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Soft computing based formulation for strength enhancement of CFRP confined
concrete cylinders
Abdulkadir Cevik a,*, M. Tolga Gögüs a, _Ibrahim H. Güzelbey b, Hüzeyin Filiz b
a Department of Civil Engineering, University of Gaziantep, Turkeyb Department of Mechanical Engineering, University of Gaziantep, Turkey
a r t i c l e i n f o
Article history:
Received 26 May 2009
Received in revised form 18 September
2009
Accepted 15 October 2009
Available online xxxx
Keywords:
Soft computing
Stepwise regression
Genetic programming
FRP confinement
Concrete cylinder
Strength enhancement
a b s t r a c t
This study presents the application of soft computing techniques namely as genetic programming (GP)
and stepwise regression (SR) for formulation of strength enhancement of carbon-fiber-reinforced poly-
mer (CFRP) confined concrete cylinders. The proposed soft computing based formulations are based on
experimental results collected from literature. The accuracy of the proposed GP and SR formulations
are quite satisfactory as compared to experimental results. Moreover, the results of proposed soft com-
puting based formulations are compared with 15 existing models proposed by various researchers so far
and are found to be more accurate.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
With over fifty years of excellent performance records in the
aerospace industry, fiber-reinforced-polymer (FRP) composites
have been introduced with confidence to the construction indus-
try. These high-performance materials have been accepted by civil
engineers and have been utilized in different construction applica-
tions such as repair and rehabilitation of existing structures as well
as in new construction applications. One of the successful and most
popular structural applications of FRP composites is the external
strengthening, repair and ductility enhancement of reinforced con-
crete (RC) columns in both seismic and corrosive environments [1].
Main types of FRP composites used in external strengthening and
repair of RC columns are: Glass-fiber-reinforced polymers (GFRP),carbon-fiber-reinforced polymers (CFRP), and aramid-fiber-rein-
forced polymers (AFRP). Types of FRP confinement can be spiral,
wrapped and tube. FRP composites offer several advantages due
to extremely high strength-to-weight ratio, good corrosion behav-
iour, and electromagnetic neutrality. Thus the effect of FRP con-
finement on the strength and deformation capacity of concrete
columns has been extensively studied and several empirical and
theoretical models have been proposed [2]. This study proposes a
new approach for the formulation of strength enhancement of
CFRP wrapped concrete cylinders using Stepwise regression and
genetic programming approach which have not been applied so
far in this field.
2. Behaviour of FRP-confined concrete
Being a frictional material, concrete is sensitive to hydrostatic
pressure. The beneficial effect of lateral stresses on the concrete
strength and deformation has been recognized nearly for a century.
In other words, when uniaxially loaded concrete is restrained from
dilating laterally, it exhibits increased strength and axial deforma-
tion capacity indicated as confinement which has been generally
applied to compression members through steel transverse rein-
forcement in the form of spirals, circular hoops or rectangular ties,
or by encasing the concrete columns into steel tubes that act as
permanent formwork [2]. Besides steel reinforcement FRPs are also
for confinement of concrete columns and offers several advantages
as compared to steel [3] such as continuous confining action to the
entire cross-section, easiness and speed of application, no change
in the shape and size of the strengthened elements, corrosive resis-
tance [2].
Typical response of FRP-confined concrete is shown in Fig. 1,
where normalized axial stress is plotted against axial, lateral, and
volumetric strains. The stress is normalized with respect to the
unconfined strength of concrete core. The figure shows that both
axial and lateral responses are bi-linear with a transition zone at
0965-9978/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.advengsoft.2009.10.015
* Corresponding author. Tel.: +90 342 3172409; fax: +90 342 3601107.
E-mail address: [email protected] (A. Cevik).
Advances in Engineering Software xxx (2009) xxx–xxx
Contents lists available at ScienceDirect
Advances in Engineering Software
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a d v e n g s o f t
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or near the peak strength of unconfined concrete core. The volu-
metric response shows a similar transition toward volume expan-
sion. However, as soon as the jacket takes over, volumetric
response undergoes another transition which reverses the dilation
trend and results in volume compaction. This behaviour is shown
to be markedly different from plain concrete and steel-confined
concrete [4].
The characteristic response of confined concrete includes three
distinct regions of un-cracked elastic deformations, crack forma-
tion and propagation, and plastic deformations. It is generally as-
sumed that concrete behaves like an elastic-perfectly plastic
material after reaching its maximum capacity, and that the failure
surface is fixed in the stress space. Constitutive models for concrete
should be concerned with pressure sensitivity, path dependence,
stiffness degradation and cyclic response. The existing plasticity
models range from nonlinear elasticity, endo-chronic plasticity,
classical plasticity, and multi-laminate or micro-plane plasticity
to bounding surface plasticity. Many of these models, however,
are only suitable in a specific application and loading system for
which they are devised and may give unrealistic results in other
cases. Also, some of these models require several parameters to
be calibrated based on experimental results [4]. Considerable
experimental research has been performed on the behaviour of
CFRP confined concrete columns [5–11]. Several models are pro-
posed in literature for the strength enhancement of FRP confine-ment effect of concrete columns given in Table 1. Apart from
models given in Table 1, there are also studies on design-oriented
stress–strain model for FRP-confined concrete [24,25]. On the
other hand, Rousakis and Karabinis recently proposed an effective
model for FRP confining effects of substandard reinforced concrete
members subjected to compression [26]. One of the most compre-
hensive studies on empirical modelling for predicting the mechan-
ical properties of FRP-confined concrete was performed by
Vintzileou and Panagiotidou where a database of 1074 t results
were used to assess existing models that predict the strength of
confined concrete [27]. Apart from regression models, Neural Net-
works are also used effectively to predict the strength of FRP-con-
fined concrete [28].
3. Soft computing
The definition of soft computing is not precise. Lotfi A. Zadeh,
the inventor of the term soft computing, describes it as follows
[29]:
‘‘Soft computing is a collection of methodologies that aim to
exploit the tolerance for imprecision and uncertainty to achieve
tractability, robustness, and low solution cost. Its principal con-
stituents are fuzzy logic, neurocomputing, and probabilistic rea-
soning. Soft computing is likely to play an increasingly
important role in many application areas, including software
engineering. The role model for soft computing is the human
mind.”
Soft computing can be seen as an attempt of collection of tech-
niques that mimic natural creatures: plants, animals, human
beings, which are soft, flexible, adaptive and clever. It can be de-
scribed as a family of problem-solving methods that have analogy
with biological reasoning and problem solving. It includes basicmethods such as fuzzy logic (FL), neural networks (NN), genetic
algorithms (GA) and genetic programming – the methods which
do not derive from classical theories. Soft computing can also be
seen as a foundation for the growing field of computational intel-
ligence (CI) as an alternative to traditional artificial intelligence
(AI) which is based on hard computing [30].
In many ways, soft computing represents a significant paradigm
shift in the aims of computing – a shift which reflects the fact that
the human mind, unlike present day computers, possesses a
remarkable ability to store and process information which is per-
vasively imprecise, uncertain and lacking in categorisation [31].
Two soft computing approaches based on stepwise regression
and genetic programming is the scope of this study which will be
described in the following sections.
3.1. Brief overview of stepwise regression
While dealing with large number of independent variables, it is
of significance to determine the best combination of these vari-
ables to predict the dependent variable. Stepwise regression serves
as a robust tool for the selection of best subset models, i.e. the best
combination of independent variables that best fits the dependent
variable with considerably less computing than is required for all
possible regressions [32].
The determination of subset models are based on consecutively
by adding or deleting, the variable/variables that has the greatest
impact on the residual sum of squares. The selection of variables
may be either forward, backward or a combination of them. In for-ward selection, the subset models are chosen by adding one
Nomenclature
f 0co compressive strength of the unconfined concrete cylin-der
f 0cc compressive strength of the confined concrete cylinder pu ultimate confinement pressureE l confinement modulus or lateral modulus
E f modulus of elasticity of the FRP laminatent total thickness of FRP layerD diameter of the concrete cylinderL length of the concrete cylinder f fu tensile strength of the FRP laminate
Fig. 1. Typical response of FRP-confined concrete [4].
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variable at a time to the previously chosen subset. At each succes-
sive step, the variable in the subset of variables not already in the
model that causes the largest decrease in the residual sum of
squares is added to the subset. Without a termination rule, forward
selection continues until all variables are in the model. On the
other hand, backward stepwise selection of variables chooses thesubset models by starting with the full model and then eliminating
at each step the one variable whose deletion will cause the residual
sum of squares to increase the least and continues until the subset
model contains only one variable [33].
Regarding forward and backward procedures, it should be noted
that the effect of adding or deleting a variable on the contributions
of other variables to the model is not being considered. Thus step-
wise regression is actually a forward selection process that re-
checks at each step the importance of all previously included
variables. If the partial sums of squares for any previously included
variables do not meet a minimum criterion to stay in the model,
the selection procedure changes to backward elimination and vari-
ables are dropped one at a time until all remaining variables meet
the minimum criterion. Stepwise selection of variables requires
more computing than forward or backward selection but has an
advantage in terms of the number of potential subset models
checked before the model for each subset size is decided. It is rea-
sonable to expect stepwise selection to have a greater chance of
choosing the best subsets in the sample data, but selection of the
best subset for each subset size is not guaranteed. The stopping
rule for stepwise selection of variables uses both the forward and
backward elimination criteria. The variable selection process ter-
minates when all variables in the model meet the criterion to stay
and no variables outside the model meet the criterionto enter [33].
3.2. Overview of genetic programming
Genetic programming (GP) proposed by Koza [34] is an exten-
sion to genetic algorithms (GA). Koza defines GP as a domain-inde-
pendent problem-solving approach in which computer programs
are evolved to solve, or approximately solve, problems based on
the Darwinian principle of reproduction and survival of the fittest
and analogs of naturally occurring genetic operations such as
crossover (sexual recombination) and mutation.
When the genetic algorithm is implemented it is usually done
in a manner that involves the following cycle: Evaluate the fitness
of all of the individuals in the population. Create a new population
by performing operations such as crossover, fitness-proportionatereproduction and mutation on the individuals whose fitness has
just been measured. Discard the old population and iterate using
the new population. GP reproduces computer programs to solve
problems by executing the following steps:
(1) Generate an initial population of random compositions of
the functions and terminals of the problem (computer
programs).
(2) Execute each program in the population and assign it a fit-
ness value according to how well it solves the problem.
(3) Create a new population of computer programs.
(i) Copy the best existing programs (reproduction).
(ii) Create new computer programs by mutation.
(iii) Create new computer programs by crossover (sexualreproduction).
(iv) Select an architecture-altering operation from the pro-
grams stored so far.(4) The best computer program that appeared in any generation,
the best-so-far solution, is designated as the result of genetic
programming [34].
Gene expression programming (GEP) software which is used in
this study is an extension to GP that evolves computer programs of
different sizes and shapes encoded in linear chromosomes of fixed
length. The chromosomes are composed of multiple genes, each
gene encoding a smaller sub-program. Furthermore, the structural
and functional organization of the linear chromosomes allows the
unconstrained operation of important genetic operators such asmutation, transposition, and recombination. One strength side of
Table 1
Models for strength enhancement of FRP-confined concrete cylinders.
Model Expression ( f 0cc = f 0co)
Fardis and Khalili [12] f 0cc f 0co
¼ 1 þ 4:1 pu
f 0coð1Þ
f 0cc f 0co
¼ 1 þ 3:7 pu
f 0co
0:86
ð2Þ
Saadatmanesh et al. [13] f 0cc f 0co
¼ 2:254
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 7:94
pu
f 0co
r 2
pu
f 0co 1:254 ð3Þ
Miyauchi et al. [5] f 0cc
f 0co¼ 1 þ 3:485
pu
f 0coð4Þ
Kono et al. [6] f 0cc
f 0co¼ 1 þ 0:0572 pu ð5Þ
Saaman et. al. [14] f 0cc
f 0co¼ 1 þ 6:0
p0:7u
f 0coð6Þ
Tountanji [15] f 0cc f 0co
¼ 1 þ 3:5 pu
f 0co
0:85
ð7Þ
Saafi et al. [16] f 0cc f 0co
¼ 1 þ 2:2 pu
f 0co
0:84
ð8Þ
Spoelstra and Monti [17] f 0cc f 0co
¼ 0:2 þ 3 pu
f 0co
0:5
ð9Þ
Xiao and Wu [18] f 0cc
f 0co¼ 1:1 þ 4:1 0:75
f 0co2
E 1
pu
f 0coð10Þ
Karabinis and Rousakis [19] f 0cc f 0co
¼ 1 þ 2:1 pu
f 0co
0:87
ð11Þ
Lam and Teng [20] f 0cc
f 0co¼ 1 þ 2:0
pu
f 0co
ð12Þ
Shehata et al. [21] f 0cc
f 0co¼ 1 þ 1:25
pu
f 0co
ð13Þ
Matthys et al. [22] f 0cc f 0co
¼ 1 þ 2:3 pu
f 0co
0:85
ð14Þ
Kumutha et al. [23] f 0cc f 0co
¼ 1 þ 0:93 pu
f 0co
ð15Þ
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the GEP approach is that the creation of genetic diversity is extre-
mely simplified as genetic operators work at the chromosome le-
vel. Another strength side of GEP consists of its unique,
multigenic nature which allows the evolution of more complex
programs composed of several sub-programs. As a result GEP sur-
passes the old GP system in 100–10,000 times [35–37]. APS 3.0
[38], a GEP software developed by Candida Ferreira is used in this
study.
The fundamental difference between GA, GP and GEP is due to
the nature of the individuals: in GAs the individuals are linear
strings of fixed length (chromosomes); in GP the individuals are
nonlinear entities of different sizes and shapes (parse trees); and
in GEP the individuals are encoded as linear strings of fixed length
(the genome or chromosomes) which are afterwards expressed as
nonlinear entities of different sizes and shapes (i.e., simple diagram
representations or expression trees). Thus the two main parame-
ters GEP are the chromosomes and expression trees (ETs). The pro-
cess of information decoding (from the chromosomes to the ETs) is
called translation which is based on a set of rules. The genetic code
is very simple where there exist one-to-one relationships between
the symbols of the chromosome and the functions or terminals
they represent. The rules which are also very simple determine
the spatial organization of the functions and terminals in the ETs
and the type of interaction between sub-ETs [25–27]. That’s why
two languages are utilized in GEP: the language of the genes and
the language of ETs. A significant advantage of GEP is that it en-
ables to infer exactly the phenotype given the sequence of a gene,
and vice versa which is termed as Karva language. For each prob-
lem, the type of linking function, as well as the number of genes
and the length of each gene, are a priori chosen for each problem.
While attempting to solve a problem, one can always start by using
a single-gene chromosome and then proceed by increasing the
length of the head. If it becomes very large, one can increase the
number of genes and obviously choose a function to link the sub-
ETs. One can start with addition for algebraic expressions or OR
for Boolean expressions, but in some cases another linking function
might be more appropriate (like multiplication or IF, for instance).The idea, of course, is to find a good solution, and GEP provides the
means of finding one very efficiently [36].
As an illustrative example consider the following case where
the objective is to show how GEP can be used to model complex
realities with high accuracy. So, suppose one is given a sampling
of the numerical values from the curve (remember, however, that
in real-world problems the function is obviously unknown):
y ¼ 3a2 þ 2a þ 1 ð16Þ
over 10 randomly chosen points in the real interval [10, +10] andthe aim is to find a function fitting those values within a certain er-
ror. In this case, a sample of data in the form of 10 pairs ( ai, y i) is
given where ai is the value of the independent variable in the given
interval and yi is the respective value of the dependent variable (ai
values: 4.2605, 2.0437, 9.8317, 8.6491, 0.7328, 3.6101,
2.7429, 1.8999, 4.8852, 7.3998; the corresponding yi values
can be easily evaluated). These 10 pairs are the fitness cases (the in-
put) that will be used as the adaptation environment. The fitness of
a particular program will depend on how well it performs in this
environment [36].
There are five major steps in preparing to use gene expression
programming. The first is to choose the fitness function. For this
problem one could measure the fitness f i of an individual program
i by the following expression:
f i ¼XC t j¼1
M jC ði; jÞ T jj ð17Þ
where M is the range of selection, C (i, j) the value returned by the
individual chromosome i for fitness case j (out of C t fitness cases)
and T j is the target value for fitness case j. If, for all j, |C (i, j) T j|
(the precision) less than or equal to 0.01, then the precision is equal
to zero, and f i = f max = C t M . For this problem, use an M = 100 and,
therefore, f max = 1000. The advantage of this kind of fitness function
is that the system can find the optimal solution for itself. However,
there are other fitness functions available which can be appropriate
for different problem types [36].
The second step is choosing the set of terminals T and the set
of functions F to create the chromosomes. In this problem, the
terminal set consists obviously of the independent variable, i.e.,T = {a}. The choice of the appropriate function set is not so obvious,
but a good guess can always be done in order to include all the
Fig. 2. ET for the problem of Eq. (13).
Table 2
Experimental database and ranges of variables.
Ref. Number of specimen D (mm) nt (mm) E f (MPa) f 0co(MPa)
Miyauchi et al. [5] 10 100, 150 0.11–0.33 3481 31.2–51.9
Kono et al. [6] 17 100 0.167–0.501 3820 32.3–34.8
Matthys et al. [7] 2 150 0.117, 0.235 2600, 1100 34.9
Shahawy et al. [8] 9 153 0.36–1.25 2275 19.4–49
Rochette and Labossiere [9] 7 100, 150 0.6–5.04 230, 1265 42–43
Micelli et al. [10] 8 100 0.16, 0.35 1520, 3790 32–37
Rousakis [11] 48 150 0.169–0.845 2024 25.15–82.13
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necessary functions. In this case, to make things simple, use the
four basic arithmetic operators. Thus, F = {+, , , /}. It should be
noted that there many other functions that can be used.
The third step is to choose the chromosomal architecture, i.e.,
the length of the head and the number of genes.
The fourth major step in preparing to use gene expression pro-
gramming is to choose the linking function. In this case we will link
the sub-ETs by addition. Other linking functions are also available
such as subtraction, multiplication and division.
And finally, the fifth step is to choose the set of genetic opera-
tors that cause variation and their rates. In this case one can use
a combination of all genetic operators (mutation at pm
= 0.051; IS
and RIS transposition at rates of 0.1 and three transposons of
length 1, 2, and 3; one-point and two-point recombination at rates
of 0.3; gene transposition and gene recombination both at rates of
0.1). To solve this problem, Lets choose an evolutionary time of 50
generations and a small population of 20 individuals in order to
simplify the analysis of the evolutionary process and not fill this
text with pages of encoded individuals. However, one of the advan-
tages of GEP is that it is capable of solving relatively complex prob-
lems using small population sizes and, thanks to the compact
Karva notation; it is possible to fully analyze the evolutionary his-
tory of a run. A perfect solution can be found in generation 3 which
has the maximum value 1000 of fitness. The sub-ETs codified by
each gene are given in Fig. 2. Note that it corresponds exactly to
the same test function given above in Eq. (16) [36].
Thus expressions for each corresponding Sub-ET can be given as
follows:
y ¼ ða2 þ aÞ þ ða þ 1Þ þ ð2a2Þ ¼ 3a2 þ 2a þ 1 ð18Þ
4. Numerical application
One of the main issues in modelling experimental data is the
determination of variables that will be used in the modelling. In
this study, prior to the modelling phase the correlation of each var-
iable on output which is the confined strength has been deter-
mined. As a result of these analyses, diameter of the concrete
cylinder (D), total thickness of FRP layer (nt ), tensile strength of
the FRP laminate ( f fu) and compressive strength of the unconfinedconcrete cylinder ( f co) was used in the modelling. Strain at failure
of FRP was excluded in the variables as tensile strength of the
FRP laminate ( f fu) was used instead. The use of strain at failure of
FRP together with the modulus of elasticity of the FRP laminate
Table 3
Parameters of the GEP models.
P1 Function set +, -, *, /, ffi p
, e x, ln( x), power
P2 Chromosomes 30–60
P3 Head size 6, 8, 10
P4 Number of genes 3
P5 Linking function Addition, multiplication
P6 Fitness function error type MAE (mean absolute error), custom fitness function
P7 Mutation rate 0.044P8 Inversion rate 0.1
P9 One-point recombination rate 0.3
P10 Two-point recombination rate 0.3
P11 Gene recombination rate 0.1
P12 Gene transposition rate 0.1
Table 4
Statistical parameters of testing and training sets.
Testing set (SR) Training set (SR) Testing set (GP) Training set (GP)
Mean 1.01 0.99 1.00 0.96
Std. Dev. 0.12 0.13 0.09 0.14
R 0.95 0.93 0.94 0.92
Fig. 3. Expression tree (ET) of the proposed GP formulation.
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was also evaluated before modelling phase and the effect of tensile
strength of the FRP laminate ( f fu) was observed to be more
significant.
4.1. Numerical application of GP
The main aimin this study is to obtain an empirical formulation
using stepwise regression and genetic programming for enhanced
strength of CFRP confined concrete cylinders based on test results
available in literature as a function the following parameters:
f 0cc ¼ f ðD;nt ; f fu; f coÞ ð19ÞTherefore, an extensive literature review on experimental stud-
ies related to strength enhancement of CFRP wrapped concrete cyl-
inders has been carried out and an experimental database has been
gathered. It should be noted that all specimen used in the database
have a length to diameter ratio of 2 (L/D = 2). A total of 101 speci-
mens from 7 separate studies with the ranges of variables were in-
cluded in the database shown in Table 2. Further details of the
experimental database are given in Table A.1.
To achieve generalization capability for the formulations, the
experimental database is divided into two sets as training and
test sets. The formulations are based on training sets and are fur-
ther tested by test set values to measure their generalization
capability. The patterns used in test and training sets are ran-domly selected. For example, regarding the ETF formulation,
among 101 tests 18 tests were used as test set given in bold
and the remaining as training set (Table A.1). Related parameters
for the training of the GP models are given in Table 3. Detailed
information on values given in Table 3 can be found in Section
3.2. Statistical parameters of test and training sets of GP formula-
tions are presented in Table 4 where R corresponds to the coeffi-
cient of correlation and Std. Dev. refers to standard deviation of
the mean of test/predicted values.
The results of the proposed GP formulations vs. actual experi-
mental values are given in Tables 8. The expression tree of the for-
mulation obtained from APS 3.0 is shown in Fig. 3 whichcorresponds to the following equation:
f cc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi f fu nt
q e
ffiffiffiffiffiffiffi1= f fu
p þ tanð1000 þ 1=nt Þ
þ tan 455= ffiffiffiffiffi f fu
q þ f co tan f co þ
ffiffiffiffiffiffiffiffiffi1=D
p ð20Þ
4.2. Numerical application of SR
Possible forms for all combinations of independent variables
used for the stepwise selection process are given as follows:
X i; 1= X i; X 2; ln
ð X
Þ; 1=ln
ð X
Þwhere X i stands for the independent variables given in Eq. (19).
Table 5
Models considered in SR process (inputs vs. equations).
Model Inputs Equation
Linear x1, x2 y = b0 + b1 x1 + b2 x2
Linear + interaction x1, x2, x1 x2 y = b0 + b1 x1 + b2 x2 + b3 x1 x2
Full quadratic x1, x2, x1 x1, x2 x2, x1 x2 y = b0 + b1 x1 + b2 x2 + b3 x1 x1 + b4 x1 x2 + b5 x2 x2
Squared + interaction x1, x2, x1 x1, x2 x2, x1 x2, x1 x1 x2, x1 x2 x2
y = b0 + b1 x1 + b2 x2 + b3 x1 x1 + b4 x1 x2 + b5 x2 x2 + b6 x1 x1 x2 + b7 x1 x2 x2
Table 6
Statistical details and equations of best subsets for each stepwise regression model.
Model Equation of best subset Constants R COV
Linear f cc = b0 + b1 ln E f + b2 ln 1/ f co + b3 ln 1/nt + b4 nt + b5 1/D + b6 D + b7 1/nt + b8 E f + b9 1/E f b0 = 6766.2 0.95 0.12
b1 = 174.93
b2 = 39.70
b3 = 53.46
b4 = 14.49
b5 = 964,690
b6 = 16.08
b7 = 5.402
b8 = 0.05178
b9 = 54536.1
Linear + interaction f cc = b0 + b1 f co ln E f + b2 1/nt 1/E f + b3 1/D 1/ f co b0 = 91.78 0.893 0.14
b1 = 0.08324
b2 = 15308.6
b3 = 182,923
Full quadratic f cc = b0 + b1 f co ln E f + b2 1/nt 1/E f + b3 1/D 1/ f co + b4 f co f co + b5 E f ln nt b0 = 56.17 0.905 0.13
b1 = 0.231
b2 = 12038.5
b3 = 100,483
b4 = 0.00984
b5 = 0.00201
Squared + interaction f cc = b0 + b1 D f co ln E f + b2 f co 1/nt 1/E f + b3 nt 1/nt 1/ f co + b4 1/ f co ln nt ln 1/ f co + b5 1/
D 1/E f ln E f + b6 f co f co 1/E f + b7 1/nt 1/E f 1/ f co
b0 = 249.07 0.931 0.12
b1 = 0.000592
b2 = 101.31
b3 = 2279.6
b4 = 339.11
b5 = 3842947.468b6 = 21.68
b7 = 358,381
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Table 7
Statistics of performance and accuracy of ( f 0cc / f 0
co) of proposed GP, SR formulations and existing models compared to experimental results.
Model Test/
SR
Test/
GP
Test/Eq.
(1)
Test/Eq.
(2)
Test/Eq.
(3)
Test/Eq.
(4)
Test/Eq.
(5)
Test/Eq.
(6)
Test/Eq.
(7)
Test/Eq.
(8)
Test/Eq.
(9)
Test/Eq.
(10)
Test/Eq.
(11)
Test/
(12)
Mean 1.00 0.99 1.23 0.78 0.82 0.88 1.06 0.97 0.80 0.99 1.03 1.30 1.02 1.10
Std.
Dev.
0.12 0.10 0.19 0.15 0.12 0.18 0.20 0.15 0.15 0.15 0.18 0.90 0.16 0.18
COV 0.12 0.10 0.15 0.19 0.15 0.20 0.19 0.15 0.18 0.15 0.17 0.69 0.15 0.16
R 0.95 0.94 0.87 0.87 0.85 0.86 0.77 0.87 0.87 0.87 0.87 0.87 0.87 0.87
P l e a s e c i t e t h i s a r t i c l e i npr e s s a s : C e vi k A e t a l .S of t c omput i ngb a s e d f or m
ul a t i onf or s t r e ngt h e nh a nc e me nt of C F R P c
onfi ne d c onc r e t e c yl i nd e r s .A d vE ng
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Models considered for the stepwise regression process are given
in Table 5 for 2 independent variables ( x1, x2) and 1 dependent
variable ( y) with possible corresponding equations. All possible
combinations of independent variables with models considered
and corresponding equation of best subset are given in Table 6.
The stepwise regression analysis in this study is performed by SPSS
and the following SR equation has been obtained for the best sub-
set (R
= 0.95):
f cc ¼ 6766 þ 174:9 lnð f fuÞ 39:7 lnð1= f coÞ 53:5
lnð1=nt Þ 14:5 nt 964; 690 1=D 16 D þ 5:4
1=nt 0:052 f fu þ 54; 536 1= f fu ð21ÞStatistical parameters of test and training sets of GP formula-
tions are presented in Table 4. The results of the proposed SR for-
mulation vs. actual experimental values are given in Table A.1.
Statistical parameters of proposed GP and SR formulations com-
pared with existing models are presented in Table 7. It should be
noted that the proposed GP and SR formulations presented above
are valid only for the ranges of experimental database given in Ta-
ble 2 and for specimen that have a length to diameter ratio of 2 (L/
D = 2).
5. Conclusion
This study proposes application of soft computing techniques
namely as stepwise regression and genetic programming for the
formulation of strength enhancement of CFRP confined concrete
cylinders which have not been used so far. The proposed SR and
GP formulations are actually empirical formulations based on a
wide range of experimental database collected from literature.
Both formulations are quite accurate show good agreement with
experimental results. For comparative analysis, Numerical results
of the same experimental database are obtained by existing models
and the proposed SR and GP formulations and soft computing
based formulations are found to be more accurate. It should be
noted that empirical formulations in structural engineering are
mostly based on predefined functions where regression analysis
of these functions are later performed. However, in the case of SR
and GP approach there is no predefined function to be considered,
i.e. SR and GP adds or deletes various combinations of parameters
to be considered for the formulation that best fits the experimental
results based on highest correlation coefficient. However, it should
be kept in mind that SR and GP models presented in this study are
constructed from the experimental database used in this study
which means they are valid for ranges of variables of the database.
Predictionfor tests that are not present in the database may lead to
inconsistent results. Therefore, these models should be updated
with extra test results. If a larger database is used, the models pre-
sented in this study may change considerably. This can be treated
as a disadvantage which is actually true for many regression mod-
els. It is obvious that soft computing based formulations will serve
as a robust approach for the accurate and effective explicit formu-
lation of many structural engineering problems in the future.
Appendix A
Table A.1.
Table A.1
Results of the SR and GP formulations vs. experimental and theoretical results.
Ref. Code D (mm) nt (mm) f fu (MPa) f co (MPa) f cc test (MPa) f cc SR (MPa) f cc GP (MPa) Test/SR Test/GP
Miyauchi et al. [5] MI1 150 0.11 3481 45.2 59.4 68.53 67.53 0.87 0.88
MI2 150 0.22 3481 45.2 79.4 79.44 78.16 1.00 1.02MI3 150 0.11 3481 31.2 52.4 53.81 56.98 0.97 0.92
MI4 150 0.22 3481 31.2 67.4 64.72 67.60 1.04 1.00
MI5 150 0.33 3481 31.2 81.7 76.62 71.94 1.06 1.14
MI6 100 0.11 3481 51.9 75.2 74.47 84.92 1.01 0.88
MI7 100 0.22 3481 51.9 104.6 85.38 95.54 1.22 1.10
MI8 100 0.11 3481 33.7 69.6 57.32 60.32 1.22 1.15
MI9 100 0.22 3481 33.7 88 68.23 70.95 1.28 1.23
MI10 150 0.11 3481 45.2 59.4 68.53 67.53 0.87 0.88
Kono et al. [6] KO1 100 0.167 3820 34.3 57.4 60.07 61.02 0.95 0.94
KO2 100 0.167 3820 34.3 64.9 60.07 61.02 1.08 1.06
KO3 100 0.167 3820 32.3 58.2 57.69 57.42 1.01 1.01
KO4 100 0.167 3820 32.3 61.8 57.69 57.42 1.08 1.08
KO5 100 0.167 3820 32.3 57.7 57.69 57.42 1.00 1.00
KO6 100 0.334 3820 32.3 61.8 76.15 68.30 0.81 0.85
KO7 100 0.334 3820 32.3 80.2 76.15 68.30 1.05 0.90
KO8 100 0.334 3820 32.3 58.2 76.15 68.30 0.76 1.18
KO9 100 0.501 3820 32.3 86.9 90.02 94.15 0.96 0.93KO10 100 0.501 3820 32.3 90.1 90.02 94.15 1.00 0.96
KO11 100 0.167 3820 34.8 57.8 60.65 61.00 0.95 0.94
KO12 100 0.167 3820 34.8 55.6 60.65 61.00 0.92 0.91
KO13 100 0.167 3820 34.8 50.7 60.65 61.00 0.83 0.83
KO14 100 0.334 3820 34.8 82.7 79.11 71.89 1.04 1.15
KO15 100 0.334 3820 34.8 81.4 79.11 71.89 1.03 1.14
KO16 100 0.501 3820 34.8 103.3 92.98 97.74 1.11 1.05
KO17 100 0.501 3820 34.8 110.1 92.98 97.74 1.19 1.12
Matthys et al. [7] MA1 150 0.117 2600 34.9 46.1 58.4 46.99 0.79 0.98
MA2 150 0.235 1100 34.9 45.8 26.59 53.98 1.72 0.85
Shahawy et al. [8] SH1 153 0.36 2275 19.4 33.8 29.26 46.91 1.15 0.72
SH2 153 0.66 2275 19.4 46.4 50.5 60.05 0.92 0.78
SH3 153 0.9 2275 19.4 62.6 61.42 65.75 1.02 0.95
SH4 153 1.08 2275 19.4 75.7 67.56 69.91 1.12 1.09
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Table A.1 (continued)
Ref. Code D (mm) nt (mm) f fu (MPa) f co (MPa) f cc test (MPa) f cc SR (MPa) f cc GP (MPa) Test/SR Test/GP
Shahawy et al. [8] SH5 153 1.25 2275 19.4 80.2 72.23 73.60 1.11 1.09
SH6 153 0.36 2275 49 59.1 66.04 79.84 0.89 0.74
SH7 153 0.66 2275 49 76.5 87.28 92.99 0.88 0.82
SH8 153 0.9 2275 49 98.8 98.21 98.68 1.01 1.00
SH9 153 1.08 2275 49 112.7 104.35 102.84 1.08 1.10
Rochette and Labossiere [9] RL1 100 0.6 1265 42 73.5 74.67 70.18 0.98 1.05
RL2 100 0.6 1265 42 73.5 74.67 70.18 0.98 1.05
RL3 100 0.6 1265 42 67.62 74.67 70.18 0.91 0.96
RL4 150 1.26 230 43 47.3 49.93 55.75 0.94 0.85
RL5 150 2.52 230 43 58.91 66.59 62.88 0.88 0.93
RL6 150 3.78 230 43 70.95 69.3 68.52 1.02 1.03
RL7 150 5.04 230 43 74.39 66.07 73.32 1.12 1.01
Micelli et al. [10] MC1 100 0.35 1520 32 54 56.8 52.66 0.95 1.02
MC2 100 0.35 1520 32 48 56.8 52.66 0.85 0.91
MC3 100 0.35 1520 32 54 56.8 52.66 0.95 1.02
MC4 100 0.35 1520 32 50 56.8 52.66 0.88 0.95
MC5 100 0.16 3790 37 60 62.6 64.17 0.96 0.93
MC6 100 0.16 3790 37 62 62.6 64.17 0.99 0.97
MC7 100 0.16 3790 37 59 62.6 64.17 0.94 0.92
MC8 100 0.16 3790 37 57 62.6 64.17 0.91 0.88
Rousakis [11] RO1 150 0.169 2024 25.15 44.13 42.08 43.66 1.05 1.01
RO2 150 0.169 2024 25.15 41.56 42.08 43.66 0.99 0.95
RO3 150 0.169 2024 25.15 38.75 42.08 43.66 0.92 0.88RO4 150 0.338 2024 25.15 60.09 60.7 51.84 0.99 1.16
RO5 150 0.338 2024 25.15 55.93 60.7 51.84 0.92 1.08
RO6 150 0.338 2024 25.15 61.61 60.7 51.84 1.01 1.19
RO7 150 0.507 2024 25.15 67 74.61 70.73 0.90 0.94
RO8 150 0.507 2024 25.15 67.27 74.61 70.73 0.90 0.95
RO9 150 0.507 2024 25.15 70.18 74.61 70.73 0.94 0.99
RO10 150 0.169 2024 47.44 72.26 67.27 65.63 1.08 1.10
RO11 150 0.169 2024 47.44 64.4 67.27 65.63 0.96 0.98
RO12 150 0.169 2024 47.44 66.19 67.27 65.63 0.98 1.01
RO13 150 0.338 2024 47.44 82.36 85.9 73.82 0.96 1.11
RO14 150 0.338 2024 47.44 82.35 85.9 73.82 0.96 1.11
RO15 150 0.338 2024 47.44 79.11 85.9 73.82 0.92 1.08
RO16 150 0.507 2024 47.44 96.29 99.8 92.70 0.96 1.04
RO17 150 0.507 2024 47.44 95.22 99.8 92.70 0.95 1.03
RO18 150 0.507 2024 47.44 103.9 99.8 92.70 1.04 1.12
RO19 150 0.169 2024 51.84 78.65 70.79 86.68 1.11 0.91
RO20 150 0.169 2024 51.84 79.18 70.79 86.68 1.12 0.92RO21 150 0.169 2024 51.84 72.76 70.79 86.68 1.03 0.84
RO22 150 0.338 2024 51.84 95.4 89.42 94.86 1.06 1.01
RO23 150 0.338 2024 51.84 90.3 89.42 94.86 1.01 0.95
RO24 150 0.338 2024 51.84 90.65 89.42 94.86 1.01 0.95
RO25 150 0.507 2024 51.84 110.5 103.32 113.75 1.06 0.97
RO26 150 0.507 2024 51.84 103.6 103.32 113.75 1.00 0.91
RO27 150 0.507 2024 51.84 117.2 103.32 113.75 1.14 1.03
RO28 150 0.845 2024 51.84 112.6 121.48 112.03 0.93 1.01
RO29 150 0.845 2024 51.84 126.6 121.48 112.03 1.04 1.14
RO30 150 0.845 2024 51.84 137.9 121.48 112.03 1.14 1.23
RO31 150 0.169 2024 70.48 87.29 82.99 82.36 1.05 1.06
RO32 150 0.169 2024 70.48 84.03 82.99 82.36 1.01 1.02
RO33 150 0.169 2024 70.48 83.22 82.99 82.36 1.00 1.01
RO34 150 0.338 2024 70.48 94.06 101.62 90.55 0.93 1.04
RO35 150 0.338 2024 70.48 98.13 101.62 90.55 0.96 1.09
RO36 150 0.338 2024 70.48 107.2 101.62 90.55 1.05 1.19
RO37 150 0.507 2024 70.48 114.1 115.52 109.43 0.99 1.04RO38 150 0.507 2024 70.48 108 115.52 109.43 0.93 0.99
RO39 150 0.507 2024 70.48 110.3 115.52 109.43 0.95 1.01
RO40 150 0.169 2024 82.13 94.08 89.06 100.16 1.05 0.94
RO41 150 0.169 2024 82.13 97.6 89.06 100.16 1.10 0.97
RO42 150 0.169 2024 82.13 95.83 89.06 100.16 1.08 0.95
RO43 150 0.338 2024 82.13 97.43 107.69 108.35 0.90 0.90
RO44 150 0.338 2024 82.13 98.85 107.69 108.35 0.92 0.91
RO45 150 0.338 2024 82.13 98.24 107.69 108.35 0.91 0.91
RO46 150 0.507 2024 82.13 124.2 121.59 127.23 1.02 0.98
RO47 150 0.507 2024 82.13 129.5 121.59 127.23 1.06 1.02
RO48 150 0.507 2024 82.13 120.3 121.59 127.23 0.99 0.94
Mean 1.00 0.99
Std. Dev. 0.12 0.10
R 0.95 0.94
Bold sets are test sets.
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