior and develops
a model using Bayesian optimization algorithm-based support
vector regression (BOA–SVR) for forecasting the shear capacity
of these types of members. A 5-fold cross validation approach
was used to avoid overfitting. A comprehensive dataset of test
results for simply supported beams and unidirectional slabs was
obtained from the literature for the development of the model.
The performance of the model was assessed utilizing the different
performance-measuring indicators. The predicted results were
also compared with some traditional design codes and guide-
lines of JSCE, BISE, CSA, and ACI and with AI models of Nehdi
et al. [11], Lee and Lee [14], and Jumaa and Yousif [15]. From the
investigation, the following conclusions can be made:
1. The anticipated shear capacity of FRP-reinforced members
using the hybrid BOA–SVR model closely matches the ex-
perimental results. The mean, median, and standard devi-
ation of the experimental and predicted shear capacities
are 58.85, 38.06, and 51.69 and 58.63, 39.28, and 49.11,
respectively. The interquartile ranges of the experimental
and predicted results are also very close, which are 48.94
and 50.28, respectively.
2. The forecasted results overlap the laboratory results since
the R2as well as the adjusted R2values between the
experimental and the forecasted values are about 95.5%.
3. The proposed model shows that the relative deviations are
well dispersed around the zero line with low deviation. The
residual data points also lie around the zero line, which
further validates the reliability of the proposed model.
4. The comparisons between the hybrid BOA–SVR model and
other traditional approaches (e.g., JSCE, BISE, CSA, and ACI
codes and guidelines) show that the hybrid model predic-
tions are more accurate, consistent, and uniform than those
of the other four methods. The mean, standard deviation,
and coefficient of variations of the proposed model are
0.99, 0.14, and 13.6%, respectively, which are lower than
those of other methods.
5. The statistical error parameters (MAE, RMSE) were ob-
served to be the lowest compared to the other codes and
guidelines. Additionally, the value of fractional bias (|FB|)
is 0.0033, which is close to zero, indicating that the model
is robust and reliable.
6. In addition, the proposed BOA–SVR model shows the low-
est mean, standard deviation, and coefficient of variation
compared to the other artificial intelligence (AI) models
mentioned in this study.
7. The BOA–SVR method can be used to develop the design
code for FRP-reinforced RC members.
8. The proposed method is investigated for FRP-reinforced
members only. Further investigation is needed to utilize/
combine this for steel-reinforced concrete members.
Nomenclature
Symbol Meaning
FB Fractional Bias
BOA Bayesian Optimization Algorithm
MAE Mean Absolute Error
MSE Mean Squared Error
RMSE Root Mean Square Error
RSM Response Surface Methodology
SVR Support Vector Regression
dDepth of Beams
a/dShear Span to Depth Ratio
ρfReinforcement Ratio
f′
cConcrete Compressive Strength
EsElastic modulus of steel
EfElastic modulus of FRP
CRediT authorship contribution statement
Md Shah Alam: Formal analysis, Writing - original draft. N.
Sultana: Formal analysis, Methodology, Writing - review & edit-
ing. S.M. Zakir Hossain: Conceptualization, Writing - review &
editing.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgment
The authors acknowledge the support provided by University
of Bahrain, Bahrain and Imam Abdulrahman Bin Faisal University,
Saudi Arabia to conduct this research.
9
M.S. Alam, N. Sultana and S.M.Z. Hossain Applied Soft Computing Journal 105 (2021) 107281
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