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doi:10.1016/j.bu

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Building and Environment 43 (2008) 751–763

www.elsevier.com/locate/buildenv

Neural network modeling of strength enhancementfor CFRP confined concrete cylinders

Abdulkadir Cevika,�, Ibrahim H. Guzelbeyb

aDepartment of Civil Engineering, University of Gaziantep, 27310, TurkeybDepartment of Mechanical Engineering, University of Gaziantep, 27310, Turkey

Received 17 March 2006; received in revised form 15 January 2007; accepted 17 January 2007

Abstract

This study presents the application of neural networks (NN) for the modeling of strength enhancement of CFRP (carbon fiber-

reinforced plastic) confined concrete cylinders. The proposed NN model is based on experimental results collected from literature. It

represents the ultimate strength of concrete cylinders after CFRP confinement which is also given in explicit form in terms of diameter,

unconfined concrete strength, tensile strength CFRP laminate and total thickness of CFRP layer used. The accuracy of the proposed NN

model is quite satisfactory as compared to experimental results. Moreover the results of proposed NN model are compared with 10

different theoretical models proposed by researchers so far and are found to be by far more accurate.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Neural network; CFRP confinement; Concrete cylinder; Strength enhancement

1. Introduction

With over 50 years of excellent performance records inthe aerospace industry, fiber-reinforced-polymer (FRP)composites have been introduced with confidence to theconstruction industry. These high-performance materialshave been accepted by civil engineers and have beenutilized in different construction applications such as repairand rehabilitation of existing structures as well as in newconstruction applications. One of the successful and mostpopular structural applications of FRP composites is theexternal strengthening, repair and ductility enhancement ofreinforced concrete (RC) columns in both seismic andcorrosive environments [1]. Main types of FRP compositesused in external strengthening and repair of RC columnsare: glass-fiber-reinforced polymers (GFRP), carbon-fiber-reinforced polymers (CFRP), and aramid-fiber-reinforcedpolymers (AFRP). Types of FRP confinement can bespiral, wrapped and tube. FRP composites offer severaladvantages due to extremely high strength-to-weight ratio,

e front matter r 2007 Elsevier Ltd. All rights reserved.

ildenv.2007.01.036

ing author. Tel.: +90342 3601200x2409;

01107.

ess: akcevik@gantep.edu.tr (A. Cevik).

good corrosion behavior, electromagnetic neutrality. Thus,the effect of FRP confinement on the strength anddeformation capacity of concrete columns has beenextensively studied and several empirical and theoreticalmodels have been proposed [2]. This study proposes a newapproach for the modeling of strength enhancement ofCFRP wrapped concrete cylinders using NNs. Theproposed NN model for the compressive strength of theconfined concrete cylinder is presented in explicit form.

2. Behavior of FRP confined concrete

Being a frictional material, concrete is sensitive tohydrostatic pressure. The beneficial effect of lateral stresseson the concrete strength and deformation has beenrecognized nearly for a century. In other words whenuniaxially loaded, concrete is restrained from dilatinglaterally, it exhibits increased strength and axial deforma-tion capacity indicated as confinement which has beengenerally applied to compression members through steeltransverse reinforcement in the form of spirals, circularhoops or rectangular ties, or by encasing the concretecolumns into steel tubes that act as permanent formwork

ARTICLE IN PRESS

Nomenclature

f 0co compressive strength of the unconfined con-crete cylinder

f 0cc compressive strength of the confined concretecylinder

pu ultimate confinement pressureEl confinement modulus or lateral modulusEf modulus of elasticity of the FRP laminatent total thickness of FRP layerD diameter of the concrete cylinderL Length of the concrete cylinder

Table 1

Models for strength enhancement of FRP confined concrete cylinders

Model Expression ðf 0cc=f 0coÞ

Fardis and Khalili [12] f 0ccf 0co¼ 1þ 4:1

puf 0co

(1)

f 0ccf 0co¼ 1þ 3:7

puf 0co

� �0:86

(2)

Saadatmanesh et al. [13] f 0ccf 0co¼ 2:254

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 7:94

puf 0co

r� 2

puf 0co� 1:254 (3)

Miyauchi et al. [5] f 0cc pu

A. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763752

[2]. Besides steel reinforcement FRPs are also for confine-ment of concrete columns and offers several advantages ascompared to steel [3] such as continuous confining actionto the entire cross-section, easiness and speed of applica-tion, no change in the shape and size of the strengthenedelements, corrosive resistance [2].

Typical response of FRP-confined concrete is shown inFig. 1, where normalized axial stress is plotted againstaxial, lateral, and volumetric strains. The stress is normal-ized with respect to the unconfined strength of concretecore. The figure shows that both axial and lateral responsesare bi-linear with a transition zone at or near the peakstrength of unconfined concrete core. The volumetricresponse shows a similar transition toward volume expan-sion. However, as soon as the jacket takes over, volumetricresponse undergoes another transition which reverses thedilation trend and results in volume compaction. Thisbehavior is shown to be markedly different from plainconcrete and steel-confined concrete [4].

The characteristic response of confined concrete includesthree distinct regions of un-cracked elastic deformations,crack formation and propagation, and plastic deforma-tions. It is generally assumed that concrete behaves like anelastic-perfectly plastic material after reaching its max-imum capacity, and that the failure surface is fixed in thestress space. Constitutive models for concrete should beconcerned with pressure sensitivity, path dependence,stiffness degradation and cyclic response. The existingplasticity models range from nonlinear elasticity, endo-chronic plasticity, classical plasticity, and multi-laminate or

2.5

2.0

1.5

1.0

0.5

0

-0.02 -0.01 0 0.01 0.02 0.03 0.04

Lateral or Volumetric Strain Axial Volumetric Strain

Nor

mal

ized

Axi

al S

tres

s (f

c/f' c

o)

LateralResponse

VolumetricResponse Axial

Response

Concrete

FRPTube

Volume CompactionVolume Expantion

Fig. 1. Typical response of FRP-confined concrete [4].

micro-plane plasticity to bounding surface plasticity. Manyof these models, however, are only suitable in a specificapplication and loading system for which they are devisedand may give unrealistic results in other cases. Also, someof these models require several parameters to be calibratedbased on experimental results [4]. Considerable experi-mental research has been performed on the behavior ofCFRP confined concrete columns [5–11]. Several modelsare proposed in literature for the strength enhancementof FRP confinement effect of concrete columns given inTable 1.This study aims to propose an alternative approach

and a new formulation by means of NNs for the predictionof strength enhancement of CFRP confined concretecylinders.

f 0co¼ 1þ 3:485

f 0co(4)

Kono et al. [6] f 0ccf 0co¼ 1þ 0:0572pu (5)

Samaan et al. [14] f 0ccf 0co¼ 1þ 6:0

p0:7u

f 0co(6)

Toutanji [15] f 0ccf 0co¼ 1þ 3:5

puf 0co

� �0:85

(7)

Saafi et al. [16] f 0ccf 0co¼ 1þ 2:2

puf 0co

� �0:84

(8)

Spoelstra and Monti [17] f 0ccf 0co¼ 0:2þ 3

puf 0co

� �0:5

(9)

Xiao and Wu [18] f 0ccf 0co¼ 1:1þ 4:1� 0:75

f 0co2

E1

!puf 0co

(10)

ARTICLE IN PRESS

Table 2

Experimental database and ranges of variables

Reference Number of specimen D (mm) nt (mm) Ef (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

+

DENDRITE CELL BODY

AXON

THRESHOLD

A. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763 753

3. Experimental database

In this study an extensive literature review on experi-mental studies related to strength enhancement of CFRPwrapped concrete cylinders has been carried out and anexperimental database has been gathered. It should benoted that all specimen used in the database have a lengthto diameter ratio of 2 ðL=D ¼ 2Þ. A total of 101 specimensfrom seven separate studies with the ranges of variableswere included in the database shown in Table 2. Furtherdetails of the experimental database are given in Table 6.

wi1

wi2

wi3

wij

x1

x2

x3

xi

N

i=1ui = ∑ wijxi + bi Yi=f(ui)

Bias Activation Function

Output••

Fig. 3. Basic elements of an artificial neuron.

SUMMATION

Fig. 2. Artificial neuron model.

4. Brief overview of NNs

Artificial Neural Networks (NNs) are computer modelsthat mimic the biological nervous system. An NN can bedefined as a massively parallel distributed processor thathas a natural propensity for storing experiential knowledgeand making it available for use [19]. The main componentof this model is the structure of its information processingunit. A biological neuron is made up of four main parts:dendrites, synapses, axon and the cell body. The dendritesreceive signals from other neurons. The axon of a singleneuron serves to form synaptic connections with otherneurons. The cell body of a neuron sums the incomingsignals from dendrites. If input signals are sufficient tostimulate the neuron to its threshold level, the neuron sendsan impulse to its axon. On the other hand if the inputs donot reach the required level, no impulse will occur. Theanalogy between a biological neuron model and anartificial neuron model is shown in Fig. 2.

The basic element of an NN is the artificial neuron asshown in Fig. 3 which consists of three main componentsnamely as weights, bias, and an activation function. Eachneuron receives inputs x1;x2; . . . ; xn, attached with a weightwi which shows the connection strength for that input foreach connection. Each input is then multiplied by thecorresponding weight of the neuron connection. A bias bi

can be defined as a type of connection weight with aconstant nonzero value added to the summation of inputsand corresponding weights u, given by

ui ¼XHj¼1

wijxj þ bi. (11)

The summation ui is transformed using a scalar-to-scalarfunction called an ‘‘activation or transfer function’’, F ðuiÞ

yielding a value called the unit’s ‘‘activation’’, given by

Y i ¼ f ðuiÞ. (12)

Activation functions serve to introduce nonlinearity intoNNs which makes NNs so powerful.NNs are commonly classified by their network topology

(i.e. feedback, feed forward) and learning or trainingalgorithms (i.e. supervised, unsupervised). For example, amultilayer feed forward NN with back propagationindicates the architecture and learning algorithm of theNN.

5. Optimal NN model selection

The performance of an NN model mainly depends onthe network architecture and parameter settings. One of

ARTICLE IN PRESSA. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763754

the most difficult tasks in NN studies is to find this optimalnetwork architecture which is based on determination ofnumbers of optimal layers and neurons in the hidden layersby trial and error approach. The assignment of initialweights and other related parameters may also influencethe performance of the NN in a great extent. However,there is no well-defined rule or procedure to have optimalnetwork architecture and parameter settings where trialand error method still remains valid. This process is verytime consuming.

In this study Matlab NN toolbox is used for NNapplications. Various back propagation training algorithmsare used given in Table 3. Matlab NN toolbox randomlyassigns the initial weights for each run each time whichconsiderably changes the performance of the trained NNeven all parameters and NN architecture are kept constant.This leads to extra difficulties in the selection of optimalnetwork architecture and parameter settings. To overcomethis difficulty a program has been developed in Matlab whichhandles the trial and error process automatically. Theprogram tries various number of layers and neurons in thehidden layers both for first and second hidden layers for aconstant epoch for several times and selects the best NNarchitecture with the minimum MAPE (mean absolute %error) or RMSE (root mean squared error) of the testing set,

Table 3

Back propagation training algorithms used in NN training

MATLAB

function name

Algorithm

trainbfg BFGS quasi-Newton back propagation

traincgf Fletcher–Powell conjugate gradient back propagation

traincgp Polak–Ribiere conjugate gradient back propagation

traingd Gradient descent back propagation

traingda Gradient descent with adaptive linear back

propagation

traingdx Gradient descent w/momentum and adaptive linear

back propagation

trainlm Levenberg–Marquardt back propagation

trainoss One step secant back propagation

trainrp Resilient back propagation (Rprop)

trainscg Scaled conjugate gradient back propagation

bias bias

Inputs

D

nt

Ef

f'co

f'cc

Output

Fig. 4. Optimum NN

as the training of the testing set is more critical. For instance,an NN architecture with 1 hidden layer with 7 nodes is tested10 times and the best NN is stored where in the second cyclethe number of hidden nodes is increased up to 8 and theprocess is repeated. The best NN for cycle 8 is comparedwith cycle 7 and the best one is stored as best NN. Thisprocess is repeated N times where N denotes the number ofhidden nodes for the first hidden layer. This whole process isrepeated for changing number of nodes in the second hiddenlayer. More over this selection process is performed fordifferent back propagation training algorithms such astrainlm, trainscg and trainbfg given in Table 3. The programbegins with simplest NN architecture i.e. NN with onehidden node for the first and second hidden layers and endsup with optimal NN architecture as shown in Fig. 4. Theflowchart is of the whole process is shown in Fig. 5.

6. Results of NN model

The experimental database is randomly divided astraining and test set given in Table 6. Among 101 datasets 10 data sets were used as test and the rest as trainingset. The optimal NN architecture in this part was found tobe 4–15–1 NN architecture with hyperbolic tangentsigmoid transfer function (tansig). The training algorithmwas quasi-Newton back propagation (BFGS). The opti-mum NN model is given in Fig. 6. Statistical parameters oflearning and training sets of NN model are presented inTable 4. The % errors and prediction of NN and actualvalues of learning and testing sets and their correspondingcorrelation are given in Figs. 7–10. Results of NN model ispresented in Table 6. The results of the proposed NNmodels are quite satisfactory (R ¼ 0:98, std. dev: ¼ 0:06).Moreover, the results of proposed NN model are comparedwith 10 different FRP confinement models and are foundto be by far more accurate as given in Table 5. Results ofthe proposed NN model and 10 different FRP confinementmodels are given in Table 6.The outcomes of this study offer original contributions.

The FRP confinement models in the literature areproposed as a function of unconfined compressive strengthand ultimate confinement pressure where the NN model

Inputs

D

nt

Ef

F'co

f'cc

Output

biasbiasbias

selection process.

ARTICLE IN PRESS

START

CHOOSE TRAINING]ALGORITH

NODE(2)=0(2. HIDDEN LAYER)

NODE(1)=1(1. HIDDEN LAYER)

EPOCH=50

KTRIAL=1

TRAIN NET

SIMULATENETWORK

COMPUTE TESTSET ERROR

NO

ERR<ERR(TARGET)

YES

NET=BESTNET

STORE BESTNET

YES YES YESYES

NO NO NO NO

KTRIAL=10 EPOCH=100 NODE(1)=30 NODE(2)=30

KTRIAL=KTRAIL+1

EPOCH=EPOCH+10

NODE(1)=NODE(1)+1

NODE(2)=NODE(2)+1

END

Fig. 5. Flowchart of optimal NN selection.

A. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763 755

proposed in this study considers the effects of diameter ofthe concrete cylinder, total thickness of FRP layer,modulus of elasticity of the FRP laminate and unconfinedcompressive strength at the same time. This leads to moreaccurate modeling. NNs are treated as black box in theliterature. However, in this study the proposed NN modelis no more a black box as it is given in explicit form as amathematical function.

7. Explicit formulation of the NN model

The main focus is to obtain the explicit formula-tion of the compressive strength of CFRP confinedconcrete cylinder as a function of variables given asfollows:

f 0cc ¼ f ðD; nt;Ef ; f coÞ. (13)

ARTICLE IN PRESS

Table 4

Statistical parameters of the proposed NN model

MSE RMSE SSE MAPE (%)

NN training set 18.76 4.33 1895.6 3.88

NN test set 77.06 8.77 770.6 8.41

Inputs

D

nt

Ef

f'co

f'cc

Output

biasbias

Fig. 6. Proposed NN model for the prediction of f 0cc.

Fig. 7. % error of test set for NN model.

Fig. 8. % error of train set for NN model.

Fig. 9. Performance of test set for NN model.

A. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763756

The explicit formulation for the proposed NN model isobtained by using the well-trained NN parameters whichare biases, and weights for the input and hidden layer andthe normalization factors both for inputs and outputproposed NN model. An important step in the explicitformulation of NN models is the normalization process.

The output has been normalized by 150 to make the outputless than 1 in the calculations. That is why the output ismultiplied by 150 given in Eq. (14). The normalizationvalues of the inputs have been directly multiplied withweights given in Eqs. (15)–(29). Thus they are not givenexplicitly as the derivation of the explicit formulation is toocomplex particularly for those who do not have an NNbackground. Detailed information about the derivationcan be found in Refs. [20,21]. The same steps can be givenin a simpler form as follows:

f 0cc ¼ 150 �2

1þ e�2W� 1

� �, (14)

ARTICLE IN PRESSA. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763 757

where

W ¼ ð0:63Þ �2

1þ e�2U1� 1

� �þ ð0:74Þ �

2

1þ e�2U2� 1

� �

þ ð�3:16Þ �2

1þ e�2U3� 1

� �

þ ð�2:62Þ �2

1þ e�2U4� 1

� �

þ ð�0:68Þ �2

1þ e�2U5� 1

� �

þ ð1:16Þ �2

1þ e�2U6� 1

� �

þ ð�1:36Þ �2

1þ e�2U7� 1

� �

þ ð0:61Þ �2

1þ e�2U8� 1

� �þ ð0:83Þ �

2

1þ e�2U9� 1

� �

þ ð�0:52Þ �2

1þ e�2U10� 1

� �

þ ð�0:61Þ �2

1þ e�2U11� 1

� �

þ ð�2:57Þ �2

1þ e�2U12� 1

� �

Table 5

Statistics of performance and accuracy of ðf 0cc=f 0coÞ of proposed NN model an

Model NN/test Eq. (1)/test Eq. (2)/test Eq. (3)/test Eq. (4)/test Eq.

Mean 1.00 1.31 1.33 1.25 1.20 0.98

Std. dev. 0.06 0.34 0.28 0.19 0.28 0.18

R 0.98 0.87 0.87 0.85 0.86 0.77

Fig. 10. Performance of train set for NN model.

þ ð1:6Þ �2

1þ e�2U13� 1

� �

þ ð1:84Þ �2

1þ e�2U14� 1

� �

þ ð�3:77Þ �2

1þ e�2U15� 1

� �þ 0:25,

U1 ¼ ð0:024 �DÞ þ ð0:59 � ntÞ þ ð0:0004 � Ef Þ

þ ð0:037 � f coÞ þ 14:02, ð15Þ

U2 ¼ ð0:0217 �DÞ þ ð1:56 � ntÞ þ ð�0:0003 � Ef Þ

þ ð0:0346 � f coÞ � 4:42, ð16Þ

U3 ¼ ð�0:07 �DÞ þ ð�0:1 � ntÞ þ ð�0:0013 � Ef Þ

þ ð0:073 � f coÞ þ 16:12, ð17Þ

U4 ¼ ð0:058 �DÞ þ ð�0:96 � ntÞ þ ð�0:0028 � Ef Þ

þ ð�0:041 � f coÞ þ 1:42, ð18Þ

U5 ¼ ð0:061 �DÞ þ ð�0:138 � ntÞ þ ð�0:0006 � Ef Þ

þ ð�0:06 � f coÞ � 4:02, ð19Þ

U6 ¼ ð�0:0639 �DÞ þ ð�0:6017 � ntÞ þ ð�0:0014 � Ef Þ

þ ð�0:0327 � f coÞ þ 15:32, ð20Þ

U7 ¼ ð�0:0365 �DÞ þ ð�0:4598 � ntÞ þ ð�0:0004 � Ef Þ

þ ð0:0691 � f coÞ þ 2:07, ð21Þ

U8 ¼ ð�0:0684 �DÞ þ ð0:1734 � ntÞ þ ð0:0006 � Ef Þ

þ ð�0:0381 � f coÞ þ 9:33, ð22Þ

U9 ¼ ð0:0044 �DÞ þ ð0:2966 � ntÞ þ ð0:0008 � Ef Þ

þ ð0:0736 � f coÞ � 6:82, ð23Þ

U10 ¼ ð0:0559 �DÞ þ ð1:3957 � ntÞ þ ð0:0004 � Ef Þ

þ ð0:0160 � f coÞ � 10:58, ð24Þ

U11 ¼ ð0:0434 �DÞ þ ð�0:7968 � ntÞ þ ð0:0014 � Ef Þ

þ ð�0:0164 � f coÞ þ 3:41, ð25Þ

d various models compared to experimental results

(5)/test Eq. (6)/test Eq. (7)/test Eq. (8)/test Eq. (9)/test Eq. (10)/test

1.06 1.29 1.03 1.01 1.00

0.17 0.26 0.16 0.19 0.46

0.87 0.87 0.87 0.87 0.87

ARTIC

LEIN

PRES

STable 6

Results of the NN model vs. experimental and theoretical results

Ref. Code D (mm) nt (mm) Ef

(MPa)

f co

(MPa)

f cc test

(MPa)

f cc NN

(MPa)f 0cc=f 0cotest

f 0cc=f 0coNN

f 0cc=f 0coEq. (1)

f 0cc=f 0coEq. (2)

f 0cc=f 0coEq. (3)

f 0cc=f 0coEq. (4)

f 0cc=f 0coEq. (5)

f 0cc=f 0coEq. (6)

f 0cc=f 0coEq. (7)

Miyauchi et al. [5] MI1 150 0.11 3481 45.2 59.4 59.73 1.31 1.32 1.46 1.57 1.61 1.38 1.29 1.42 1.54

MI2 150 0.22 3481 45.2 79.4 78.27 1.76 1.73 1.93 2.03 2.08 1.80 1.58 1.68 2.00

MI3 150 0.11 3481 31.2 52.4 51.99 1.68 1.67 1.67 1.78 1.82 1.56 1.29 1.60 1.74

MI4 150 0.22 3481 31.2 67.4 67.07 2.16 2.15 2.34 2.42 2.37 2.15 1.58 1.98 2.36

MI5 150 0.33 3481 31.2 81.7 82.10 2.62 2.63 3.01 3.01 2.75 2.71 1.88 2.30 2.91

MI6 100 0.11 3481 51.9 75.2 73.66 1.45 1.42 1.61 1.71 1.78 1.52 1.44 1.48 1.70

MI7 100 0.22 3481 51.9 104.6 106.9 2.02 2.06 2.21 2.30 2.29 2.05 1.88 1.78 2.26

MI8 100 0.11 3481 33.7 69.6 73.95 2.07 2.19 1.93 2.03 2.08 1.80 1.44 1.74 2.00

MI9 100 0.22 3481 33.7 88 87.89 2.61 2.61 2.86 2.88 2.67 2.57 1.88 2.20 2.78

MI10 150 0.11 3481 45.2 59.4 59.73 1.31 1.32 1.46 1.57 1.61 1.38 1.29 1.42 1.54

Kono et al. [6] KO1 100 0.167 3820 34.3 57.4 58.85 1.67 1.72 2.53 2.58 2.48 2.29 1.73 2.04 2.50

KO2 100 0.167 3820 34.3 64.9 58.85 1.89 1.72 2.53 2.58 2.48 2.29 1.73 2.04 2.50

KO3 100 0.167 3820 32.3 58.2 56.36 1.8 1.74 2.62 2.66 2.55 2.39 1.73 2.10 2.61

KO4 100 0.167 3820 32.3 61.8 56.36 1.91 1.74 2.62 2.66 2.55 2.39 1.73 2.10 2.61

KO5 100 0.167 3820 32.3 57.7 56.36 1.79 1.74 2.62 2.66 2.55 2.39 1.73 2.10 2.61

KO6 100 0.334 3820 32.3 61.8 72.90 1.8 2.26 4.24 4.02 3.24 3.75 2.46 2.79 3.86

KO7 100 0.334 3820 32.3 80.2 72.90 1.91 2.26 4.24 4.02 3.24 3.75 2.46 2.79 3.86

KO8 100 0.334 3820 32.3 58.2 72.90 2.48 2.26 4.24 4.02 3.24 3.75 2.46 2.79 3.86

KO9 100 0.501 3820 32.3 86.9 91.28 2.69 2.83 5.86 5.28 3.65 5.15 3.19 3.38 5.06

KO10 100 0.501 3820 32.3 90.1 91.28 2.79 2.83 5.86 5.28 3.65 5.15 3.19 3.38 5.06

KO11 100 0.167 3820 34.8 57.8 59.55 1.66 1.71 2.50 2.56 2.48 2.29 1.73 2.02 2.50

KO12 100 0.167 3820 34.8 55.6 59.55 1.6 1.71 2.50 2.56 2.48 2.29 1.73 2.02 2.50

KO13 100 0.167 3820 34.8 50.7 59.55 1.46 1.71 2.50 2.56 2.48 2.29 1.73 2.02 2.50

KO14 100 0.334 3820 34.8 82.7 80.61 2.38 2.32 4.01 3.83 3.16 3.54 2.46 2.66 3.68

KO15 100 0.334 3820 34.8 81.4 80.61 2.34 2.32 4.01 3.83 3.16 3.54 2.46 2.66 3.68

KO16 100 0.501 3820 34.8 103.3 103.9 2.97 2.99 5.51 5.02 3.58 4.83 3.19 3.21 4.80

KO17 100 0.501 3820 34.8 110.1 103.9 3.16 2.99 5.51 5.02 3.58 4.83 3.19 3.21 4.80

Matthys et al. [7] MA1 150 0.117 2600 34.9 46.1 46.33 1.32 1.33 1.48 1.58 1.66 1.42 1.23 1.46 1.58

MA2 150 0.235 1100 34.9 45.8 46.03 1.31 1.32 1.41 1.51 1.57 1.35 1.20 1.41 1.49

Shahawy et al. [8] SH1 153 0.36 2275 19.4 33.8 33.00 1.74 1.70 3.27 3.23 2.87 2.92 1.61 2.63 3.11

SH2 153 0.66 2275 19.4 46.4 46.78 2.39 2.41 5.16 4.75 3.51 4.55 2.13 3.49 4.56

SH3 153 0.9 2275 19.4 62.6 62.49 3.23 3.22 6.67 5.89 3.78 5.81 2.54 4.09 5.60

SH4 153 1.08 2275 19.4 75.7 73.61 3.9 3.79 7.81 6.72 3.91 6.79 2.84 4.52 6.38

Shahawy et al. [8] SH5 153 1.25 2275 19.4 80.2 81.79 4.13 4.22 8.88 7.49 3.99 7.69 3.13 4.90 7.09

SH6 153 0.36 2275 49 59.1 59.74 1.21 1.22 1.90 2.00 2.04 1.77 1.61 1.65 1.97

SH7 153 0.66 2275 49 76.5 78.83 1.56 1.61 2.65 2.69 2.55 2.39 2.13 1.99 2.61

SH8 153 0.9 2275 49 98.8 95.61 2.02 1.95 3.25 3.21 2.87 2.92 2.54 2.23 3.11

SH9 153 1.08 2275 49 112.7 113.8 2.3 2.32 3.70 3.58 3.06 3.30 2.84 2.39 3.46

RL1 100 0.6 1265 42 73.5 71.46 1.75 1.70 2.48 2.54 2.45 2.25 1.87 1.96 2.47

Rochette and Labossiere [9] RL2 100 0.6 1265 42 73.5 71.46 1.75 1.70 2.48 2.54 2.45 2.25 1.87 1.96 2.47

RL3 100 0.6 1265 42 67.62 71.46 1.61 1.70 2.48 2.54 2.45 2.25 1.87 1.96 2.47

RL4 150 1.26 230 43 47.3 47.52 1.1 1.11 1.37 1.47 1.52 1.31 1.22 1.36 1.45

RL5 150 2.52 230 43 58.91 59.16 1.37 1.38 1.74 1.85 1.90 1.63 1.44 1.58 1.81

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PRES

SRL6 150 3.78 230 43 70.95 70.61 1.65 1.64 2.11 2.20 2.20 1.94 1.66 1.78 2.15

RL7 150 5.04 230 43 74.39 74.63 1.73 1.74 2.47 2.54 2.45 2.25 1.88 1.95 2.47

Micelli et al. [10] MC1 100 0.35 1520 32 54 51.50 1.69 1.61 2.36 2.44 2.37 2.15 1.61 1.98 2.36

MC2 100 0.35 1520 32 48 51.50 1.5 1.61 2.36 2.44 2.37 2.15 1.61 1.98 2.36

MC3 100 0.35 1520 32 54 51.50 1.69 1.61 2.36 2.44 2.37 2.15 1.61 1.98 2.36

MC4 100 0.35 1520 32 50 51.50 1.56 1.61 2.36 2.44 2.37 2.15 1.61 1.98 2.36

MC5 100 0.16 3790 37 60 59.60 1.62 1.61 2.34 2.42 2.37 2.15 1.69 1.93 2.36

MC6 100 0.16 3790 37 62 59.60 1.68 1.61 2.34 2.42 2.37 2.15 1.69 1.93 2.36

MC7 100 0.16 3790 37 59 59.60 1.59 1.61 2.34 2.42 2.37 2.15 1.69 1.93 2.36

MC8 100 0.16 3790 37 57 59.60 1.54 1.61 2.34 2.42 2.37 2.15 1.69 1.93 2.36

Rousakis [11] RO1 150 0.169 2024 25.15 44.13 43.02 1.75 1.71 1.74 1.85 1.90 1.63 1.26 1.69 1.81

RO2 150 0.169 2024 25.15 41.56 43.02 1.65 1.71 1.74 1.85 1.90 1.63 1.26 1.69 1.81

RO3 150 0.169 2024 25.15 38.75 43.02 1.54 1.71 1.74 1.85 1.90 1.63 1.26 1.69 1.81

RO4 150 0.338 2024 25.15 60.09 55.94 2.39 2.22 2.49 2.55 2.45 2.25 1.52 2.12 2.47

RO5 150 0.338 2024 25.15 55.93 55.94 2.22 2.22 2.49 2.55 2.45 2.25 1.52 2.12 2.47

RO6 150 0.338 2024 25.15 61.61 55.94 2.45 2.22 2.49 2.55 2.45 2.25 1.52 2.12 2.47

RO7 150 0.507 2024 25.15 67 69.94 2.66 2.78 3.23 3.19 2.85 2.88 1.78 2.49 3.07

RO8 150 0.507 2024 25.15 67.27 69.94 2.67 2.78 3.23 3.19 2.85 2.88 1.78 2.49 3.07

RO9 150 0.507 2024 25.15 70.18 69.94 2.79 2.78 3.23 3.19 2.85 2.88 1.78 2.49 3.07

RO10 150 0.169 2024 47.44 72.26 65.46 1.52 1.38 1.39 1.49 1.57 1.35 1.26 1.37 1.49

RO11 150 0.169 2024 47.44 64.4 65.46 1.36 1.38 1.39 1.49 1.57 1.35 1.26 1.37 1.49

RO12 150 0.169 2024 47.44 66.19 65.46 1.4 1.38 1.39 1.49 1.57 1.35 1.26 1.37 1.49

RO13 150 0.338 2024 47.44 82.36 83.93 1.74 1.77 1.79 1.90 1.94 1.66 1.52 1.59 1.85

RO14 150 0.338 2024 47.44 82.35 83.93 1.74 1.77 1.79 1.90 1.94 1.66 1.52 1.59 1.85

RO15 150 0.338 2024 47.44 79.11 83.93 1.67 1.77 1.79 1.90 1.94 1.66 1.52 1.59 1.85

RO16 150 0.507 2024 47.44 96.29 95.75 2.03 2.02 2.18 2.27 2.26 2.01 1.78 1.79 2.22

RO17 150 0.507 2024 47.44 95.22 95.75 2.01 2.02 2.18 2.27 2.26 2.01 1.78 1.79 2.22

RO18 150 0.507 2024 47.44 103.9 95.75 2.19 2.02 2.18 2.27 2.26 2.01 1.78 1.79 2.22

RO19 150 0.169 2024 51.84 78.65 73.73 1.52 1.42 1.36 1.46 1.52 1.31 1.26 1.33 1.45

RO20 150 0.169 2024 51.84 79.18 73.73 1.53 1.42 1.36 1.46 1.52 1.31 1.26 1.33 1.45

RO21 150 0.169 2024 51.84 72.76 73.73 1.4 1.42 1.36 1.46 1.52 1.31 1.26 1.33 1.45

RO22 150 0.338 2024 51.84 95.4 93.79 1.84 1.81 1.72 1.83 1.90 1.63 1.52 1.54 1.81

RO23 150 0.338 2024 51.84 90.3 93.79 1.74 1.81 1.72 1.83 1.90 1.63 1.52 1.54 1.81

RO24 150 0.338 2024 51.84 90.65 93.79 1.75 1.81 1.72 1.83 1.90 1.63 1.52 1.54 1.81

RO25 150 0.507 2024 51.84 110.5 106.7 2.13 2.06 2.08 2.18 2.17 1.91 1.78 1.72 2.11

RO26 150 0.507 2024 51.84 103.6 106.7 2 2.06 2.08 2.18 2.17 1.91 1.78 1.72 2.11

RO27 150 0.507 2024 51.84 117.2 106.7 2.26 2.06 2.08 2.18 2.17 1.91 1.78 1.72 2.11

RO28 150 0.845 2024 51.84 112.6 120.6 2.17 2.33 2.80 2.83 2.64 2.53 2.30 2.03 2.74

RO29 150 0.845 2024 51.84 126.6 120.6 2.44 2.33 2.80 2.83 2.64 2.53 2.30 2.03 2.74

RO30 150 0.845 2024 51.84 137.9 120.6 2.66 2.33 2.80 2.83 2.64 2.53 2.30 2.03 2.74

RO31 150 0.169 2024 70.48 87.29 85.95 1.24 1.22 1.27 1.35 1.36 1.21 1.26 1.25 1.32

RO32 150 0.169 2024 70.48 84.03 85.95 1.19 1.22 1.27 1.35 1.36 1.21 1.26 1.25 1.32

RO33 150 0.169 2024 70.48 83.22 85.95 1.18 1.22 1.27 1.35 1.36 1.21 1.26 1.25 1.32

RO34 150 0.338 2024 70.48 94.06 98.05 1.33 1.39 1.53 1.64 1.70 1.45 1.52 1.40 1.62

RO35 150 0.338 2024 70.48 98.13 98.05 1.39 1.39 1.53 1.64 1.70 1.45 1.52 1.40 1.62

RO36 150 0.338 2024 70.48 107.2 98.05 1.52 1.39 1.53 1.64 1.70 1.45 1.52 1.40 1.62

RO37 150 0.507 2024 70.48 114.1 111.5 1.62 1.58 1.80 1.90 1.94 1.66 1.78 1.53 1.85

RO38 150 0.507 2024 70.48 108.0 111.5 1.53 1.58 1.80 1.90 1.94 1.66 1.78 1.53 1.85

RO39 150 0.507 2024 70.48 110.3 111.5 1.57 1.58 1.80 1.90 1.94 1.66 1.78 1.53 1.85

RO40 150 0.169 2024 82.13 94.08 94.01 1.14 1.14 1.23 1.31 1.36 1.21 1.26 1.21 1.32

RO41 150 0.169 2024 82.13 97.6 94.01 1.16 1.14 1.23 1.31 1.36 1.21 1.26 1.21 1.32

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STable 6 (continued )

Ref. Code D (mm) nt (mm) Ef

(MPa)

f co

(MPa)

f cc test

(MPa)

f cc NN

(MPa)f 0cc=f 0cotest

f 0cc=f 0coNN

f 0cc=f 0coEq. (1)

f 0cc=f 0coEq. (2)

f 0cc=f 0coEq. (3)

f 0cc=f 0coEq. (4)

f 0cc=f 0coEq. (5)

f 0cc=f 0coEq. (6)

f 0cc=f 0coEq. (7)

RO42 150 0.169 2024 82.13 95.83 94.01 1.15 1.14 1.23 1.31 1.36 1.21 1.26 1.21 1.32

RO43 150 0.338 2024 82.13 97.43 104.10 1.15 1.27 1.46 1.56 1.61 1.38 1.52 1.34 1.54

RO44 150 0.338 2024 82.13 98.85 104.1 1.2 1.27 1.46 1.56 1.61 1.38 1.52 1.34 1.54

RO45 150 0.338 2024 82.13 98.24 104.1 1.16 1.27 1.46 1.56 1.61 1.38 1.52 1.34 1.54

RO46 150 0.507 2024 82.13 124.2 122.8 1.51 1.50 1.68 1.79 1.86 1.59 1.78 1.46 1.78

RO47 150 0.507 2024 82.13 129.5 122.8 1.58 1.50 1.68 1.79 1.86 1.59 1.78 1.46 1.78

RO48 150 0.507 2024 82.13 120.3 122.8 1.47 1.50 1.68 1.79 1.86 1.59 1.78 1.46 1.78

Ref. Code f 0cc=f 0coEq. (8)

f 0cc=f 0coEq. (9)

f 0cc=f 0coEq. (10)

NN/

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

Miyauchi et al. [5] MI1 1.34 1.19 1.05 1.01 1.12 1.20 1.23 1.06 0.99 1.08 1.17 1.03 0.91 0.80

MI2 1.64 1.64 1.52 0.98 1.09 1.15 1.18 1.02 0.90 0.95 1.14 0.93 0.93 0.86

MI3 1.47 1.40 1.41 0.99 0.99 1.06 1.08 0.93 0.77 0.95 1.03 0.88 0.83 0.84

MI4 1.87 1.92 2.10 1.00 1.08 1.12 1.10 1.00 0.73 0.92 1.09 0.86 0.89 0.97

MI5 2.21 2.30 2.76 1.00 1.15 1.15 1.05 1.03 0.72 0.88 1.11 0.84 0.88 1.05

MI6 1.45 1.36 1.12 0.98 1.11 1.18 1.23 1.05 0.99 1.02 1.17 1.00 0.94 0.77

MI7 1.80 1.84 1.73 1.02 1.09 1.14 1.13 1.01 0.93 0.88 1.12 0.89 0.91 0.86

MI8 1.64 1.64 1.66 1.06 0.93 0.98 1.00 0.87 0.69 0.84 0.97 0.79 0.79 0.80

MI9 2.12 2.21 2.57 1.00 1.10 1.10 1.02 0.98 0.72 0.84 1.06 0.81 0.85 0.98

MI10 1.34 1.19 1.05 1.01 1.12 1.20 1.23 1.06 0.99 1.08 1.17 1.03 0.91 0.80

Kono et al. [6] KO1 1.95 2.02 2.20 1.03 1.51 1.55 1.48 1.37 1.04 1.22 1.50 1.17 1.21 1.32

KO2 1.95 2.02 2.20 0.91 1.34 1.37 1.31 1.21 0.92 1.08 1.32 1.03 1.07 1.16

KO3 2.02 2.10 2.34 0.97 1.46 1.48 1.42 1.33 0.96 1.17 1.45 1.12 1.17 1.30

KO4 2.02 2.10 2.34 0.91 1.37 1.40 1.34 1.25 0.91 1.10 1.36 1.06 1.10 1.23

KO5 2.02 2.10 2.34 0.97 1.46 1.49 1.43 1.34 0.97 1.18 1.46 1.13 1.17 1.31

KO6 2.80 2.87 3.95 1.25 2.36 2.23 1.80 2.09 1.37 1.55 2.15 1.56 1.59 2.19

KO7 2.80 2.87 3.95 1.18 2.22 2.11 1.70 1.97 1.29 1.46 2.02 1.47 1.50 2.07

KO8 2.80 2.87 3.95 0.91 1.71 1.62 1.31 1.51 0.99 1.13 1.56 1.13 1.16 1.59

KO9 3.55 3.47 5.58 1.05 2.18 1.96 1.36 1.91 1.19 1.26 1.88 1.32 1.29 2.08

KO10 3.55 3.47 5.58 1.01 2.10 1.89 1.31 1.84 1.14 1.21 1.81 1.27 1.24 2.00

KO11 1.95 2.02 2.19 1.03 1.51 1.54 1.49 1.38 1.04 1.22 1.51 1.18 1.22 1.32

KO12 1.95 2.02 2.19 1.07 1.56 1.60 1.55 1.43 1.08 1.27 1.56 1.22 1.27 1.37

KO13 1.95 2.02 2.19 1.17 1.71 1.75 1.70 1.57 1.18 1.39 1.71 1.34 1.39 1.50

KO14 2.69 2.76 3.67 0.97 1.68 1.61 1.33 1.49 1.03 1.12 1.55 1.13 1.16 1.54

KO15 2.69 2.76 3.67 0.99 1.71 1.64 1.35 1.51 1.05 1.14 1.57 1.15 1.18 1.57

KO16 3.38 3.35 5.19 1.01 1.86 1.69 1.20 1.63 1.07 1.08 1.61 1.14 1.13 1.75

KO17 3.38 3.35 5.19 0.95 1.74 1.59 1.13 1.53 1.01 1.02 1.52 1.07 1.06 1.64

Matthys et al. [7] MA1 1.37 1.24 1.27 1.01 1.12 1.20 1.25 1.07 0.93 1.10 1.19 1.04 0.94 0.96

MA2 1.32 1.15 1.45 1.01 1.07 1.15 1.19 1.03 0.91 1.08 1.14 1.01 0.88 1.11

Shahawy et al. [8] SH1 2.33 2.42 2.96 0.98 1.88 1.85 1.65 1.68 0.93 1.51 1.78 1.34 1.39 1.70

SH2 3.24 3.23 4.88 1.01 2.16 1.99 1.47 1.91 0.89 1.46 1.91 1.35 1.35 2.04

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ARTIC

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PRES

SSH3 3.88 3.72 6.36 1.00 2.07 1.82 1.17 1.80 0.79 1.27 1.73 1.20 1.15 1.97

SH4 4.37 4.07 7.51 0.97 2.00 1.72 1.00 1.74 0.73 1.16 1.64 1.12 1.04 1.92

Shahawy et al. [8] SH5 4.81 4.36 8.57 1.02 2.15 1.81 0.97 1.86 0.76 1.19 1.72 1.16 1.05 2.08

SH6 1.62 1.61 0.99 1.01 1.57 1.66 1.69 1.46 1.33 1.36 1.63 1.34 1.33 0.82

SH7 2.02 2.10 1.73 1.03 1.70 1.72 1.64 1.53 1.36 1.27 1.67 1.29 1.34 1.11

SH8 2.33 2.42 2.34 0.97 1.61 1.59 1.42 1.44 1.26 1.10 1.54 1.15 1.20 1.16

SH9 2.55 2.64 2.79 1.01 1.61 1.56 1.33 1.43 1.24 1.04 1.50 1.11 1.15 1.21

RL1 1.93 2.00 2.10 0.97 1.42 1.45 1.40 1.29 1.07 1.12 1.41 1.10 1.14 1.20

Rochette and Labossiere [9] RL2 1.93 2.00 2.10 0.97 1.42 1.45 1.40 1.29 1.07 1.12 1.41 1.10 1.14 1.20

RL3 1.93 2.00 2.10 1.06 1.54 1.58 1.52 1.40 1.16 1.22 1.53 1.20 1.24 1.30

RL4 1.29 1.10 0.92 1.00 1.24 1.33 1.38 1.19 1.11 1.24 1.32 1.17 1.00 0.84

RL5 1.52 1.47 1.29 1.00 1.27 1.35 1.39 1.19 1.05 1.16 1.32 1.11 1.08 0.94

RL6 1.73 1.76 1.66 1.00 1.28 1.33 1.33 1.18 1.01 1.08 1.30 1.05 1.07 1.01

RL7 1.93 2.00 2.03 1.00 1.43 1.47 1.42 1.30 1.09 1.13 1.43 1.12 1.16 1.17

Micelli et al. [10] MC1 1.87 1.92 1.95 0.95 1.40 1.44 1.41 1.27 0.95 1.17 1.40 1.10 1.14 1.16

MC2 1.87 1.92 1.95 1.07 1.58 1.62 1.58 1.43 1.07 1.32 1.58 1.24 1.28 1.30

MC3 1.87 1.92 1.95 0.95 1.40 1.44 1.41 1.27 0.95 1.17 1.40 1.10 1.14 1.16

MC4 1.87 1.92 1.95 1.03 1.51 1.56 1.52 1.38 1.03 1.27 1.52 1.20 1.23 1.25

MC5 1.87 1.92 1.99 0.99 1.45 1.49 1.47 1.33 1.05 1.19 1.46 1.15 1.19 1.23

MC6 1.87 1.92 1.99 0.96 1.40 1.44 1.41 1.28 1.01 1.15 1.41 1.11 1.14 1.18

MC7 1.87 1.92 1.99 1.01 1.47 1.52 1.49 1.35 1.07 1.21 1.49 1.17 1.21 1.25

MC8 1.87 1.92 1.99 1.05 1.52 1.57 1.54 1.40 1.10 1.25 1.54 1.21 1.25 1.29

Rousakis [11] RO1 1.52 1.47 1.52 0.98 1.00 1.06 1.09 0.93 0.72 0.97 1.04 0.87 0.84 0.87

RO2 1.52 1.47 1.52 1.04 1.06 1.12 1.15 0.99 0.76 1.02 1.10 0.92 0.89 0.92

RO3 1.52 1.47 1.52 1.11 1.13 1.20 1.23 1.06 0.82 1.10 1.18 0.99 0.96 0.99

RO4 1.93 2.00 2.26 0.93 1.04 1.07 1.03 0.94 0.64 0.89 1.03 0.81 0.84 0.94

RO5 1.93 2.00 2.26 1.00 1.12 1.15 1.11 1.02 0.69 0.96 1.11 0.87 0.90 1.02

RO6 1.93 2.00 2.26 0.91 1.02 1.04 1.00 0.92 0.62 0.87 1.01 0.79 0.82 0.92

RO7 2.31 2.40 2.99 1.05 1.21 1.20 1.07 1.08 0.67 0.94 1.16 0.87 0.90 1.13

RO8 2.31 2.40 2.99 1.04 1.21 1.20 1.07 1.08 0.67 0.93 1.15 0.87 0.90 1.12

RO9 2.31 2.40 2.99 1.00 1.16 1.14 1.02 1.03 0.64 0.89 1.10 0.83 0.86 1.07

RO10 1.32 1.15 0.88 0.91 0.92 0.98 1.03 0.89 0.83 0.90 0.98 0.87 0.76 0.58

RO11 1.32 1.15 0.88 1.01 1.03 1.10 1.15 0.99 0.93 1.00 1.10 0.97 0.84 0.65

RO12 1.32 1.15 0.88 0.99 1.00 1.07 1.12 0.96 0.90 0.98 1.07 0.94 0.82 0.63

RO13 1.55 1.51 1.28 1.02 1.03 1.09 1.11 0.96 0.87 0.92 1.07 0.89 0.87 0.73

RO14 1.55 1.51 1.28 1.02 1.03 1.09 1.11 0.96 0.87 0.92 1.07 0.89 0.87 0.73

RO15 1.55 1.51 1.28 1.06 1.07 1.14 1.16 1.00 0.91 0.95 1.11 0.93 0.90 0.76

RO16 1.78 1.82 1.68 0.99 1.08 1.12 1.11 0.99 0.88 0.88 1.09 0.88 0.89 0.83

RO17 1.78 1.82 1.68 1.00 1.09 1.13 1.13 1.00 0.89 0.89 1.11 0.88 0.90 0.83

RO18 1.78 1.82 1.68 0.92 1.00 1.04 1.03 0.92 0.81 0.82 1.01 0.81 0.83 0.77

RO19 1.29 1.10 0.79 0.94 0.90 0.96 1.00 0.86 0.83 0.88 0.96 0.85 0.72 0.52

RO20 1.29 1.10 0.79 0.93 0.89 0.95 0.99 0.86 0.82 0.87 0.95 0.84 0.72 0.52

RO21 1.29 1.10 0.79 1.02 0.97 1.04 1.08 0.94 0.90 0.95 1.04 0.92 0.79 0.56

RO22 1.52 1.47 1.16 0.98 0.94 0.99 1.03 0.88 0.83 0.84 0.99 0.83 0.80 0.63

RO23 1.52 1.47 1.16 1.04 0.99 1.05 1.09 0.94 0.87 0.89 1.04 0.87 0.85 0.67

RO24 1.52 1.47 1.16 1.03 0.98 1.05 1.09 0.93 0.87 0.88 1.04 0.87 0.84 0.66

RO25 1.71 1.73 1.51 0.97 0.98 1.02 1.02 0.89 0.84 0.81 0.99 0.80 0.81 0.71

RO26 1.71 1.73 1.51 1.03 1.04 1.09 1.09 0.95 0.89 0.86 1.06 0.85 0.86 0.76

RO27 1.71 1.73 1.51 0.91 0.92 0.96 0.96 0.84 0.79 0.76 0.94 0.76 0.77 0.67

RO28 2.10 2.19 2.24 1.07 1.29 1.30 1.22 1.17 1.06 0.94 1.26 0.97 1.01 1.03

A.

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ARTIC

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PRES

STable 6 (continued )

Ref. Code f 0cc=f 0coEq. (8)

f 0cc=f 0coEq. (9)

f 0cc=f 0coEq. (10)

NN/

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

RO29 2.10 2.19 2.24 0.95 1.15 1.16 1.08 1.04 0.94 0.83 1.12 0.86 0.90 0.92

RO30 2.10 2.19 2.24 0.88 1.05 1.06 0.99 0.95 0.87 0.76 1.03 0.79 0.82 0.84

RO31 1.21 0.93 0.51 0.98 1.02 1.09 1.10 0.98 1.02 1.01 1.06 0.97 0.75 0.41

RO32 1.21 0.93 0.51 1.02 1.06 1.14 1.15 1.02 1.06 1.05 1.11 1.01 0.79 0.43

RO33 1.21 0.93 0.51 1.03 1.07 1.15 1.16 1.02 1.07 1.06 1.12 1.02 0.79 0.43

RO34 1.40 1.28 0.72 1.05 1.15 1.23 1.28 1.09 1.14 1.05 1.22 1.05 0.96 0.54

RO35 1.40 1.28 0.72 1.00 1.10 1.18 1.22 1.05 1.09 1.01 1.16 1.00 0.92 0.52

RO36 1.40 1.28 0.72 0.92 1.01 1.08 1.12 0.96 1.00 0.92 1.06 0.92 0.84 0.48

RO37 1.55 1.51 0.99 0.98 1.11 1.17 1.20 1.03 1.10 0.95 1.14 0.95 0.93 0.61

RO38 1.55 1.51 0.99 1.03 1.17 1.24 1.27 1.09 1.17 1.00 1.21 1.01 0.99 0.65

RO39 1.55 1.51 0.99 1.01 1.14 1.21 1.23 1.06 1.14 0.98 1.18 0.98 0.96 0.63

RO40 1.21 0.93 0.21 1.00 1.08 1.15 1.20 1.06 1.11 1.06 1.16 1.06 0.82 0.18

RO41 1.21 0.93 0.21 0.99 1.06 1.13 1.18 1.04 1.09 1.04 1.14 1.04 0.81 0.18

RO42 1.21 0.93 0.21 1.00 1.07 1.14 1.19 1.05 1.10 1.05 1.15 1.05 0.81 0.18

RO43 1.34 1.19 0.51 1.10 1.27 1.36 1.40 1.20 1.32 1.17 1.34 1.17 1.04 0.44

RO44 1.34 1.19 0.51 1.06 1.21 1.30 1.34 1.15 1.27 1.12 1.28 1.12 1.00 0.42

RO45 1.34 1.19 0.51 1.09 1.25 1.34 1.39 1.19 1.31 1.16 1.32 1.16 1.03 0.44

RO46 1.50 1.44 0.72 0.99 1.11 1.19 1.23 1.05 1.18 0.96 1.18 0.99 0.95 0.48

RO47 1.50 1.44 0.72 0.95 1.07 1.13 1.18 1.01 1.13 0.92 1.12 0.95 0.91 0.46

RO48 1.50 1.44 0.72 1.02 1.14 1.22 1.27 1.08 1.21 0.99 1.21 1.02 0.98 0.49

Mean 1.00 1.31 1.33 1.25 1.20 0.98 1.06 1.29 1.03 1.01 1.00

Std. dev. 0.06 0.34 0.28 0.19 0.28 0.18 0.17 0.26 0.16 0.19 0.46

R 0.98 0.87 0.87 0.85 0.86 0.77 0.87 0.87 0.87 0.87 0.87

Bold sets are test sets.

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ARTICLE IN PRESSA. Cevik, I.H. Guzelbey / Building and Environment 43 (2008) 751–763 763

U12 ¼ ð�0:0309 �DÞ þ ð�1:0127 � ntÞ þ ð�0:0014 � Ef Þ

þ ð�0:0326 � f coÞ þ 6:09, ð26Þ

U13 ¼ ð�0:0638 �DÞ þ ð0:7750 � ntÞ þ ð0:0003 � Ef Þ

þ ð0:0633 � f coÞ þ 1:34, ð27Þ

U14 ¼ ð0:0920 �DÞ þ ð�0:6339 � ntÞ þ ð0:0007 � Ef Þ

þ ð�0:0032 � f coÞ � 2:86, ð28Þ

U15 ¼ ð�0:0288 �DÞ þ ð�0:1202 � ntÞ þ ð0:0028 � Ef Þ

þ ð0:0137 � f coÞ � 5:91. ð29Þ

It should be noted that the proposed explicit formulation ofthe NN GEP models presented above are valid only for theranges of experimental database given in Table 2 and forspecimen that have a length to diameter ratio of 2 ðL=D ¼ 2Þ.

8. Conclusion

This study presents an alternative formulation forstrength enhancement of CFRP wrapped concrete cylin-ders based on experimental results by means of NNs. Theoptimum NN architecture is obtained by a MATLABprogram that automatically finds the best NN model. Theresults of the proposed NN model compared to experi-mental results are found to be quite satisfactory (R ¼ 0:98,std. dev: ¼ 0:06). Moreover, the accuracy of the proposedNN model is compared with several FRP confinementmodels proposed in literature and is found to be by farmore accurate. The proposed NN model is also presentedin explicit form which enables the NN model to be used forpractical applications.

Appendix A

The details of the experimental databases are given inTable 6.

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