Computers and Structures 162 (2016) 28–37

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Computers and Structures

journal homepage: www.elsevier .com/locate/compstruc

Evaluation of ultimate conditions of FRP-confined concrete columnsusing genetic programming

http://dx.doi.org/10.1016/j.compstruc.2015.09.0050045-7949/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +61 8 8313 6471; fax: +61 8 8313 4359.E-mail address: murat.karakus@adelaide.edu.au (M. Karakus).

Jian C. Lim, Murat Karakus ⇑, Togay OzbakkalogluSchool of Civil, Environmental and Mining Engineering, University of Adelaide, South Australia 5005, Australia

a r t i c l e i n f o

Article history:Received 11 February 2015Accepted 22 September 2015

Keywords:Genetic programming (GP)Fiber reinforced polymer (FRP)ConfinementConcreteCompressive strengthUltimate axial strain

a b s t r a c t

A large database consisting of 832 axial compression tests results of fiber reinforced polymer (FRP)-confined concrete specimens was assembled. Using the test database, existing conventional and evolu-tionary algorithm models developed for FRP-confined concrete were then assessed. New genetic pro-gramming (GP) models for predicting the ultimate condition of FRP-confined concrete were developed.The predictions of the proposed models suggest that more accurate results can be achieved in explainingand formulating the ultimate condition of FRP-confined concretes by GP. The model assessment alsoillustrates the influences of the size of the databases and the selected parameters used in the GP models.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

It is now well understood that the confinement of concrete withfiber reinforced polymer (FRP) composites can substantiallyenhance concrete strength and deformability. A large number ofstudies undertaken to date have produced over 3000 test resultson FRP-confined concrete and resulted in the development of over90 models. The conventional models that were developed usingregression analysis can be classified into two broad categoriesnamely the design-oriented models and the analysis-orientedmodels. In predictions of the ultimate condition of FRP-confinedconcrete, the design-oriented models use closed-form empiricalexpressions that were derived directly from database results. Theanalysis-oriented models, on the other hand, use a combinationof empirical and theoretical expressions through an incrementalprocedure to consider the interaction mechanism between theexternal FRP jacket and the internal concrete core. The analysis-oriented models are built on the assumption of stress path inde-pendence, which assumes that the axial stress and axial strain ofFRP-confined concrete at a given lateral strain are the same asthose of the same concrete actively confined with a constant con-fining pressure equal to that supplied by the FRP shell.

As indicated by the assessment results of these models using acomprehensive experimental database, the performances of a largeproportion of the conventional models were compromised when

they were assessed against a large database covering parametricranges that are much wider than the original databases used todevelop these models [1,2]. This can be attributed to the limitedcapability of the design-oriented models in handling uncertaintiesin complex experimental database, whereas the assumptionadopted by the analysis-oriented models has recently been shownto be inaccurate [3,4]. In addition, the development of these exist-ing conventional models is often based on the expressions pro-posed by Richart et al. [5], with refinement subsequently appliedto those earlier expressions to incorporate new research findings.Given the dependencies of the conventional models on the baseexpressions and the gradual refinement process, an efficient alter-native approach is therefore required. Recently, new models basedon artificial neural network, genetic programming, stepwiseregression, and fuzzy logic algorithms have been proposed bymany researchers [6–18]. Models in this category could handlecomplex databases containing large number of independent vari-ables, identify the sensitivity of input parameters, and providemathematical solutions between dependent and independent vari-ables. However, the complexity of modeling frameworks and thedependency of these models on computers have significantlyreduced their versatility in design applications. The computer gen-erated statistical solutions have also compromised the physicalsignificance unfolding the structural behavior of FRP-confined con-crete. In addition, several common modeling issues identified fromthe assessment of these models include: (i) limited size of databaseresults, (ii) overfitted with redundant test parameters that causeunreliable prediction beyond their original observation range,

J.C. Lim et al. / Computers and Structures 162 (2016) 28–37 29

and (iii) lack of consideration of important test parameters, includ-ing the ultimate rupture strain of FRP jacket.

With proper treatments given to the aforementioned modelingissues, genetic programming (GP) can be a potential candidate toaddress these shortcomings. GP is an evolutionary algorithmattempting to find key variables for a problem in a given searchspace, and generates mathematical expressions to explain the rela-tions between the variables. By using GP based on the principles ofsymbolic regression (SR) analysis, the relationships between thedependent and independent variables of a complex databaseinvolving uncertainties and variability can be solved. SR is a pro-cess of finding a mathematical expression by minimizing the errorsbetween given finite data set as well as providing a method offunction identification [19]. Symbolic regression, as opposed toother regression techniques, discovers both the form of the modeland its parameters from the search space. In other words, a mea-sured dataset is fitted to an appropriate mathematical formula bya fitness function. Determination or identification of key variablesand variable combinations, providing comprehension of developedmodels are among the benefits of symbolic regression analysis. Inthis research, SR analysis was conducted using GP approach whichfits well to wide range of engineering problems.

In the recent years, the use of GP for optimum solution andfunction identification of engineering problems have been gainingacceleration as the approach is capable of dealing with complexdatabase that contains a large number of parameters. GP progres-sively improves solution while it maintains the versatility of themodel in closed-form expressions (e.g. [20–25]). For example,Johari et al. [26] has successfully applied GP for the prediction ofsoil–water characteristic curve. Baykasoglu et al. [27] applied multiexpression programming (MEP), gene expression programming(GEP) and linear genetic programming (LGP) to estimate compres-sive and tensile strength of limestone for the first time with goodpredictions. Javadi et al. [28] introduced a new technique basedon GP for the determination of liquefaction induced lateral spread-ing. Cabalar and Cevik [29] applied GP for the prediction of peakground acceleration using strong-ground-motion data from Tur-key. The use of GP for the prediction of axial compressive strengthof FRP-confined concrete has also been demonstrated by Cevik 3[11], Cevik and Cabalar [6], and Cevik et al. 1 [8]. Results from thesestudies [6–8] proved that highly accurate prediction models can bedeveloped by using genetic programming.

In the present study, the GP approach was used to establishmodels to predict the ultimate conditions of FRP-confined concretecolumns under concentric compression. Based on a comprehensiveexperimental database that was carefully assembled using a set ofselection criteria to ensure the reliability and consistency of thedatabase, three closed-form expressions are proposed for the pre-dictions of the compressive strength, ultimate axial strain andhoop rupture strain of FRP-confined concrete. This is the first studyin the literature to establish expressions for the ultimate axialstrain and hoop rupture strain of FRP-confined concrete on thebasis of evolutionary algorithms. Details of the adopted approachare discussed in Section 2. A summary of the experimental data-base is provided in Section 3. The selection process of independentvariables, functions and fitness rule, together with the proposedexpressions are presented in Section 4. The predictions statisticsof the proposed and the existing models with experimental resultsare presented in Section 5.

Fig. 1. Flow chart of genetic programming [21].

2. Overview of genetic programming

In this section, the GP paradigm is discussed and the essentialsof GP are highlighted. Further concepts and terminology behind GPcan be found from the inventor of this paradigm [19]. However, it

is advised that the genetic algorithm concept developed in 1975 byHolland [30] and work from his student, Goldberg [31], can also bevisited for further insight.

2.1. Basic concepts of GP

Genetic programming is an extension of the conventionalgenetic algorithm (GA), generating novel solutions to complexproblems, developed by Koza [19]. Unlike GA which uses a stringof numbers to represent the solution, the GP automatically createsseveral computer programs (CP) with a parse tree structure tosolve the problem considered. The process of solving the problemswith GP is equivalent to searching a space of possible computerprograms for the fittest individual computer program [19]. Thegenerated CP is based on Darwinian concept of survival and repro-duction of the fittest as well as appropriate mating of CPs. As illus-trated in the flow chart in Fig. 1, the problem will be solved usingthe Darwinian genetic operators such as reproduction, crossover,and mutation. Initial population consists of randomly generatedCPs, which is composed of functions and terminals appropriate tothe characteristics of the problem. If the functions and terminalsselected are not appropriate for the problem, the desired solutioncannot be achieved. A basic flow chart of the genetic programmingparadigm is shown in Fig. 1.

As stated earlier, CP is composed of functions and terminals. Thefunctions can be standard arithmetic operators, trigonometric log-arithmic or power, e.g. f ¼ f=;�;þ;�; sin; cos; log 2; power; . . .gand/or any mathematical functions, logical functions, as well asuser-defined operators. Depending on the nature of the probleminvestigated, the computer program might be Boolean-valued,integer-valued, real-valued, complex-valued, vector-valued,symbolic-valued, or multiple-valued [19]. A typical expression tree,representing the computer program is shown in Fig. 2. In this exam-ple, the function set (F) is composed of multiplication, division,

Fig. 2. A typical expression tree.

30 J.C. Lim et al. / Computers and Structures 162 (2016) 28–37

addition, subtraction and the sine function, F = {�, /,�, sin}. The ter-minal set (T) is composed of N = 3 variable as T = {x,y,z}. The func-tions and terminals must fulfill two important properties in orderto solve the problemwith an appropriate representation [19]. Theseparameters are closure property and sufficiency property. The closureproperty includes protection of the function set and the terminal setagainst all possible argument values, e.g. protection of negativesquare root. Sufficiency property is the selection of the appropriatefunctions and terminals to solve the problem at hand.

2.2. Genetic operations

Genetic operations used in GP are composed of; reproduction,crossover, and mutation. Reproduction operation involves selecting,in proportion to fitness, a computer program from the current pop-ulation of programs, and allowing it to survive by copying it intothe new population. Several different types of reproduction opera-tions such as fitness proportionate reproduction or roulette wheelalgorithm, tournament selection and lexicographic parsimonypressure selection are commonly used in GP. In this study Lexico-graphic parsimony pressure selection was used, which is a multi-objective method similar to tournament selection. This particularmethod optimizes both fitness and parse tree size. The shortestindividual, the tree with fewer nodes, is selected as the fittest whentwo individuals are equally fit. Silva and Almeida [32] reported thatthis technique is effective in controlling the bloat which is a phe-nomenon consisting of an excessive code growth without the cor-responding improvement in fitness. The theory of ParsimonyPressure are discussed in detail by Poli and McPhee [33]. The stan-dard method of controlling bloat is to set up a maximum depth ontrees in the proposed GP model.

Crossover operation involves choosing random nodes in twoparent trees and swapping respective branches creating two newoffspring. Fig. 3 illustrates the crossover operation. Mathematicalexpressions for parent I and parent II are as follows beforecrossover;

Parent I :xy

sinðzÞ þ ðx� zÞ Parent II :x2þ ey � z2

Resulting offspring after crossover operation are:

Offspring I :x2þ ðx� zÞ Offspring II :

xysinðzÞ þ ey � z2

As can be seen from the above expressions, crossover betweenS-expressions consists of swapping the randomly selected subS-expressions.

In mutation operation, ‘‘a random node is selected from the par-ent tree and substituted by a new random tree created with theterminals and the functions available” as described by Silva [32].

This is known as tree mutation. However, Koza [19] stated thatmutation plays a minor role in GP. Therefore, it can be disregardedin most cases.

3. Database of FRP-confined concrete

The database of FRP-confined concrete was assembled throughan extensive review of the literature that covered 3042 test resultsfrom 253 experimental studies published between 1991 and 2013.The suitability of the results was then assessed using a set of care-fully established selection criteria to ensure the reliability and con-sistency of the database. Only monotonically loaded circularspecimens with unidirectional fibers orientated in the hoop direc-tion and an aspect ratio (H/D) of less than three were included inthe database. Specimens containing internal steel reinforcementor partial FRP confinement were not included. This resulted in afinal database size of 832 datasets collected from 99 experimentalstudies. The complete database of experimental results used in thepresent study can be found in Ozbakkaloglu and Lim [34]. Thedatabase consists of specimens confined by five main types ofFRP materials (carbon FRP (CFRP), high-modulus carbon FRP (HMCFRP), ultra-high-modulus carbon FRP (UHM CFRP), S- or E-glassFRP (GFRP), and aramid FRP (AFRP)) and two confinement tech-niques (wraps and tubes). The carbon FRPs were categorized intothree subgroups on the basis of their elastic modulus of fibers(Ef) (i.e., carbon FRP with Ef 6 270 GPa is categorized as CFRP; fol-lowed by 270 < Ef 6 440 GPa as HM CFRP; and Ef > 440 GPa asUHM CFRP). 755 specimens in the database were FRP-wrapped,whereas 77 specimens were confined by FRP tubes. 495 of thespecimens were confined by CFRP; 206 by GFRP; 79 by AFRP; 40by HM CFRP; and 12 by UHM CFRP. The unconfined concretestrength (f0co) and strain (eco), as obtained from concrete cylindertests, varied from 6.2 to 55.2 MPa and 0.14% to 0.70%, respectively.The diameters of the specimens (D) included in the test databasevaried between 47 and 600 mm, with the majority of the speci-mens having a diameter of 150 mm. The hoop rupture strain (eh,rup)of the FRP-confined concrete specimens varied from 0.09% to3.21%. The actual confinement ratio, defined as the ratio of the ulti-mate confining pressure of the FRP jacket at rupture to the com-pressive strength of an unconfined concrete specimen (flu,a/f 0co),varied from 0.02 to 4.74. Figs. 4 and 5, respectively, show the rela-tionship of the strength enhancement ratio (f 0cc/f 0co) and strainenhancement ratio (ecu/eco) versus the actual confinement ratio(flu,a/f0co), established from the database.

3.1. Parameter selection

The GP analysis is based on three main stages. Firstly, the termi-nal sets are selected. The term terminal refers to independent vari-ables used to approximate dependent variables. Parameters thatwere identified to be non-influential were either eliminated orcombined with several other parameters until a strong trend ofinfluence was observed. One of such input parameters that combi-nes several other input parameters is the confinement stiffness ofFRP jacket (Kl), defined by Eq. (2), which was presented later in Sec-tion 4.1. In the experimental database [34], it has already beenobserved that strong exponential or power relations exist betweenthe independent variable Kl and the dependent variables f0cc andecu. With the use of Kl as an independent variable, GP alsogenerated similar exponential and/or power formulations whichunderscore the experimental observations. Other influential inde-pendent parameters selected for the GP analyses include theunconfined strength of concrete (f0co), elastic modulus of fiber(Ef), hoop rupture strain of FRP jacket (eh,rup), and ultimate tensilestrength of fiber (ef). A summary of the selected input parameters

Fig. 3. Crossover operation in genetic programming.

(f'cc / f'co)= 1.00 + 4.22( flu,a / f'co) Data = 8 R2 = 0.482 (f'cc / f'co)= 1.00 + 4.08( flu,a / f'co) Data = 34 R2 = 0.712 (f'cc / f'co)= 1.00 + 3.64( flu,a / f'co) Data = 426 R2 = 0.873 (f'cc / f'co)= 1.00 + 3.18( flu,a / f'co) Data = 67 R2 = 0.840(f'cc / f'co)= 1.00 + 2.64( flu,a / f'co) Data = 149 R2 = 0.746

0

2

4

6

8

10

0 0.5 1 1.5 2

Stre

ngth

Enh

ance

men

t (f' c

c/ f'

co)

Confinement Ratio (flu,a / f'co)

UHM CFRP wrappedHM CFRP wrappedCFRP wrappedAFRP wrappedGFRP wrapped

Fig. 4. Variation of strength enhancement ratio (f0cc/f0co) of FRP-confined concretewith confinement ratio (flu,a/f0co).

(εcu / εco)= 2.00 + 24.91( flu,a / f'co) Data = 36 R2 = 0.521(εcu / εco)= 2.00 + 24.47( flu,a / f'co) Data = 109 R2 = 0.783(εcu / εco)= 2.00 + 17.41( flu,a / f'co) Data = 282 R2 = 0.641(εcu / εco)= 2.00 + 11.69( flu,a / f'co) Data = 30 R2 = 0.844(εcu / εco)= 2.00 + 7.67( flu,a / f'co) Data = 5 R2 = 0.803

0

10

20

30

40

50

0 0.5 1 1.5 2

Stra

in E

nhan

cem

ent (

ε cu

/ εco

)

Confinement Ratio ( flu,a / f'co)

AFRP wrappedGFRP wrappedCFRP wrappedHM CFRP wrappedUHM CFRW wrapped

Fig. 5. Variation of strain enhancement ratio (ecu/eco) of FRP-confined concrete withconfinement ratio (flu,a/f0co).

J.C. Lim et al. / Computers and Structures 162 (2016) 28–37 31

related to FRP-Confined concrete and GP parameters are given inTable 1.

In the second stage, a set of functions which are related to thenature of the problem or data set was determined. As can be seenfrom the Table 1, additional to the basic arithmetic operations

(+, �, /, �), exponential (exp), square root (sqrt) and power functionsare also included. We have also included logarithmic functionswhich did not improve the fitness. Therefore it was removed fromthe function set.

In the third stage, a symbolic expression (S-expression), a list oran atom in LISP, was generated by GP in terms of the fitness func-

Table 1Parameters used in genetic programming.

Parameters Values

f0cc f0co, Kl, eh,rupTerminal sets (T) for ecu f0co, Ef, eh,rup

eh,rup f0co, Ef, efFunction set (F) +, �, /, �, exp, sqrt, powerPopulation size 300Maximum generation 50Maximum tree depth 17Selection method LexictourTermination criterion Maximum generationType of elitism Half elitismMutation probability RandomCrossover probability Random

32 J.C. Lim et al. / Computers and Structures 162 (2016) 28–37

tion adopted. The S-expression is the only syntactic form of theLISP programming language. For example, (+(� 3 4) 5) is a LISPS-expression. In this S-expression the atoms or individuals (3 and4) are subtracted first and the result (�1) is added to 5 yieldingin value 4. S-expressions were chosen according to the lower fit-ness value. The lower the fitness value, the better the model is.According to the selected fitness (AAE), the lowest value indicatesa small error between the measured and predicted data. Thegenetic programming will run until the termination criterion(stopping condition) is satisfied. This can be done either by deter-mining a maximum generation limit or a tolerated error limit. Theprogram can also be terminated by the variation in fitnessobserved during the run. We used maximum generation limit asthe termination criterion for the all GP runs.

Some of the mathematical functions included in the GP are pro-tected against zero division or negative square root. In the divisionoperation, if the denominator is equal to zero then the resultsreturns to the numerator. In the power operation xx21 returns zeroif xx21 is not a number (NaN) or infinity (1), or has an imaginarypart, otherwise it returns xx21 . In the function square root,

ffiffiffix

p

returns zero if x 6 0 andffiffiffix

potherwise. Table 1 summarizes the

parameters adopted in GP analysis.

4. Proposed model for ultimate condition of FRP-confinedconcrete

The ultimate condition of FRP-confined concrete is often char-acterized as the compressive strength and the corresponding axialstrain of concrete and hoop strain recorded at the rupture of theFRP jacket. This makes the relationship between the ultimate axialstress (f0cc), ultimate axial strain (ecu) and hoop rupture strain(eh,rup) an important one. Using the comprehensive experimentaldatabase [34], a model consisting of three expressions for thepredictions of the compressive strength (f0cc), ultimate axial strain(ecu) and hoop rupture strain (eh,rup) was developed using GP and ispresented in this section. The model is applicable to FRP-confinedconcrete with unconfined concrete strength up to 55 MPa. Table 1summarizes the parameters that were used in GP analysis. As GPanalysis does not directly reveal the underlying physical relation-ships of a given dataset, the search of a physically meaningfulmodel structure relies on users’ engineering knowledge of FRP-confined concrete systems. This includes the understanding ofexperimentally influential parameters for implementation in GPformulation, and the careful selection of population size, functionsand model structure that will result in practical closed-formexpressions. These are no easy tasks since both the structure andparameters of the physical model must be determined [35]. Withthe help of GP approach, a large number of potential model compo-nents and structures can be tested, while the best parts of thesestructures and combinations can be retained to produce new and

possibly better expressions. In addition, the expressions resultingfrom GP formulation process are often useful in revealing pertinentaspects that are physically meaningful. Hence, with carefullyselected functions for these pertinent aspects, an accurate andphysically meaningful model can be established. In our trial runsof this approach, larger population sizes up to 1500 were tested.However, the use of the larger population sizes tends to createbloated/large number of nodes leading to extremely complex for-mulation. Therefore, we gradually reduced the population size to300 in the actual analysis. Mutation and crossover probabilitiesare not fixed but random. In tree crossover, random nodes are cho-sen from both parent trees, and the respective branches areswapped creating two offspring. It should be noted that not allthe datasets included in the database contained all the relevantdetails required for the model development. As a result, out of832 results, 753, 511, and 325 were used in the development ofthe expressions of the compressive strength (f0cc), ultimate axialstrain (ecu), and hoop rupture strain (eh,rup), respectively. The exper-imental values of the compressive strength (f0co) and the corre-sponding axial strain (eco) of unconfined concrete were based onthe test results of the unconfined cylinders. In order to avoid over-fitting random data division method was used, in which 70% of thetotal data were randomly selected for training and the remaining30% were used for testing purposes. This approach was previouslyused by a number of existing studies [23,36,37].

4.1. Compressive strength of FRP-confined concrete

As was discussed earlier, the majority of the conventional mod-els are based on the expression forms proposed by Richart et al. [5].GP, on the other hand, initiates the formulation of expressionsfrom a few random seeds, of which the evolution took place artifi-cially and terminated when a robust solution was found. In the GPformulation process, the selection of independent variables, func-tions and model structure for a given terminal set relies signifi-cantly on the users’ knowledge of the FRP-confined concretesystems. Overfitting of model with redundant independent vari-ables and functions results in a complex mathematical solutionrather than a physically meaningful model structure, whereasunderfitting reduces the model accuracy. In this process, findingthe potential combination of several input parameter to form a sin-gle representative input parameter, such as the confinement stiff-ness of FRP jacket (Kl) used in this study, is important. As presentedin Table 1, only parameters that had been observed experimentallyto influence the terminal sets [2] were selected as independentvariables for GP formulation. In addition, only simple model struc-ture yielding practical closed-formed expressions suitable for engi-neering application were adopted by the authors. The convergedexpression for the prediction of the compressive strength of FRP-confined concrete (f0cc) is presented in Eq. (1).

f 0cc ¼ f 0co þ Kleh;rup þ K1:5l e2h;rup þ a where a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKl � f 0coffiffiffiffiffiffiffiffiffiffieh;rup

ps

P 0

ð1Þwhere f0co is the unconfined concrete strength, Kl is the confinementstiffness of the FRP jacket, to be calculated using Eq. (2), and eh,rup isthe hoop rupture strain of the FRP jacket to be calculated using Eq.(8) presented later in Section 4.3. Kl and f0co are in MPa.

Kl ¼ 2Ef tfD

ð2Þ

Ef is the elastic modulus of fibers, tf is the total nominal fiber thick-ness of the FRP jacket, D is the diameter of the concrete specimen.

To demonstrate an example of the GP formulation process,Fig. 6 shows the relationship between accuracy and complexity

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Stre

ngth

Enh

ance

men

t (f' c

c/ f'

co) m

odel

Strength Enhancement ( f'cc / f'co)exp

AAE = 10.5%M = 100.9%SD = 13.5%

Fig. 7. Comparison of model predictions of strength enhancement ratios (f0cc/f0co)with experimental results.

J.C. Lim et al. / Computers and Structures 162 (2016) 28–37 33

of the GP formulation of the compressive strength expression. Asillustrated by the figure, the fitness or accuracy of the expressionimproves significantly after the first few generations. After which,the complexity of the expression represented by the level andnodes in Fig. 6 continue to increase with marginal improvementsin its fitness in the proceeding generations. At the 28th generation,where a low level and a small number of nodes were found atlower fitness, the final expression is selected and presented asEq. (1). Level is the depth of parse tree which controls the numberof nodes. It is used to avoid bloating, an excessive code growthwithout corresponding improvement in fitness.

Fig. 7 shows the comparison of the strength enhancement ratiopredictions (f0cc/f0co) of the proposed expression (Eq. (1)) with the30% testing datasets of the experimental database. The comparisonindicates that the model predictions are in close agreement withthe test results, which are quantified through the use of statisticalindicators: average absolute error (AAE) to establish overall modelaccuracy; mean (M) to establish average overestimation or under-estimation of the model; and standard deviation (SD) to establishthe magnitude of the associated scatter of the model prediction.These indicators are defined by Eqs. (3)–(5). As evident from thefigure, 10.5% of AAE, 100.9% of M, and 13.5% of SD were achieved.

AAE ¼Pn

i¼1modeli�expi

expi

��� ���n

ð3Þ

M ¼Pn

i¼1modeliexpi

nð4Þ

SD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1

modeliexpi

�M� �2n� 1

vuutð5Þ

4.2. Ultimate axial strain of FRP-confined concrete

Based on the GP formulation process discussed earlier, theexpression for the prediction of the ultimate axial strain (ecu) ofFRP-confined concrete is established and presented in Eq. (6).

ecu ¼ ðeco þ bÞðecco þKl

f 0coð2eco þ bÞÞ

where b ¼ eh;rup � eKlf 0coh;rup and c ¼ f 0coðeco þ eh;rup þ eeh;rup Þ

Klð6Þ

where eco is the axial strain corresponding to the compressivestrength of unconfined concrete, to be calculated using Eq. (7) pro-

0

20

40

60

80

0 5 10 15 20 25 30

Fitn

ess,

Leve

l, an

d N

ode

Generation

FitnessLevelNodes

Fig. 6. Accuracy versus complexity for the compressive strength expression. Pleasenote that each point on the fitness and nodes curves correspond to the lowestfitness values.

posed by Lim and Ozbakkaloglu [38], eh,rup is the hoop rupture strainof the FRP jacket, to be calculated using Eq. (8) presented in the nextsection, and Kl is the confinement stiffness of the FRP jacket, to becalculated using Eq. (2).

eco ¼ f 00:225co

1000kska ð7Þ

where ks ¼ 152D

� �0:1

and ka ¼ 2DH

� �0:13

where f0co is in MPa, and ks and ka, respectively, are the coefficientsto allow for the specimens size and specimen aspect ratio.

Fig. 8 shows comparisons of the strain enhancement ratio pre-dictions (ecu/eco) of the proposed model with testing datasets. Thecomparison indicates that the model predictions are in close agree-ment with the test results, of which AAE, M, and SD of 25.7%,101.1%, and 33.1%, respectively, were achieved.

4.3. Hoop rupture strain of FRP jacket

The hoop rupture strains (eh,rup) of FRP jackets are commonlyreported to be lower than the ultimate tensile strain of the fibermaterial (ef) [39–43]. As previously reported in Ozbakkaloglu andAkin [44], an increase in the compressive strength of concrete,which alters the concrete cracking pattern from heterogenic micro-cracks to localized macrocracks, has an adverse influence on thehoop rupture strain (eh,rup) of the FRP jacket. An increase in theelastic modulus of fibers (Ef) of the FRP jacket was also reported

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Stra

in E

nhan

cem

ent (

ε cu

/ εco

) mod

el

Strain Enhancement (εcu / εco)exp

AAE = 25.7%M = 101.1%SD = 33.1%

Fig. 8. Comparison of model predictions of strain enhancement ratios (ecu/eco) withexperimental results.

(a)

(b)

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Hoo

p R

uptu

re S

train

(ε h

,rup)

mod

el(%

)

Hoop Rupture Strain (εh,rup)exp (%)

AAE = 22.7%M = 109.6%SD = 34.2%

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Hoo

p R

uptu

re S

train

(ε h

,rup)

mod

el(%

)

Hoop Rupture Strain (εh,rup)exp (%)

AAE = 22.5%M = 112.3%SD = 33.1%

Fig. 9. Comparison of model predictions of hoop rupture strains (eh,rup) withexperimental results: (a) simplified function, (b) complex function.

34 J.C. Lim et al. / Computers and Structures 162 (2016) 28–37

to decrease the hoop rupture strain (eh,rup) [45]. To account for suchdependencies of the hoop rupture strain on the FRP and concretematerials, the elastic modulus of fibers (Ef), ultimate tensile strainof fibers (ef), and unconfined concrete strength (f0co) were consid-

Table 2Statistics of strength enhancement ratio (f0cc/f0co) predictions of best performing

Model Model categorya Prediction o

Test data

Proposed model GP 753Proposed model (30% testing dataset) GP 226Ozbakkaloglu and Lim [34] DO 753Teng et al. [47] AO 753Lam and Teng [48] DO 753Wu and Zhou [49] DO 753Wu and Wang [50] DO 753Al-Salloum and Siddiqui [51] DO 753Wei and Wu [52] DO 753Realfonzo and Napoli [53] DO 753Bisby et al. [54] DO 753Jiang and Teng [1] AO 753Cevik 1 [11] NN 753Elsanadedy et al. [13] NN 753Cevik and Cabalar [6] GP 753Cevik 2 [11] SW 753Cevik 3 [11] GP 753Cevik et al. 1 [8] GP 753Cevik et al. 2 [8] SW 753Naderpour et al. [9] NN 753Cevik and Guzelbey [7] NN 753Jalal and Ramezanianpour [14] NN 753

a Conventional models – AO: Analysis-oriented model, DO: Design-oriented mNeural network model, SW: Stepwise regression model.

ered as separate input variables for the development of the hooprupture strain expression (eh,rup). The expression established usingthe GP formulation process for the prediction of the hoop rupturestrain of FRP (eh,rup) is presented in Eq. (8).

eh;rup ¼ eff 00:125co

ð8Þ

To demonstrate an example of overfitted expression, the GP formu-lation process was allowed to continue until a next expression(Eq. (9)) with a slightly higher accuracy. Fig. 9(a) and (b) showthe comparisons of the results from testing dataset with the hooprupture strains (eh,rup) predicted using Eqs. (8) and (9), respectively.The comparison shows that the model predictions are in closeagreement with the test results, of which AAE, M, and SD of 22.7%,109.6%, and 34.2% were achieved in the first expression, while22.5%, 112.2%, and 33.1% were achieved in the second expression.With only a small margin of improvement achieved, the form ofthe expression of Eq. (9) became relatively complex in comparisonto the earlier form shown in Eq. (8). On this basis, Eq. (8) is recom-mended for its simplicity. In addition, as Eq. (9) was overfitted forthe current size of test database, its performance is likely to degradeat out-of-range prediction, due to its increased sensitivity to theparametric ranges of the current database.

eh;rup ¼ efEf þ f 0co

!0:5edf

where d ¼ ef

f0ð f 0coefef þf 0co

Þco

ð9Þ

where Ef and f0co are in MPa.

5. Model validation and comparisons with existing models

To establish the relative performance of the proposed model, itsprediction statistics were compared with those of the 10 best per-forming conventional models identified in a recent comprehensivereview study reported in Ozbakkaloglu et al. [34]. In addition, themodel was also compared with seven artificial intelligence (AI)models currently available in the literature that were developedusing evolutionary programming techniques, including neural net-work (NN) [7,9,11,13,14], genetic programming (GP) [6,8,11], and

models.

f f0cc/f0co

Average absolute error (%) Mean (%) Standard deviation (%)

10.8 102.1 14.410.5 100.9 13.511.2 99.6 13.711.8 98.8 14.512.4 99.4 15.312.4 102.1 15.512.7 101.4 15.712.7 101.7 15.812.7 101.5 15.712.7 103.2 15.812.8 101.9 15.812.9 93.9 14.615.6 96.2 18.518.4 100.4 31.019.6 115.1 22.620.1 95.7 29.921.0 116.1 23.223.9 108.0 33.733.3 108.8 42.240.2 127.6 50.575.6 154.3 110.779.2 178.3 79.1

odel. Artificial intelligence models – GP: Genetic programming model, NN:

0% 10% 20% 30%

Proposed modelCevik 1 [11]Elsanadedy et al. [13]Cevik and Cabalar [6]Cevik 2 [11]Cevik 3 [11]Cevik et al. 1 [8]Cevik et al. 2 [8]Naderpour et al. [9]Cevik and Guzelbey [7]Jalal and Ramezanianpour [14]

AAE in predictions of strength enhancementratio ( f'cc/f'co)

10.8% (10.5% based on testing dataset)

18.4%

75.6%

33.3%

21.0%20.1%

19.6%

15.6%

79.2%

40.2%

23.9%

Fig. 11. Average absolute error in strength enhancement ratios (f0cc/f0co) predictionsof artificial intelligence models.

Table 3Statistics of strain enhancement ratio (ecu/eco) predictions of best performing models.

Model Model categorya Prediction of ecu/eco

Test data Average absolute error (%) Mean (%) Standard deviation (%)

Ozbakkaloglu and Lim [34] DO 511 21.7 100.5 27.2Proposed model GP 511 23.8 98.5 30.1Proposed model (30% testing dataset) GP 153 25.7 101.1 33.1Tamuzs et al. [55] DO 511 26.3 108.4 35.0Wei and Wu [52] DO 511 28.7 98.0 35.8Binici [56] AO 511 29.2 92.3 34.8Jiang and Teng [57] DO 511 29.5 116.1 38.5Youssef et al. [58] DO 511 30.0 112.5 39.0Teng et al. [59] DO 511 30.2 117.6 39.0Fahmy and Wu [60] DO 511 30.5 99.5 38.9Teng et al. [47] AO 511 30.5 117.0 39.3De Lorenzis and Tepfers [61] DO 511 31.3 77.9 27.9

a Conventional models – AO: Analysis-oriented model, DO: Design-oriented model. Artificial intelligance models – GP: Genetic programming model, NN: Neural networkmodel, SW: Stepwise regression model.

(a)

(b)

10% 11% 12% 13% 14%

Ozbakkaloglu and Lim [34]

Teng et al. [47]

Lam and Teng [48]

Wu and Zhou [49]

Wu and Wang [50]

Al-Salloum and Siddiqui [51]

Wei and Wu [52]

Realfonzo and Napoli [53]

Bisby et al. [54]

Jiang and Teng [1]

AAE in predictions of strength enhancementratio ( f'cc/f'co)

11.2%

11.8%12.4%

12.9%

12.8%

12.7%

12.7%

12.4%

12.7%

12.7%

20% 25% 30% 35%

Ozbakkaloglu and Lim [34]

Tamuzs et al. [55]

Wei and Wu [52]

Binici [56]

Jiang and Teng [57]

Youssef et al. [58]

Teng et al. [59]

Fahmy and Wu [60]

Teng et al. [47]

De Lorenzis and Tepfers [61]

AAE in predictions of strain enhancementratio ( εcu/εco)

21.7%

26.3%

28.7%

29.2%

29.5%

30.0%

30.2%

30.5%

30.5%

31.3%

Fig. 10. Average absolute error in predictions of conventional models: (a) strengthenhancement ratios (f0cc/f0co), (b) strain enhancement ratios (ecu/eco).

J.C. Lim et al. / Computers and Structures 162 (2016) 28–37 35

stepwise regression (SW) [8,11]. It should be noted that the modelsproposed in various studies that only reports the architecture oftheir modeling framework without complete expressions for theprediction of the ultimate condition of FRP-confined concrete werenot assessed (e.g., [10,12,15,17,18]). The prediction statistics forthe strength and strain enhancement ratios (f0cc/f0co and ecu/eco)are given in Tables 2 and 3, respectively. In addition to the resultsshown in Tables 2 and 3, the graphical comparison of the AAEs forthe conventional and the AI models are given in Figs. 10 and 11,respectively. In calculating model predictions of the peak strainratios (ecu/eco), the eco values were determined using the expres-sions given in the original publication when available. If an expres-sion was not specified in the original publication, the eco values

were then based on the experimental values obtained from cylin-der tests. If an experimental eco value was not available from agiven dataset, Eq. (7) proposed by Tasdemir et al. [46] is used todetermine the eco value. For the proposed model, two sets of pre-diction statistics are presented in Tables 2 and 3. The first setwas based on the full datasets of 753 and 511 results in the exper-imental database for strength and strain enhancement ratios(f0cc/f0co and ecu/eco), while the second set was based the 30% testingdatasets.

As evident from Tables 2 and 3 and Figs. 10 and 11, the pro-posed model provides improved predictions of the strengthenhancement ratios (f0cc/f0co) compared to all of the existing mod-els. On the other hand, the performance of the model developedby Ozbakkaloglu and Lim [34] for the predictions of the strainenhancement ratios (ecu/eco) is slightly better than that of the cur-rent proposed model, given that the influence of axial strain instru-mentation methods was considered in the model [34]. For suchconsideration, a numerical input such as the gauge length of theinstrument which is necessary for GP, however, was not availablefrom the experimental database [34]. As a result, the minimumAAE of the predicted strain enhancement ratios (ecu/eco) by the pro-posed model is closer to the natural scatter of the database of 23%.It is also worthwhile noting that, none of the reviewed AI modelsproposed an expression for the prediction of the ultimate axialstrain (ecu) of FRP-confined concrete. This paper presents the firstexpression established for the prediction of the ultimate axialstrain (ecu) on the basis of GP (Eq. (6)). As evident from Fig. 8 andTable 3, the proposed expression provides reasonable predictionsof the test results, which can be further improved in the futurethrough accurate modeling of the specimen instrumentationmethods.

36 J.C. Lim et al. / Computers and Structures 162 (2016) 28–37

A close examination of the assessment results has led to a num-ber of important findings on factors influencing the performancesof existing models including the size of database, parameters con-sidered, ability to handle uncertainties, dependency on assump-tions, and the architecture of their modeling frameworks. For theconventional models, only the statistics of the model performancesare presented in Tables 2 and 3. However, a detailed review of fac-tors influencing the performances of these models was previouslydiscussed by Ozbakkaloglu et al. [2]. In this section, we only dis-cussed the factors influencing the performances of AI models. Asevident from Fig. 11, the proposed model outperformed existingAI models by a significant margin. This improvement was achievedthrough consideration of a wide range of parameters that was cov-ered by the comprehensive experimental database and carefulselection of influential input parameters in model development.In the better performing models illustrated in Fig. 11, includingthe NN models proposed by Cevik [11] and Elsanadedy et al.[13], and the GP model proposed by Cevik and Cabalar [6], it wasfound that the sizes of the databases used in their developmentwere generally larger than those of their underperforming counter-parts, but the architecture of their modeling frameworks was notnecessarily more complex. On the other hand, excessive complex-ity of modeling framework due to overfitting with redundant vari-ables, as evident from some of the underperforming models (e.g.,[9,14]), significantly undermined the modeling accuracy. On thesebases, it is recommended that comprehensive experimental data-bases should be used and selection of key parameters should becarefully implemented in the future development of AI modelsfor FRP-confined concrete.

6. Conclusions

A comprehensive experimental test database that consisted of832 test results of FRP-confined concrete has been assembled fromthe published literature. Using the test database, the performancesof a number of existing empirical, theoretical, and artificial intelli-gence models developed for FRP-confined concrete were thenassessed. A close examination of the results from model assess-ments has led to a number of important findings on factors influ-encing the strengths and weaknesses of models in each category.These findings have been summarized and discussed in detail inthe paper. On the basis of the experimental database, a new modelfor evaluating the ultimate condition of FRP-confined concrete wasdeveloped using genetic programming and has been presented inthis paper. The model is the first to establish the ultimate axialstrain and hoop rupture strain expressions for FRP-confined con-crete on the basis of evolutionary algorithms. Comparisons withexperimental test results show that the predictions from the pro-posed model are in good agreement with the test results of thedatabase. The proposed models provide improved predictionscompared to the existing artificial intelligence models. Geneticprogramming proved that more accurate results can be achievedin explaining and formulating the ultimate condition of FRP con-fined concretes. The model assessment presented in this studyclearly illustrates the important influences of the size of the testdatabases and the selected test parameters used in the develop-ment of artificial intelligence models on their overallperformances.

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