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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

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Please cite this article in press as: Cevik A et al. Soft computing based formulation for strength enhancement of CFRP confined concrete cylinders. Adv Eng

Softw (2009), doi:10.1016/j.advengsoft.2009.10.015

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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].

2 A. Cevik et al. / Advances in Engineering Software xxx (2009) xxx–xxx

ARTICLE IN PRESS







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

A. Cevik et al. / Advances in Engineering Software xxx (2009) xxx–xxx 3

ARTICLE IN PRESS







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


ARTICLE IN PRESS







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.


ARTICLE IN PRESS







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

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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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References

[1] Hollaway LC. Advanced polymer composites for structural applications in

construction: ACIC 2004. Woodhead Publishing; 2004.

[2] Lorenzis L. A comparative study of models on confinement of concrete

cylinders with FRP composites. PhD thesis, Division for Building Technology,

Chalmers University of Technology, Sweden; 2001.

[3] Fardis MN, Khalili H. FRP-encased concrete as a structural material. Mag Concr

Res 1982;34(121):191–202.

[4] Mirmiran A, Zagers K, Yuan W. Nonlinear finite element modelling of concreteconfined by fiber composites. Finite Elem Anal Design 2000;35:79–96.

[5] Miyauchi K, Nishibayashi S, Inoue S. Estimation of strengthening effects with

carbon fiber sheet for concrete column. In: Proceedings of the third

international symposium (FRPRCS-3) on non-metallic (FRP) reinforcement

for concrete structures, Sapporo, Japan, vol. 1; 1997. p. 217–24.

[6] Kono S, Inazumi M, Kaku T. Evaluation of confining effects of CFRP sheets on

reinforced concrete members. In: Proceedings of the 2nd international

conference on composites in infrastructure ICCI’98; 1998. p. 343–55.

[7] Matthys S, Taerwe L, Audenaert K. Tests on axially loaded concrete columns

confined by fiber reinforced polymer sheet wrapping. In: 4th international

symposium on fiber reinforced polymer reinforcement for reinforced concrete

structures; 1999. p. 217–28.

[8] Shahawy M, Mirmiran A, Beitelmann T. Tests and modeling of carbon-wrapped

concrete columns. Compos Part B: Eng 2000;31:471–80 [London: Elsevier

Science Ltd.].

[9] Rochette P, Labossiére P. Axial testing of rectangular column models confined

with composites. J Compos Construct, ASCE 2000;4(3):129–36.

[10] Micelli F, Myers JJ, Murthy S. Effect of environmental cycles on concrete

cylinders confined with FRP. In: Proceedings of CCC 2001 internationalconference on composites in construction, Porto, Portugal; 2001.

[11] Rousakis T. Experimental investigation of concrete cylinders confined by

carbon FRP sheets, under monotonic and cyclic axial compressive load.

Research report, Chalmers University of Technology, Göteborg, Sweden;

2001.

[12] Fardis MN, Khalili H. Concrete encased in fiberglass-reinforced-plastic. J Am

Concr Inst Proc 1981;78(6):440–6.

[13] Saadatmanesh H, Ehsani MR, Li MW. Strength and ductility of concrete

columns externally reinforced with fiber composite straps. ACI Struct J

1994;91(4):434–47.

[14] Samaan M, Mirmiram A, Shahawy M. Model of concrete confined by fiber

composites. J Struct Eng, ASCE 1998;124(9):1025–31.

[15] Toutanji H. Stress–strain characteristics of concrete columns externally

confined with advanced fiber composite sheets. ACI Mater J 1999;96(3):

397–404.

[16] Saafi M, Toutanji HA, Li Z. Behavior of concrete columns confined with fiber

reinforced polymer tubes. ACI Mater J 1999;96(4):500–9.

[17] Spoelstra MR, Monti G. FRP-confined concrete model. J Compos Construct,

ASCE 1999;3(3):143–50.

[18] Xiao Y, Wu H. Compressive behaviour of concrete confined by carbon fiber

composite jackets. J Mater Civil Eng, ASCE 2000;12(2):139–46.

[19] Karabinis AI, Rousakis TC. Carbon FRP confined concrete elements under axial

load. In: Proc int conf on FRP composites in civil engineering. Hong Kong;

2001. p. 309–16.

[20] Lam L, Teng JG. A new stress–strain model for FRP confined concrete. In: Teng

JG, editor. Proc int conf on FRP composites in civil engineering. Hong Kong;

2001. p. 283–92.

[21] Shehata LAEM, Carneiro LAV, Shehata LCD. Strength of short concrete columns

confined with CFRP sheets. Mater Struct 2002;35:50–8.

[22] Matthys S, Toutanji H, Audenaert K, Taerwe L. Axial load behavior of large-scale columns confined with FRP composites. ACI Struct J 2005;102(2):

258–67.

[23] Kumutha R, Vaidyanathan R, Palanichamy MS. Behaviour of reinforced

concrete rectangular columns strengthened using GFRP. Cem Concr Compos

2007;29(8):609–15.

[24] Teng JG, Jiang T, Lam L, Luo YZ. Refinement of a design-oriented stress–

strain model for FRP-confined concrete. J Compos Construct 2009;13(4):

269–78.

[25] Lam L, Teng JG. Design-oriented stress–strain model for FRP-confined

concrete. Construct Build Mater 2003;17(6–7):471–89.

[26] Vintzileou E, PanagiotidouE. An empiricalmodel for predictingthe mechanical

properties of FRP-confined concrete. Construct Build Mater 2008;22(5):

841–54.

[27] Rousakis TC, Karabinis AI. Substandard reinforced concrete members

subjected to compression: FRP confining effects. Mater Struct/Mater

Construct 2008;41(9):1595–611.

[28] Cevik A, Guzelbey _Ibrahim H. Neural network modeling of strength

enhancement for CFRP confined concrete cylinders. Build Environ 2008;43:

751–63.

[29] Zadeh LA. Soft computing and fuzzy logic. IEEE Software 1994;11(6):48–56.

[30] Koivo H. Soft computing in dynamical systems; 2000.

[31] Dug Hun H, Changha H. A brief introduction to soft computing proceedings of

the autumn conference. Korean Statistical Society; 2004. p. 65–6.

[32] Campbell MJ. Statistics at square two: understanding modern statistical

applications in medicine. London, GBR: BMJ Publishing Group; 2001.

[33] Rawlings JO. Applied regression analysis: a research tool. New York: Springer-

Verlag; 1998.

[34] Koza JR. Genetic programming: on the programming of computers by means of

natural selection. Cambridge, MA: MIT Press; 1992.

[35] Ferreira C. Gene expression programming: mathematical modelling by an

artificial intelligence; 2002.

[36] Ferreira C. Gene expression programming in problem solving. In: Invited

tutorial of the 6th online world conference on soft computing in industrial

applications; 2001.

[37] Ferreira C. Gene expression programming: a new adaptive algorithm for

solving problems. Complex Syst 2001;13(2):87–129.

[38] <http://www.gepsoft.com/ >.


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