applied

sciences
Article
Prediction of the Yield Strength of RC Columns Using a
PSO-LSSVM Model
Bochen Wang 1,2 , Weiming Gong 1,2, *, Yang Wang 1,2 , Zele Li 1,2 and Hongyuan Liu 3

1 Key of Laboratory for Concrete and Prestressed Concrete Structures of Ministry of Education,
Southeast University, Nanjing 211100, China
2 School of Civil Engineering, Southeast University, Nanjing 211100, China
3 School of Civil Engineering and Architecture, Jiangsu University of Science and Technology,
Zhenjiang 212110, China
* Correspondence: wmgong@seu.edu.cn

Abstract: Accuracy prediction of the yield strength and displacement of reinforced concrete (RC)
columns for evaluating the seismic performance of structure plays an important role in engineering
the structural design of RC columns. A new hybrid machine learning technique based on the least
squares support vector machine (LSSVM) and the particle swarm optimization (PSO) algorithm is
proposed to predict the yield strength and displacement of RC columns. In this PSO-LSSVM model,
the LSSVM is applied to discover the mapping between the influencing factors and the yield strength
and displacement, and the PSO algorithm is utilized to select the optimal parameters of LSSVM to
facilitate the prediction performance of the proposed model. A dataset covering the PEER database
and the available experimental data in relevant literature is established for model training and testing.
The PSO algorithm is then evaluated and compared with other metaheuristic algorithms based on
the experiment’s database. The results indicate the effectiveness of the PSO employed for improving
the prediction performance of the LSSVM model according to the evaluation criteria such as the root
mean square error (RMSE), mean absolute error (MAE) and coefficient of determination (R2 ). Overall,
the simulation demonstrates that the developed PSO-LSSVM model has ideal prediction accuracy in
Citation: Wang, B.; Gong, W.; Wang, the yield properties of RC columns.
Y.; Li, Z.; Liu, H. Prediction of the
Yield Strength of RC Columns Using Keywords: RC column; skeleton curve; yield strength; least squares support vector machine; particle
a PSO-LSSVM Model. Appl. Sci. 2022,
swarm optimization
12, 10911. https://doi.org/
10.3390/app122110911

Academic Editor: Sang Whan Han

1. Introduction
Received: 9 October 2022
Accepted: 26 October 2022 Reinforced concrete (RC) structures are the main parts of modern buildings, and
Published: 27 October 2022 RC columns are one of the crucial weight-bearing components in RC structures, which
is closely related to the seismic performances of buildings [1,2]. The post-earthquake
Publisher’s Note: MDPI stays neutral
reconnaissance [3] demonstrates that the collapse of buildings in the earthquake was
with regard to jurisdictional claims in
mostly caused by the destruction and failure of RC columns, which play an important role
published maps and institutional affil-
in resisting lateral load. It is thus necessary to evaluate the existing structural performance
iations.
and load-carrying capacity of post-earthquake RC columns, and quickly identify the most
collapse-prone RC frames across the region, providing insights for repair strategies and
useful information to emergency teams for making plans for safety rescues. Seismic
Copyright: © 2022 by the authors. resistance assessment of post-earthquake RC structures is an important issue that has
Licensee MDPI, Basel, Switzerland. received widespread attention in the structural engineering field. Several existing modeling
This article is an open access article approaches were proposed for accurately estimating the residual collapse capacity of
distributed under the terms and buildings, but they are time-consuming and have low prediction accuracy and lack general
conditions of the Creative Commons applicability [4–6]. Hence, efficient and accurate prediction of the yield strength and
Attribution (CC BY) license (https:// displacement of RC columns has great practical significance for evaluating the seismic
creativecommons.org/licenses/by/ performance and engineering design of RC structures [7].
4.0/).

Appl. Sci. 2022, 12, 10911. https://doi.org/10.3390/app122110911 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022, 12, 10911 2 of 15

Several numerical and experimental [8] methods have been extensively employed
to investigate the nonlinear behaviors of RC structures. The numerical models, based
on the finite element method, are usually used for nonlinear static or dynamic analyses.
However, many FE-based models are incapable of capturing the hysteretic behaviors
accurately, something which depends on the selection of model parameters and boundary
conditions [4–6]. It is, therefore necessary to further improve the assessment accuracy
of the behaviors of RC columns by the numerical simulation method. Additionally, the
calculation of tedious iterative processes is time-consuming. In contrast, the experimental
method is more effective and capable of identifying the real responses of a structure [9,10],
but it is quite costly, time-consuming and labor-intensive.
Over the past few decades, artificial intelligence (AI) technologies, with the merits of
their strong nonlinear learning capacities, have been widely utilized in the earthquake and
civil engineering fields [11–16]. The technologies have gradually developed into an effective
research method to study the abovementioned problems [17–24], one which enables civil
engineers to predict the structural performance of concrete members. For instance, Quar-
anta et al. [24] performed innovative work in which a machine-learning-aided approach
was proposed to improve the accuracy of the mechanics-based shear capacity equation
for RC beams and columns. On the other hand, many models based on hybrid machine
learning (ML) techniques were developed to predict structural performance straightfor-
wardly without explicit formulae. Typical of these predictive models, the support vector
machine (SVM) proposed by Vapnik [25] is a widely used machine learning technique.
Thereafter, the least squares support vector machine (LSSVM) [26], as a modified version of
SVM, is reported to be able to conduct a faster training process for tackling many complex
and non-linear problems in engineering compared to the standard SVMs, i.e., with faster
computational and generalized capacity [27,28]. Some work on the application of machine
learning (ML) to the study of RC structures is listed in Table 1, as follows:

Table 1. Application of machine learning in the study of RC structures.

Study Model Issues or Objectives

Xu et al. [18] LSSVM Predict the strength of radiation-shielding concrete
Predict the ultimate punching shear capacity of
Vu and Hoang [19] LSSVM
FRP-RC slabs.
Luo and Paal [20] ML-BCV Predict backbone curves of RC columns
Mangalathu and Jeon [21] MLs Predict shear strength for RC beam-column joints
Ning et al. [22] DE-ANN Predict the hysteresis loop of RC columns

It is worth noting that Luo and Paal [20] developed a novel machine learning–based
backbone curve (ML-BCV) model to predict backbone curves of RC columns subjected to
three different failure modes. This model consists of a modified LSSVM to address the
multioutput case, and a grid search algorithm (GSA) to facilitate the training process. The
essence of the training process is to select the optimal solution for two key parameters that
significantly affect the accuracy level of predicted results. The selection process of these
two parameters can be regarded as an optimization problem [20]. Thus, an appropriate
swarm intelligence (SI) algorithm is needed to solve the optimization problem. Several SI
algorithms have been widely applied for solving various optimization problems owing to
their powerful global search capability [29], typically genetic algorithm (GA) [30], differen-
tial evolution (DE) [31], artificial bee colony (ABC) [32] and particle swarm optimization
(PSO) [33] algorithms. The PSO algorithm developed by Kennedy and Eberhart [33], is
regarded as a reliable tool when combined with AI techniques due to its outstanding
features of high calculation efficiency, high accuracy and powerful global optimization and
search capability in searching for the optimal solution [34–37]. The PSO has been used to
enhance the training process of LSSVM or other models, and further improve the predictive
Appl. Sci. 2022, 12, 10911 3 of 15

accuracy of forecasting models in the fields of engineering [38–42]. A detailed introduction

to the PSO algorithm used in the present work can be found in Section 4.2.
To the best of our knowledge, there are still few computational models that can
rapidly calculate the strength-related properties of RC columns with high accuracy [7,43,44].
Moreover, the predictive models or formulae that enable the comprehensive covering of
most of the influencing factors are still scarce due to the complexity of multi-factor issues.
Based on the challenge, machine learning (ML) techniques are widely adopted by virtue of
their data-driven nature and their expertise in dealing with nonlinear correlations between
multiple factors.
The present work focuses on developing a new prediction model based on machine
learning techniques for achieving the goal of efficiently and accurately predicting yield
strength and displacement of RC columns with more influence factors considered. In this
model, the LSSVM is applied to establish a prediction model to estimate the nonlinear
mapping between influence factors and yield properties. The PSO algorism is adopted
to facilitate the training process of the LSSVM model for the purpose of improving the
predictive accuracy of this model through selecting the optimal solution of key parameters
i.e., regularization parameter (γ) and RBF kernel parameter (σ). For training and testing
the performance of the present model, a test database of the proposed model is built up by
an efficient program script compiled to collect and collate the existing experimental data
obtained by pseudo-static cyclic testing of RC columns. The performance of the trained
LSSVM model can be evaluated by the root mean square error (RMSE), mean absolute
error (MAE), explained variance (EV) and coefficient of determination (R2 ). Meanwhile,
the effectiveness of the PSO in enhancing the prediction accuracy of the LSSVM model is
evaluated compared to other metaheuristics.

2. Pseudo-Static Cyclic Test and Yield Point

2.1. Pseudo-Static Cyclic Test
The pseudo-static cyclic test, pseudo-dynamic test and simulated seismic shaking table
test are three test methods which are commonly used for testing the seismic performance
of building structures and their components. Therein, the pseudo-static cyclic test is an
approach which records the whole process from the elastic stage to the failure or damage by
controlling the load or displacement for low circumferential reciprocal cyclic loading [20,45].
The pseudo-static test is widely adopted by researchers in the field of study of the seismic
performance of structures or components relying on its advantages, such as the better
adaptability of the site and experimental equipment, which is convenient for conducting
full-scale experiments, and low loading rate that allows detailed observation of the whole
process of specimen failure.
Under the action of low circumferential cyclic load, a hysteresis loop denoting the
relationship between the load and the corresponding deformation of the structure is formed
during a loading and unloading process, and the series of hysteresis loops formed after
multiple cycles of loading constitute the hysteretic curves of the structure, namely the
load-displacement curves. Then, the skeleton curve of the specimen shown in Figure 1
can be obtained by connecting the peak points of each hysteretic curve of the specimen,
which reflects the relationship between the force and deformation of the experimental
model. The skeleton curve is a comprehensive reflection of the seismic performance of
the structure; it thus can be an important basis for analyzing the elastic-plastic dynamic
response of the specific structure. For the concrete specimen, its skeleton curve is taken as
the envelope connected by the peak points of the first cycle at each loading level of each
load-displacement curve, i.e., the envelope of the hysteretic curve.
In recent years, many experimental studies on the seismic performance of RC columns
have been conducted by using the pseudo-static cyclic test [46,47]. The influences of relevant
factors on their seismic performance are investigated. The load-displacement curve can be
obtained through the experimental test, and it reflects the nonlinear performance, energy
dissipation characteristics and ultimate damage mechanism of the structure or component.
ppl. Sci. 2022, 12, x FOR PEER REVIEW
Appl. Sci. 2022, 12, 10911 4 of 15

Figure
Figure 1. hysteretic
1. The The hysteretic curve and
curve and skeleton curve.skeleton curve.
2.2. Yield Point
It isIn recent
well years,
known that how tomany experimental
judge the studies
yield point on the specific on the
skeleton curveseismic
is still a perf
complicated problem, and there is no unified standard in the related research field [7,43,48].
umns have been conducted by using the pseudo-static cyclic test [46
Here are three judgment methods commonly used to determine the yield point as follows:
1.of relevant factors
For the Geometric on their
Graphic Method seismic
shown inperformance
Figure 2a, the yield are investigated.
displacement and The
curve
yieldcanpointbe areobtained through
defined as follows: Drawthetheexperimental
tangent line to the test,
curveand it The
at ‘O’. reflects t
line is extended to the intersection with a horizontal line through ‘D’ at ‘A’, where
mance, energy dissipation characteristics and ultimate damage mech
the ‘D’ corresponds to the maximum applied shear Pmax shown in Figure 2a. The
tureperpendicular
or component. of line ‘OA’ intersects the curve at ‘B’. Connect ‘O’ and ‘B’ and extend
line ‘OB’ to meet the horizontal line ‘DA’ at ‘C’, and then project onto the horizontal
axis to obtain the yield displacement ∆y and the yield point ‘E’ on the curve, where it
2.2.corresponds
Yield Point to the yield applied shear Py .
2. For R. Park Method shown in Figure 2b [48], the yield displacement and yield point are
It is well known that how to judge the yield point on the speci
defined as follows: A secant ‘OB’ is drawn to intersect the lateral load-displacement
stillrelation
a complicated problem,
at a certain proportion of theand there
maximum is no
applied unified
shear, i.e., the standard
‘B’ on the curve in the r
corresponding to shown in Figure 2b. Similarly,
[7,43,48]. Here aremaxthree judgment methods commonly used to determ
αP this extension of line ‘OB’ and
the horizontal line corresponding to the maximum applied shear Pmax intersect at ‘A’,
as follows:
and then projects onto the horizontal axis to obtain the yield displacement ∆y . The
1. intersection
For thepoint ‘C’ of the vertical and curve is defined as the yield point, which
Geometric Graphic Method shown in Figure 2a, the yiel
corresponds to the yield applied shear Py .
3. Foryield point
Equivalent are defined
Elasto-Plastic Energy asMethod
follows:shownDraw
in Figurethe2c,tangent line to the c
the yield displace-
ment and yield point are defined as follows: Determine
is extended to the intersection with a horizontal line through a point ‘B’ on the curve and ‘D’
draw secant ‘OB’ to intersect the curve for satisfying the principle that the energy
corresponds
absorbed by the ideal toelastoplastic
the maximum structureapplied
is equal to shear Pmax shown
that absorbed by the actualin Figure
ular of
structure, i.e.,line ‘OA’
the areas intersects
of shaded theand
area ‘OAB’ curve
‘BCD’at are‘B’.
equalConnect ‘O’ and
shown in Figure 2c. ‘B’ a
Similar to the R. Park Method in Figure 2b, this extension of line ‘OB’ and the hori-
to meet
zontal the horizontal
line corresponding line ‘DA’
to the maximum at ‘C’,
applied shear and then project
Pmax intersect at ‘C’, andonto th
obtain
is then the yield
projected onto thedisplacement
horizontal axis to ∆ y and
obtain thethe
yieldyield point ∆‘E’
displacement on the cu
y . The
intersection point ‘E’ of the vertical and curve is defined as the yield point, which
sponds to the yield applied shear Py.
corresponds to the yield applied shear Py .
2. Forpaper,
In this R. Park Method
the yield point of shown incurve
the skeleton Figure
of the2b [48], the yield
load-displacement displacem
curve is
are defined as follows: A secant ‘OB’ is drawn to intersect the la
calculated by using the Equivalent Elasto-Plastic Energy Method.
ment relation at a certain proportion of the maximum applied s
the curve corresponding to αPmax shown in Figure 2b. Similarly, t
‘OB’ and the horizontal line corresponding to the maximum app
sect at ‘A’, and then projects onto the horizontal axis to obtain the
2, 12, x FOR PEER REVIEW
Appl. Sci. 2022, 12, 10911
5 of 516
of 15

(a) (b) (c)

Figure 2. The yield
Figurepoint judgment
2. The methods:
yield point judgment(a) Geometric
methods: GraphicGraphic
(a) Geometric Method; (b) R.(b)
Method; Park Method;
R. Park Method;
(c) Equivalent Elasto-Plastic Energy Method.
(c) Equivalent Elasto-Plastic Energy Method.

3. Data Collection and Pre-Processing

In this paper, the yield point of the skeleton curve of the load-displacement curve is
calculated by usingIn the
this Equivalent
work, a quantity of data adopted
Elasto-Plastic for theMethod.
Energy following calculations is collected from
the PEER database [49,50] and the relevant literature [22,44]. The PEER database covers
extensive hysteretic curves of the RC columns with various material properties obtained
3. Data Collection and Pre-Processing
by pseudo-static cyclic test. As for the collected original experimental data, relevant pre-
In this work, a quantity
processing shouldof bedata adopted
conducted for the
to obtain thefollowing
corresponding calculations
yield pointiswhose
collected
values
are used to compare with predictive values (see
from the PEER database [49,50] and the relevant literature [22,44]. The PEER database Section 4 for details). The steps of data
pre-processing are as follows: Firstly, determine the skeleton curve of each hysteretic curve
covers extensive hysteretic curves of the RC columns with various material properties ob-
according to the rule seen in Section 2.1; secondly, determine the yield point on the obtained
tained by pseudo-static cyclic test. As for the collected original experimental data, relevant
skeleton curve according to the method seen in Section 2.2, and the yield strength and
pre-processingdisplacement
should be conductedare obtained. toHowever,
obtain the thecorresponding
amount of data yield involvedpoint whose
in the data values
collection
are used to compare
and pre-processing is enormous. To improve the data-processing efficiency, thedata
with predictive values (see Section 4 for details). The steps of above
pre-processingtwo aresteps are implemented
as follows: Firstly, via programming
determine calculation.
the skeleton Thus,ofa program
curve script was
each hysteretic
written
curve according to the forrule
plotting
seen theinskeleton
Sectioncurves, and then the
2.1; secondly, yield strength
determine and yield
the yield pointdisplacement
on the
are both outputted. Thereafter, a total of 382 test results were selected and organized in bulk
obtained skeleton curve according to the method seen in Section 2.2, and the yield strength
according to the parameter types, and then all the types of the parameter were categorized
and displacement
into twoare parts:
obtained.
input However,
parameters the and amount of data involved in the data collec-
output properties.
tion and pre-processing
The inputisparameters
enormous. areTo theimprove
influencing the data-processing
factors including the basic efficiency,
parametersthe of
above two steps
the RCarecolumn.
implemented The output via properties
programming denotecalculation.
performancesThus, of the RCa program script
column, e.g., yield
strength
was written for plotting and displacement,
the skeletonwhich curves, are and
codetermined
then thebyyield the influence
strength factors. In the present
and yield dis-
placement are both outputted. Thereafter, a total of 382 test results were selected and or- of
model, nine representative influence factors are identified as the input parameters
the proposed model for the prediction of output properties. The set of influence factors
ganized in bulk according to the parameter types, and then all the types of the parameter
can be expressed as {Dc , ρr , SecT, fc , fyl , fyv , ρl , ρv , ρn }, where Dc is the section size; ρr is
were categorized into two parts:
the span-to-depth ratioinput parameters
defined as L/Dc , andin which output L isproperties.
length; SecT is the section type,
The inputSecT
parameters
= 1 or SecTare = 2the influencing
denotes the square factors
or circle including the basic
section type; parameters
fc is the strength ofof con-
the RC column. Theyloutput properties denote performancesyvof the RC column, e.g., yield
crete; f is the yield strength of longitudinal bars; f is the yield strength of transverse
bars; ρl is the longitudinal
strength and displacement, which arereinforcement
codetermined ratio;
by ρthe v is influence
the transverse reinforcement
factors. In the pre- ratio;
ρ n the axial load ratio defined by ρ n = P/(A g · f
sent model, nine representative influence factors are identified as the input parameters ofand
is c ), in which P is the constant axial load
Ag is the gross section area. These input parameters, i.e., influence factors mentioned above,
the proposed are
model for the prediction of output properties. The set of influence factors
summarized in Table 2.
can be expressed as {D
The focus c, ρr, SecT, fc, fyl, fyv, ρl, ρv, ρn}, where Dc is the section size; ρr is the
of the developed model is a prediction of the yield strength (Fy ) and yield
span-to-depthdisplacement
ratio defined (∆as L/D
y ) of RCc, columns,
in whichwhich L is length; SecT is as
are considered thethesection
outputtype, SecT in
properties = 1the
or SecT = 2 denotes the square or circle section type; fc is the strength of concrete; fyl is the in
present study. The ranges of the input parameters and output properties are presented
yield strengthTable 3 in detail. Itbars;
of longitudinal can befyvseenis thethatyield
the input parameters
strength and output
of transverse bars;properties
ρl is theare well-
lon-
distributed over a wide range. Thus, each specimen in the test database can be written as
gitudinal reinforcement ratio; ρv is the transverse reinforcement ratio; ρn is the axial load
{{Dc , ρr , SecT, fc , fyl , fyv , ρl , ρv , ρn }, {Fy , ∆y }}, which involves the input parameters and output
ratio defined properties.
by ρn = P/(A g·fc), in which P is the constant axial load and Ag is the gross
Figure 3 displays the histograms of the input parameters and output properties.
section area. These input parameters, i.e., influence factors mentioned above, are summa-
rized in Table 2.

Table 2. Input parameters for LSSVM training.

Influencing Factors Parameters Description

Appl. Sci. 2022, 12, 10911 6 of 15

Table 2. Input parameters for LSSVM training.

Influencing Factors Parameters Description

Dc Section size (mm)
Geometries ρr = L/Dc Span-to-depth ratio (%)
SecT Section type
fc Strength of concrete (MPa)
Materials fyl Yield strength of longitudinal bars (MPa)
fyv Yield strength of transverse bars (MPa)
ρl Longitudinal reinforcement ratio (%)
Reinforcement
ρv Transverse reinforcement ratio (%)
Loading ρn = P/(Ag ·fc ) Axial load ratio
Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 16

Table 3. Statistical description of the input and output variables.

Table 2. Statistical description of the input and output variables. 265
Variables Min. Max. Mean Std.
Variables Min. Max. Mean Std.
Dc 80 1520 345.85 149.98
ρr Dc 180 152010 345.85 3.44 149.98 1.65
fc ρr 161 10 118 3.44 46.54 1.65 25.98
fyl fc 016 118587.1 46.54 422.84 25.98 72.41
fyv fyl 00 587.11424 422.84 454.52 72.41 206.60
ρl fyv 0.00460 1424
0.0603 454.52 0.03 206.60 0.01
ρv ρl 0.0046
0 0.0603
4.27 0.03 0.40 0.01 0.69
ρn ρv −0.0990 4.270.9 0.40 0.22 0.69 0.18
Fy ρn -0.099
16.27 0.9
2654.11 0.22 202.49 0.18 204.25
∆y Fy 16.27
0.54 2654.11
114.70 202.49 12.64 204.25 14.55
Δy 0.54 114.70 12.64 14.55
266

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)
Figure 3. Statistical distribution of the input and output variables. (a) Section size; (b) Span-to-depth 267
3. Statistical
Figureratio; distribution
(c) Strength of concrete of; (d)
theYield
inputstrength
and output variables: bars
of longitudinal (a) Section size;
; (e) Yield (b) Span-to-depth
strength of 268
ratio; (c) Strengthbars
transverse of concrete; (d) Yieldreinforcement
; (f) Longitudinal strength of longitudinal bars; (e)reinforcement
ratio; (g) Transverse Yield strength of ;transverse
ratio 269
bars; (f) Axial load ratio
(h)Longitudinal ; (i) Yield strength
reinforcement ratio; Yield
; (j) (g) displacement
Transverse .
reinforcement ratio; (h) Axial load 270ratio;

(i) Yield strength; (j) Yield displacement.

4. Model Development and Verification 271

4.1. Least Squares Support Vector Machine 272

The LSSVM first introduced by Suykens et al [23], is a supervised machine learning 273
method base on statistical learning theory. It is well-known that the LSSVM methodology 274
Appl. Sci. 2022, 12, 10911 7 of 15

4. Model Development and Verification

4.1. Least Squares Support Vector Machine
The LSSVM, first introduced by Suykens et al. [26], is a supervised machine learning
method based on statistical learning theory. It is well-known that the LSSVM methodology
possesses an outstanding feature in learning nonlinear functions, which simply solves
a set of linear equations. Therefore, LSSVM, as an extension of SVM equipped with the
advantage of fast calculation and generalized capacity, has been widely applied for practical
problems such as nonlinearly, high-dimensional input space and small training samples.
These problems are generally divided into two categories, i.e., regression problems and
classification problems. The focus of the present study is predicting the yield strength and
displacement with the LSSVM adopted in this prediction model, which is thus considered
as a regression problem. The governing equations and numerical algorithms are described
in detail as follows.
For a set of given nonlinear training samples {xi , yi }, i = 1,2, . . . ,l, where l is the number
of training samples. xi ∈Rn is the input vector, and yi ∈R is its corresponding output vector.
The regression function is obtained by mapping the input space to high-dimensional feature
space using the nonlinear mapping function φ(·), the equation of the regression function
can be written as
f ( x ) = wT φ( x ) + b (1)
where w is the weight vector; b is the bias term (also called the partial vector).
The objective of LSSVM regression can be transformed into an optimization problem
that is expressed as
l
min J (w, ξ ) = 12 kwk2 + γ
2 ∑ ξ i2
w,b,ξ i =1 (2)
s.t. yi = wT f (x i ) + b + ξi, i = 1, 2, . . . , l
where γ is the regularization parameter (also called the penalty parameter); ξ ∈R are the
error variables.
The Lagrange function of the optimization problem, i.e., Equation (2) is formulated as
follows:
l h i
L = J − ∑ αi w T · φ ( xi ) + b + ξ i − yi (3)
i =1

where αi is the Lagrange multiplier, and sample (αi 6= 0) is the support vector.
The Karush-Kuhn-Tucker (KKT) conditions are employed for the optimal solution of
the object function in a nonlinear optimization problem [51,52]. The resulting regression
model of LSSVM ultimately is obtained and can be evaluated by

l
y( x ) = ∑ αi K(x, xi ) + b (4)
i =1

where K(x, xi ) = φ(x)T φ(xi ) is the kernel function, which is used for classification by mapping
the input data from the featured space into a high-dimensional space. In the present work,
the Radial Basis Function (RBF) kernel function is adopted in the implementation of the
LSSVM and given by
k x − x i k2
−
K ( x, xi ) = e 2σ2 (5)
where σ is the RBF kernel function parameter that needs to be determined, which is
optimized via an external optimization technique during the training process. The pseudo-
code of LSSVM algorithm is presented in Algorithm 1 as follows.
Appl. Sci. 2022, 12, 10911 8 of 15

Algorithm 1: Pseudo-code of LSSVM algorithm

1 Initialize LSSVM parameters
2 Normalize data using Equation (8)
3 while stopping condition is not met do
4 Train LSSVM model with parameters (σ and γ) using each training data point
5 end while
6 Predict testing data using trained LSSVM model
7 return accuracy

In the current study, the learning performance of the LSSVM is mainly determined by
two tuning parameters, i.e., the RBF kernel function parameter (σ) and the regularization
parameter (γ). The RBF kernel function parameter (σ) influences the smoothness of the
approximated nonlinear function, and the regularization parameter (γ) controls the penalty
imposed on data point that deviates from the regression function. Particle swarm opti-
mization (PSO) is a popular approach that is employed to search for the optimal solution
of parameters (σ, γ). The minimum computational error can be obtained by selecting the
optimal solution (σ, γ) during the training process.

4.2. Particle Swarm Optimization

In this section, the basic algorithm of particle swarm optimization (PSO) [33] is intro-
duced in detail. It is well-known that the PSO is a heuristic randomized algorithm and
developed based on animal social behavior, e.g., food searching of bird flocks and fish
schools. As the LSSVM starts working for regression fitting, the solution of tuning parame-
ters (σ, γ) has a direct impact on the feature space and solution accuracy of the prediction
model. Compared with the traditional manual selection method, the PSO algorithm with
the advantages of short time consumption and high accuracy in searching for the optimal
solution is adopted in this paper. The PSO algorithm is widely proposed for simulating
population intelligence, in which swarm the individual is treated as a particle without
volume. The PSO begins with a group of particles migrating at various initial speeds and
directions. The assignment of initial speed and direction proceeds randomly in a certain
range. Thereafter, each particle iteratively updates its speed and direction according to
individual and group behavior. Finally, all the particles in the swarm migrate to the optimal
location, which is considered as the optimal solution to the corresponding optimization
problem. The PSO algorithm possesses the strong power of global optimization property
and search capability.
As mentioned above, the principle of PSO is that particles iteratively search for the
optimal solution from a random solution, and find the global optimum by following the
currently searched optimal solution. Assuming that the initial location and velocity of the
ith particle in N-dimensional space are Ui = (ui1 , ui2 , . . . , uij ) and Vi = (vi1 , vi2 , . . . ,vij ),
j = 1,2, . . . ,N; i = 1,2, . . . ,M, where M is the swarm size. In PSO, the optimal values of
location and velocity are found by tracking the extreme values of pbesti and gbest in each
iteratively updated process of the ith particle, where the pbesti is the individual optimal
experience of the ith particle denoted by Pi = [pi1 , pi2 , . . . ,pij ], and gbest is the global
optimal experience of the swarm denoted by Pg = [pg1 ,pg2 , . . . ,pgj ]. According to the fitness
value of particles, the location and velocity of each particle can be updated using the
equations below:
k +1 k k k
vid = χvid + c1 r1 ( pid − uid ) + c2 r2 ( p gd − uid ) (6)
k +1 k k +1
uid = uid + vid (7)
k +1 k are the velocities of the ith particle at kth and (k + 1)th iterations,
where vid and vid
k +1 k are the locations of the ith particle at kth and (k + 1)th iterations,
respectively; uid and uid
respectively; χ is the inertia weight coefficient; c1 and c2 are the learning coefficients, which
are two prescribed constants in PSO; r1 and r2 are two random constants in the range of [0, 1].
Appl. Sci. 2022, 12, 10911 9 of 15

In the present work, two tuning parameters (σ, γ) are required to be optimized, thus
the value of N is selected as N = 2. Considering that the assignment of M is relevant to the
specific problems, whose value is 10~50 in general. For the present problem of optimization
and prediction, the size of the solution M is set to be M = 30. The empirical values of the
learning coefficients c1 = c2 = 2 is adopted [38,42]. Since a higher value of the inertia weight
coefficient (χ) is conducive to jumping out of local convergence, while a lower value of χ
has a stronger local search capability, the value of χ is taken as 0.8 [38]. And the maximum
number of swarm evolution (i.e., iteration number) kmax = 200 is pre-set in this work. The
pseudo-code of PSO algorithm is presented in Algorithm 2 as follows.

Algorithm 2: Pseudo-code of LSSVM algorithm

1 Initialize population of particles and derive local and global best particles (pbesti and gbest )
2 for k = 1 to maximum number of iterations do
3 for i = 1 to population size do
4 Update the velocity of the ith particle (vi ) using Equation (6)
5 Update the location of the ith particle (ui ) using Equation (7)
6 if F(ui ) < F(pbesti ) then
7 pbesti = ui.
8 if F(pbesti ) < F(gbest ) then
9 gbest = pbesti
10 end if
11 end if
12 end for
13 end for
14 return gbest

When the RMSE of the LSSVM is minimal, the corresponding σ and γ can be regarded
as the optimal parameters. The detailed optimization steps are stated as follows:
(i) Initialize the parameters in PSO algorithm.
(ii) Calculate the fitness value of each particle, and evaluate their fitness with Equation (8),
i.e., F(u) = F(σ, γ) = RMSE.
(iii) Update the location and velocity of the ith particle with Equations (6) and (7) and
compare the current fitness value of each particle F(ui ) with the individual best fitness
value F(pbesti ), if satisfying F(ui ) < F(pbesti ), pbesti = ui.
(iv) Compare the current fitness values of all particles in the swarm F(ui ) with the fitness
value of the best location of the swarm F(gbest ), if satisfying F(ui ) < F(gbest ), the global
optimal solution gbest = ui.
(v) Check whether the termination condition is met. If the error accuracy is satisfied or
the maximum number of swarm evolution is reached, then the process of searching
for the optimal solution (σ, γ) ends and the optimal values of σ and γ are outputted.
Otherwise, proceed to the next step from step (ii) to continue the process of parameter
optimization.
(vi) Substitute the optimal parameters (σ, γ) into Equation (1) for predicting the yield
strength and displacement of RC columns.

4.3. Implementation of the Yield Strength and Displacement Prediction

This section introduces the implementation of the proposed PSO-LSSVM model for
the yield strength and displacement prediction in detail, as shown in Figure 4. At the
beginning of the PSO-LSSVM, to prevent the formation of larger range numerical attributes
dominating smaller range attributes, data initialization is required. The normalized data
are in the range of [0, 1] through Equation (8), which is defined by

xi − xmin
xin = (8)
xmax − xmin
 xi − xmin
Appl. Sci. 2022, 12, 10911 xin = 10 of 15
xmax − xmin

where
where xi isxany
i is any data point, xmin and xmax are the minimum and maximum
data point, xmin and xmax are the minimum and maximum values of the
entire
entire dataset;
dataset; xinnormalized
xin is the is the normalized value
value of the data of the data point.
point.

Figure
Figure4. Flowchart of PSO-LSSVM
6. Flowchart scheme.
of PSO-LSSVM scheme.
4.4. Evaluation of the PSO-LSSVM Prediction Performance
4.4. Performance
4.4.1. DiscussionEvaluation Indicators
The prediction performance of PSO-LSSVM is evaluated by the coefficient of determi-
The prediction performance of PSO-LSSVM is evaluated by the coeffi
nation (R2 ), mean absolute error (MAE) and root mean square error (RMSE). The coefficient
Rmination
2 represents (R
the ),
2 mean absolute error (MAE) and root mean square error (RM
level of explained variability between the actual values measured in the
ficient R represents the
experiment 2and predicted level
values of explained
computed variability
by the PSO-LSSVM between
model. the actual
The closer the val
coefficient R 2 varies toward 1, the more similar the actual values and predicted values are.
in the experiment and predicted values computed by the PSO-LSSVM mod
Likewise, the lower values for MAE and RMSE, the higher accuracy in the predicted values
the coefficient
indicate. R2 varies
Equations (9)–(12) toward
are the 1, the formulation
mathematical more similarof thethe actual
adopted values and pre
performance
are. Likewise,
evaluation the lower
criteria RMSE, MAE andvalues
2 for MAEasand
R , respectively, RMSE, the higher accuracy in
follows:
values indicate. Equations (9)-(11)
s are the mathematical formulation of the
1 n
n∑ i
formance evaluation criteria
RMSE RMSE,
= MAE )2
(y − ŷiand R2, respectively, as follows:
(9)
i =1

1 n
1 n
MAE = ∑ |yRMSE
n i =1 i
− ŷi | = 
n i =1
( yi − yˆ i )2 (10)

(n∑ yi × ŷi − ∑ yi ∑ ŷi )2 n

R2 =  1  (11)

ŷ2i − (=∑ ŷi )2 yi − yˆ i

n∑ y2i − (∑ yi )2 n∑ MAE
n i=1
Var(yi − ŷi )

Explained Variance = 1 − × 100% (12)
Var(yi )
( n y  yˆ −  y  yˆ )
2
i i i i
R2 =
( n y − (  y ) ) (n yˆ − (  yˆ )
2
i i
2 2
i i
2
)
where yi is the experimental value of the ith sample, yˆ i is the predicted va
 results that present quite excellent agreement with the experimental test results in both 404
The scatter
training and plots in the
testing two figures
processes. show that
Moreover, it the
alsoproposed
can be seen model thathas theyielded prediction
PSO-LSSVM earned 403 405
results thathigh
relatively presentvaluesquiteof excellent
R for testing,
2 agreement with the experimental
the corresponding values oftest theresults in both
coefficient of de-404 406
training and testing processes. Moreover, it also can be seen
termination (R ) are 0.99 and 0.96 for the training and testing cases of yield strength, re-
2 that the PSO-LSSVM earned 405 407
Appl. Sci. 2022, 12, 10911 relatively high
spectively (Figuresvalues of and
7(a) R for
2 testing,
8(a)), and the the corresponding
correspondingvalues valuesofofthe R coefficient
2 are 0.98 and of 0.97
de- for
11 of 15
406 408
termination (R 2) are 0.99 and 0.96 for the training and testing cases of yield strength, re-
the training and testing cases of yield displacement, respectively (Figures 7(b) and 8(b)). 407 409
spectively (Figures 7(a) and 8(a)), and the corresponding
The high values of R2 imply that a strong correlation exists between the predicted and values of R 2 are 0.98 and 0.97 for 408 410
the training
measured where and
yield testing
yi is
point, cases of
the experimental
which yield
value
indicates displacement,
of the
that respectively
theithPSO-LSSVM
sample, ŷi isis
the(Figures
predicted
proficient 7(b) and of
atvalue 8(b)).
capturing the ith 409
the 411
The high valuesand
sample, of nRis2 imply
the total that a strong
number correlation
of samples, and exists
Var() isbetween
the variance.the predicted and 410
underlying function of yield strength and yield displacement of RC columns. 412
measured yield point, which indicates that the PSO-LSSVM is proficient at capturing the 411
In 4.4.2.
addition, the mean
Evaluation of the absolute
PSO-LSSVM errorPrediction
(MAE) and root mean square error (RMSE) for
Results 413
underlying function of yield strength and yield displacement of RC columns. 412
evaluating the Beforepredictive
evaluating accuracy of
the proposed the PSO-LSSVM
model,and theroot model,
dataset are also
composed of presented
382 in Fig-
test results 414
In addition, the mean absolute error (MAE) mean square error (RMSE) for is413
ures 7 and 8. The divided
randomly PSO-LSSVM into two model
sets: attains pretty
thePSO-LSSVM
training setgood
(305 perdition
data samples) performances
and the in in both
testing 415
evaluating the predictive accuracy of the model, are also presented Fig- set414
yield
ures 7strength
(77 data
and 8. Theand yield With
samples).
PSO-LSSVM displacement
the optimal
model reflected
parameters
attains pretty in in
the low
Table
good 4values
adopted,
perdition of MAE
the and RMSE.
training
performances in bothNev-
performance 415
416
and
ertheless, thetesting
values results
of of theand
MAE PSO-LSSVM
RMSE in are
the displayed
testing in Figures
process of 5 and
the 6, respectively.
PSO-LSSVM areThelow 417
yield strength and yield displacement reflected in the low values of MAE and RMSE. Nev- 416
enough scatter-plots
already, in thedemonstrates
which two figures show thatthat
thethe
new proposed
model model
has a has
good yielded prediction
capability of results
predict- 418
ertheless, the values of MAE and RMSE in the testing process of the PSO-LSSVM are low 417
ing yieldthat presentand
strength quitedisplacement
excellent agreement with the experimental
accurately. test results inthe
both training
enough already, which demonstrates that the new To modelsumhas up,a it indicates
good
and testing processes. Moreover, it also can be seen that the PSO-LSSVM earned 2relatively
capabilitythat estimated
of predict- 418 419
yield strength
ing yield strength andanddisplacement
displacement are trustworthy
accurately. To andup,
sum reliable
it due tothat
indicates thethe
high R and low
estimated 420
high values of R for testing, the corresponding values of the coefficient of determination419
2
RMSE and
yield strength MAE.
(R2 ) are and Thus, the for
0.99displacement
and 0.96 proposed
are PSO-LSSVM
the trustworthy
training and and model
reliable
testing casesdue deems
of to thesuited
yield high Rfor
strength, predicting
2 and low 420
respectively 421
yield
RMSEstrength
and MAE. and displacement
Thus, the proposed of RC columns
PSO-LSSVM as a
model new 2 and
deems
(Figures 5a and 6a), and the corresponding values of R are 0.98 and 0.97 for the training effective
suited formethod.
predicting Mean- 421 422
while, the
yield strengthresultsand shown in
displacement Figuresof RC7 and 8
columns also indicate
as a new that
and the PSO
effective
and testing cases of yield displacement, respectively (Figures 5b and 6b). The high values422 algorithm
method. adopted
Mean- 423
while,
for the
searching 2 imply
of Rresults shown
optimal in Figures
that asolutions
strong to 7the
and
correlation 8 alsobetween
LSSVM
exists indicate
modelthe isthat
quitetheeffective
predicted PSO andalgorithm
and meets
measured adopted
yieldthe cal-
point,423 424
for searching
culation which
accuracy optimal
indicates solutions to
in the
that the PSO-LSSVM
requirement both LSSVM modeland istesting
is proficient
training quite
at effective
capturing
results.the and meets the
underlying cal- of424
function 425
culationyield strength
accuracy and yield displacement
requirement in both training of RCandcolumns.
testing results. 425
Table 3. Obtained optimal parameters of LSSVM by PSO. 426
Table 3. Table 4. Obtained
Obtained optimaloptimal parameters
parameters of LSSVM
of LSSVM by PSO.
by PSO. 426
σ γ
σ σ γγ
Fy 0.47 195.45
FFyy 0.47 0.47 195.45
195.45
∆yΔy 0.43 0.43 49.9349.93
Δy 0.43 49.93
427
427

(a)
(a) (b) (b)
Figure 7.Figure
Figure Training
Training performance
performance
5. Training and
andresult
performance and of
result PSO-LSSVM
of
resultPSO-LSSVM (number
of PSO-LSSVM of samples
(number
(number = 305).
ofofsamples
samples (a) Yield
== 305).
305): (a) Yield
(a) Yield428 428
strength; (b) Yield
strength;strength;
(b) Yield displacement.
(b)displacement.
Yield displacement. 429 429

(a) (b)
(a) (b)
Figure 6. Testing result of PSO-LSSVM (number of samples = 77): (a) Yield strength; (b) Yield
displacement.
Appl. Sci. 2022, 12, 10911 12 of 15

In addition, the mean absolute error (MAE) and root mean square error (RMSE) for
evaluating the predictive accuracy of the PSO-LSSVM model, are also presented in Figures 5
and 6. The PSO-LSSVM model attains good perdition performances in both yield strength
and yield displacement reflected in the low values of MAE and RMSE. Nevertheless, the
values of MAE and RMSE in the testing process of the PSO-LSSVM are low enough already,
which demonstrates that the new model has a good capability of predicting yield strength
and displacement accurately. To sum up, the results indicate that the estimated yield
strength and displacement are trustworthy and reliable due to the high R2 and low RMSE
and MAE. Thus, the proposed PSO-LSSVM model is deemed suited for predicting yield
strength and displacement of RC columns as a new and effective method. Meanwhile, the
results shown in Figures 5 and 6 also indicate that the PSO algorithm adopted for searching
optimal solutions to the LSSVM model is quite effective and meets the calculation accuracy
requirement in both training and testing results.

4.4.3. Comparative Evaluation of Model Prediction Performance

For further elaborating the effectiveness of the PSO algorithm in terms of parameter (σ, γ)
optimization, other metaheuristic algorithms such as GA [30], DE [31] and ABC [32] are
also applied to optimize the parameters of the LSSVM model. The identical training and
testing samples are employed to make a performance comparison among standard LSSVM
models without the optimized algorithm, and LSSVM optimized by GA, ABC, DE and
PSO algorithms, respectively. The configuration of the standard LSSVM model is the same
as that of the proposed model with the parameters of standard LSSVM and kernel set as
σ2 = 0.1, γ = 10 [39]. The parameter settings of the GA, ABC and DE algorithms are
identical to that used in the PSO, i.e., the size of the solution M = 30 and the maximum
iteration number kmax = 200. Table 5 is the evaluation results of the LSSVM, GA-LSSVM,
ABC-LSSVM, DE-LSSVM and PSO-LSSVM models. As shown in Table 5, it is obvious that
the prediction performance of the standard LSSVM is improved by adopting the optimized
algorithms due to the increase in R2 and EV, and the decrease in RMSE and MAE. The higher
R2 and EV, and lower RMSE and MAE imply the higher accuracy of models. Meanwhile,
comparing the evaluation indices (R2 , EV, RMSE and MAE) of testing data between the
LSSVM optimized by other metaheuristic algorithms and the proposed PSO-LSSVM model,
it can be found that the R2 of the PSO-LSSVM model in predicting yield strength (Fy ) and
yield displacement (∆y ) can be up to 0.96 and 0.97, respectively. The RMSE and MAE of
the PSO-LSSVM model are both sufficiently minimal. For the prediction of Fy , there is
little difference in the prediction accuracy between GA-LSSVM, ABC-LSSVM, DE-LSSVM
and PSO-LSSVM models. While for the prediction of ∆y , the prediction accuracy of the
PSO-LSSVM model is higher than other models. This demonstrates that the PSO-LSSVM
model performs better than GA-LSSVM, ABC-LSSVM and DE-LSSVM models in terms
of R2 , EV, RMSE and MAE, indicating the proposed PSO algorithm has an outstanding
advantage over the GA, ABC and DE algorithms in establishing the LSSVM model used
for effectively predicting the yield properties of RC columns.

Table 5. Evaluation results of standard LSSVM, GA-LSSVM, ABC-LSSVM, DE-LSSVM, PSO-LSSVM.

Training Testing
Model
RMSE MAE R2 EV RMSE MAE R2 EV
LSSVM 25.41 15.65 0.9859 98.5933 43.29 26.58 0.9248 92.6090
GA-LSSVM 20.61 13.41 0.9907 99.0746 31.92 23.93 0.9591 95.9857
Fy ABC-LSSVM 20.54 13.25 0.9908 99.0802 31.60 23.38 0.9599 96.0527
DE-LSSVM 19.90 12.94 0.9913 99.1371 31.42 23.17 0.9603 96.1449
PSO-LSSVM 20.53 13.22 0.9907 99.0821 31.45 23.25 0.9601 96.0898
LSSVM 2.067 1.282 0.9775 99.7536 7.870 2.895 0.7848 78.5773
GA-LSSVM 1.800 1.108 0.9830 98.2953 4.647 2.87 0.9249 92.5010
∆y ABC-LSSVM 1.876 1.168 0.9815 98.1484 4.943 2.674 0.9151 91.5142
DE-LSSVM 1.831 1.139 0.9823 98.2360 4.975 2.686 0.9140 91.4036
PSO-LSSVM 1.795 1.140 0.9832 98.3245 2.798 1.822 0.9728 97.2999
Appl. Sci. 2022, 12, 10911 13 of 15

5. Conclusions
A new ML-based method incorporating algorithms of the LSSVM and PSO is proposed
for predicting the yield strength and displacement of RC columns. The present PSO-LSSVM
model is capable of learning the nonlinear regression function that controls the mapping
between the influencing factors and the yield strength and displacement with the PSO
utilized for assisting to adaptively and quickly determine two optimal tuning parameters
(σ, γ). A test database of 382 pseudo-static cyclic test of RC columns is built, then the
effects of the column featured parameters on the yield-related properties are investigated
concerning the related research results. The proposed PSO-LSSVM model is then compared
with the standard LSSVM and the LSSVM optimized by other metaheuristic algorithms,
such as GA, ABC and DE algorithms, with the identical experimental database used for
model training and testing. The comparative results show that the PSO empowers the
LSSVM model to efficiently and accurately predict the target data with a self-optimized
machine learning algorithm based on the collected data samples. Based on the analysis
results about the prediction performance of the PSO-LSSVM for the yield strength and
displacement of RC columns, the following conclusions are drawn:
(1) The determination coefficient R2 is 0.96, if 80% of the whole dataset is used for training
the PSO-LSSVM model, which means a low prediction error. Meanwhile, the RMSE
and MAE are 31.45 and 23.25, respectively, which indicates that the prediction model
has a low prediction deviation.
(2) The PSO adopted for parameter optimization of σ and γ that are embedded in the
LSSVM model, can quickly find the optimal parameters to effectively assist the LSSVM
model in prediction work.
(3) The proposed PSO-LSSVM model can predict the yield strength and displacement of
RC columns with high efficiency and accuracy by comparing them with the LSSVM
models optimized by other metaheuristic algorithms.
Hence, the developed PSO-LSSVM model can be a useful tool to provide effective
guidance for the structural design of RC columns in practical engineering applications.
Essentially, the proposed model based on the ML technique is a data-driven approach that
focuses on correlations between research objects, different from the traditional research
method that is more concerned with cause-effect relationships. Moreover, ML methods
characterized by data-driven approaches are very powerful at solving regression and
optimization problems, as long as sufficient data can be provided. In our future work, the
model will be extended to study the deformation properties of other key component units
of building structures, such as RC beams and shear walls, and, the data collection and
processing of the relevant experimental database will be conducted.

Author Contributions: Conceptualization, B.W., W.G. and Z.L.; methodology, B.W. and W.G.; validation,
B.W., W.G. and Y.W.; formal analysis, Y.W. and Z.L.; investigation, B.W. and H.L.; writing—review and
editing, B.W and W.G. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by National Natural Science Foundation of China (51808112,
51878160, 52078128), Natural Science Foundation of Jiangsu Province (BK20180155) and Postgraduate
Research & Practice Innovation Program of Jiangsu Province (SJCX21_1794).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
Appl. Sci. 2022, 12, 10911 14 of 15

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