New model of cfrp confined circular concrete columns ann appr

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME
98
NEW MODEL OF CFRP-CONFINED CIRCULAR CONCRETE
COLUMNS: ANN APPROACH
Dr. Salim T. Yousif
Assistant Prof, Civil Engineering Dept., College of Engineering/University of Mosul, Iraq
ABSTRACT
The application of fiber-reinforced polymer (FRP) composites in civil engineering
works has increased in recent years, especially in the area of strengthening concrete columns.
The objective of this research is to develop new mathematical models for predicting the
confined compressive strength of carbon FRP (CFRP) circular concrete columns using
artificial neural networks (ANNs), which is done using 208 excremental data results collected
from the literature. Two mathematical models were developed: one depended on six input
parameters, whereas the other depended only on three important parameters, namely,
unconfined compressive strength of concrete, total thickness of the CFRP, and tensile
strength of CFRP along the hoop direction. Comparison of the new two models using
experimental data showed a good agreement and accuracy of the developed ANN models in
predicting the CFRP-confined compressive strength of circular concrete columns. The new
models were also used to perform a parametric study to evaluate the effect of the input
parameters on the CFRP-confined compressive strength of circular concrete columns.
Key Words: Artificial neural networks, Concrete, Compressive strength, Fiber-reinforced
polymer confinement, Mathematical modeling
1. INTRODUCTION
Externally bonded carbon fiber-reinforced polymer (CFRP) composite sheets and
laminates have been used widely in civil engineering construction to strengthen reinforced
concrete (RC) components [1] because of their high strength, light weight, ease in use,
durability against weather conditions, chemical resistance properties, relative low cost, and
ease in repair.
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 4, Issue 3, May - June (2013), pp. 98-110
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99
These CFRP composites are used for strengthening of columns [2, 3]. Confinement of
the columns using CFRP jackets is done by wrapping the fibers along the hoop direction of
the concrete columns.
Concrete expands laterally when subjected to axial compression. The FRP jacket
provides a confining pressure to the concrete to resist the expansion caused by the axial
compression. Ultimate failure occurs when the FRP jacket ruptures because of the tensile
stress along the hoop direction [3]. Because of the FRP confinement, both the compressive
strength and ultimate strain of the concrete can be improved [1].
Artificial neural networks (ANNs) have experienced increased interest over the last
years and have been successfully applied across a range of engineering problems, including
the strengthening of columns [4, 5], increasing the capacity of RC beams strengthened with
FRP reinforcements [6, 7], prediction of the compressive strength of concrete [8, 9],linear
and nonlinear model updating of RC T-beams [10], predicting the bond strength of FRP-to-
concrete joints [11], and many other engineering applications.
Naderpour et al. [2] employed the ANN to generate a model for predicting the
compressive strength of FRP-confined concrete independently from the network. The model
consisted of an empirical chart and seven mathematical equations.
In the present study, new mathematical models are developed based on ANNs using a
database built from existing tests on CFRP-confined circular concrete specimens. This new
model is then compared with the experimental data. Finally, the trained network model is
used to perform a parametric study to evaluate the effect of various parameters on the CFRP-
confined compressive strength of concrete.
2. AVAILABLE EMPIRICAL FRP-CONFINED MODELS
The confining pressure provided by the FRP jacket, as derived from empirical models,
is a function of the column’s diameter, stiffness of the FRP jacket, and compressive strength
of the unconfined concrete. A lateral confining stress f1 is produced in the concrete when the
confining jacket and the member is loaded such that the concrete starts to dilate and expands
laterally. The stress is related to the thickness and strength of the FRP by [3]:
fଵ ൌ
ଶ୤ూ౎ౌ୲
ୢ
(1)
where f୊ୖ୔ is the tensile strength of the FRP along the hoop direction, t is the total thickness
of the FRP, and d is the diameter of the confined concrete.
Several existing strength models for FRP-confined concrete take the following form
[3]:
୤ౙౙ
୤ౙ
′ ൌ 1 ൅ kଵ
୤భ
୤ౙ
′ (2)
wherefୡୡ and fୡ
′
are the compressive strength of the confined and unconfined concrete,
respectively,fଵ is the lateral confining pressure, and kଵis the confinement effectiveness
coefficient.
A number of strength models have been proposed specifically for the FRP-confined
concrete, which employ Eq. 2 with modified expressions for k1. The details of the models can
be seen in [12]

100
3. ARTIFICIAL NEURAL NETWORKS
A neural network is a computer model whose architecture essentially mimics the
knowledge acquisition and organizational skills of the human brain [13].The function of artificial
neurons is similar to that of real neurons [14]; they are able to communicate via signals sent
among them by a large number of biased and weighted connections. Each neuron has its own
transfer function, which describes how to convert a weighted sum of input to output.
The multi-layer perceptron is the most widely used type of ANN [15]. It is both simple
and based on solid mathematical grounds. The input quantities are processed through successive
layers of “neurons.”An input layer (with the number of neurons equal to the number of variables
in the problem) and an output layer always exist. The layers in between are called “hidden”
layers. Without a hidden layer, the perceptron can only perform linear tasks. All problems, which
can be solved by a perceptron, can be solved with only one hidden layer; however, using two or
more hidden layers is sometimes more efficient.
3.1 Back-propagation neural network
The back-propagation (BP) neural network is a multi-layered feed-forward [15, 16]. The
BP neural network adjusts internally the weight values to set the non-linear relationships between
the input and the output without giving explicitly the function expression. Further, the BP neural
network can be generalized for the input that is not included in the training patterns.
The BP algorithm is used to train the BP neural networks. This algorithm looks for the
minimum error function in the weight space using the method of gradient descent. The
combination of weights that minimizes the error function is considered to be a solution to the
learning problem. The input feed forward can be described by the following steps [15, 17]:
Once the input vector ix is introduced into the input layer, it can calculate the input to the
hidden layer H
Jh as
i
NI
i
jij
H
J xwbh ∑=
+=
1
(3)
where jb is the bias and jiw is the synaptic weight that connects input neuron i to hidden neuron j.
Each neuron of the hidden layer takes its input H
jh , uses it as the argument for a function,
and produces an output H
jy given by
)( H
j
H
j hfy = (4)
The input to the neurons of output layer o
kh is calculated as
∑=
+=
NH
j
H
jkjk
o
k ywbh
1
(5)
and the network output ky is given by
)( o
kk hfy = (6)
where f represents the activation function. Then

101
݂ሺ o
kh ሻ ൌ
ଵ
ଵା௘
ష
o
kh
. (7)
The complete algorithm can be found in [15].
3.2 Neural network design and training
A set of test results were collected from the literature for the axial compressive
strength of the circular confined concrete columns [18–41].The selected database contains
208 test results.
The data collected from the field were divided randomly into two groups. The first
group, which contained 188 results, was used in the training of the neural network, and the
other data group, which contained 20 results, was used to test the obtained networks. The
multi-layer feed-forward BP technique was implemented in the current research to develop
and train the neural network, where the sigmoid transform function was adopted.
Different training functions are available in MATLAB [42].The Levenberg–
Marquardt (LM) technique have been proven to be an efficient training function and are
therefore used to construct the ANN model. This training function is one of the conjugate
gradient algorithms that start the training by searching in the steepest descent direction
(negative of the gradient) on the first iteration. The LM algorithm is known to be significantly
faster than the more traditional gradient descent-type algorithms for training ANNs.
The input, as well as the output, was scaled in the range of 0.1 to 0.9. The scaling of
the training data sets was carried out using the following equation:
‫ݕ‬ ൌ
଴.଼ሺ௫ି௫೘ೌೣሻ
௫೘ೌೣି௫೘೔೙
൅ 0.9.. (8)
Any new input data should be scaled before being introduced to the network and the
corresponding predicted values should be unscaled before use.
For each model, several architectures of the ANN models were examined by varying
the number of hidden layers and the training function parameters to establish a suitable and
stable network for the project. Each network must be tested and analyzed, and the most
appropriate network must be chosen for a particular project.
The parameters used for the input nodes in the ANN modeling were as follows:
diameter (d) of the circular concrete specimen (mm), height (L) of the circular concrete
specimen (mm), compressive strength (fୡ
′
) of the unconfined concrete (MPa), total thickness
(t) of the CFRP (mm), tensile strength (f୊ୖ୔) of the CFRP along the hoop direction (MPa),
and elastic modulus (EFRP) of the CFRP (MPa).The target node was the compressive strength
of the confined concrete (fୡୡ).
The range of the input data used is listed in Table 1. The architecture of the developed
ANN model is shown in Fig. 1.
Table 1: Range of input data used in the ANN models
Input parameter d (mm) h (mm) fୡ
′
(MPa) t(mm) f୊ୖ୔(MPa) EFRP(GPa)
Minimum
Maximum
70
200
140
788
18
169.7
0.11
2.06
580
4400
38
415

102
40 60 80 100 120 140 160
40
60
80
100
120
140
160
Experimental Compressive Strength (MPa)
ANNPredictedCompressiveStrength(MPa)
R=0.964
Test data
0 50 100 150 200 250 300
0
50
100
150
200
250
300
Train data
R=0.977
A regression analysis was conducted between the network response and the
corresponding targets and a correlation coefficient was found. This option is a measure of
how well the variation in the output is explained by the targets. If this number is equal to one,
then a perfect correlation exists between the target and output predictions.
Fig. 2 shows the plot of the experimental compressive strength against the
corresponding ANN predictions for the test data. A linear correlation can be observed, and
the correlation coefficients are 0.977 and 0.964 for the training and the test data, respectively.
Therefore, we can conclude that the model successively predicts accurately the compressive
strength of the confined concrete.
Fig 1. Architecture of the first ANN model
Fig 2. Experimental and corresponding ANN compressive strength for the training and test
data of ANN Model
d
h
fc’
t
EFRP
fFRP
fcc
Input layer Hidden layer Output layer

103
4. IMPORTANCE OF INPUT PARAMETERS
Because the weight of the BP neural network cannot be easily understood in a
numerical matrix form, it could be transformed into code values in percentage form by
dividing the weights by the sum for all the input parameters, which yields the relative
importance of each input parameter to the output parameter. The method of partitioning
weights, proposed by Garson [17] and adopted by Goh [13], was used in this study to
determine the relative importance of the various input parameters (Fig. 3). The major
important parameter that influences the compressive strength of the confined concrete (fୡୡ) is
the tensile strength (f୊ୖ୔) of the CFRP along the hoop direction with an importance of
30.67%, followed by the total thickness (t) of CFRP with an importance of 20.78% and the
compressive strength (fୡ
′
) of the unconfined concrete with an importance of 19.13%. The
diameter (d) of the circular concrete specimen does not affect the compressive strength of the
confined concrete (fୡୡ) because its importance is only 4.162%. Most mathematical models
consider the column diameter as one of the main factors.
Fig 3. Importance of input parameters of the first ANN model
d L fc' t f (fbr) E (fbr)
0
5
10
15
20
25
30
35
%Importance
Input Factors
4.162%
16.00% 19.134%
20.783%
30.677%
9.200%

104
5. MODEL DEVELOPMENTS FOR COMPRESSIVE STRENGTH OF
CONFINED CONCRETE
Another application of ANNs is in building a mathematical model. The present study
contains six input and one output parameters. A model equation can be established using the
weights as the model parameters [43]. The mathematical equation can be written as
݂௖௖ ൌ
ଵ
ଵା௘
ష ሾഇరశ൬
ೢళర
భశ೐షೣభ൰శ൬
ೢఴర
భశ೐షೣమ൰శ൬
ೢవర
భశ೐షೣయ൰ሿ
(9)
where:
‫1ݔ‬ ൌ ߠଵ ൅ ‫ݓ‬ଵଵ ൈ d ൅ ‫ݓ‬ଶଵ ൈ h ൅ ‫ݓ‬ଷଵ ൈ fୡ′ ൅ ‫ݓ‬ସଵ ൈ t ൅ ‫ݓ‬ହଵ ൈ f୊ୖ୔ ൅ ‫ݓ‬଺ଵ ൈ E୊ୖ୔ (10)
‫2ݔ‬ ൌ ߠଶ ൅ ‫ݓ‬ଵଶ ൈ d ൅ ‫ݓ‬ଶଶ ൈ h ൅ ‫ݓ‬ଷଶ ൈ fୡ′ ൅ ‫ݓ‬ସଶ ൈ t ൅ ‫ݓ‬ହଶ ൈ f୊ୖ୔ ൅ ‫ݓ‬଺ଶ ൈ E୊ୖ୔ ሺ11)
‫3ݔ‬ ൌ ߠଷ ൅ ‫ݓ‬ଵଷ ൈ d ൅ ‫ݓ‬ଶଷ ൈ h ൅ ‫ݓ‬ଷଷ ൈ fୡ′ ൅ ‫ݓ‬ସଷ ൈ t ൅ ‫ݓ‬ହଷ ൈ f୊ୖ୔ ൅ ‫ݓ‬଺ଷ ൈ E୊ୖ୔ (12)
The values of weights ‫ݓ‬௜௝ and threshold ߠ௝ are shown in Table 2.
Table 2: Weights and threshold levels of the ANN model
a. Weights from node i in the input layer to node j in the hidden layer
nodes
‫ݓ‬௜௝
i=1 i=2 i=3 i=4 i=5 i=6 Hidden threshold ߠ௝
J=1
J=2
J=3
0.28
-0.10
-2.07
-7.90
1.00
3.82
8.93
-5.71
2.71
3.36
-7.00
6.15
10.19
-17.30
-2.85
1.53
7.58
-0.60
-5.73
17.60
0.68
b. Weights from node i in the hidden layer to node j in the output layer
node
Wij
i=7 i=8 i=9 output threshold θj
J=4 1.53 -16.41 2.51 12.53
Equation (9) is long and complex because it contains six independent variables. On
the other hand, it can predict accurately the compressive strength (fୡୡ) of the confined
concrete (Fig.2) with a correlation coefficient equal to 0.964.
The equation length depends on the number of nodes in the input and hidden layers.
To simplify the equation, the most importance input parameters, which are the compressive
strength (fୡ
′
), of the unconfined concrete the total thickness(t) of CFRP, and the tensile
strength(f୊ୖ୔) of the CFRP along the hoop direction, were used in training the second ANN
model with two nodes in the hidden layer. The result was the development of an ANN model
with a regression of 0.9259 (Fig. 4).The small number of connection weights of the neural
network enables the ANN model to be translated into a relatively simple formula in which the
compressive strength of the confined concrete (fୡୡ) can be expressed as follows:

105
fୡୡ ൌ
ଵ
ଵା௘
ష మఴభ.రఱశ
భయళ.ళఱ
భశ೐షೣభ శ
మఴఴ.ఴల
భశ೐షೣమ
(13)
where:
‫1ݔ‬ ൌ െ2.50 ൅ 0.40fୡ′ െ 0.64t ൅ 1.90f୊ୖ୔ (14)
‫2ݔ‬ ൌ 2.90 െ 0.36fୡ′ ൅ 0.34t െ 1.42f୊ୖ୔. (15)
Before using Eqs. 10–12, 14, and 15, all input variables must be scaled between 0.1
and 0.9 using Eq. 8 for the data ranges shown in Table 1. The predicted values obtained from
Eqs. 9 and 13 are scaled between 0.1 and 0.9. To obtain the actual values, these had to be
unscaled using Eq. 8.
In contrast to all previous models, the second ANN model depends on the
compressive strength (fୡ
′
) of the unconfined concrete, the total thickness (t) of CFRP, and the
tensile strength(f୊ୖ୔) of the CFRP in the hoop direction, whereas the geometry of the column
is not considered in this ANN model.
Fig. 4. Experimental and corresponding ANN compressive strength of the test data of second
Model
Gaussian distributions are perhaps the most important model for studying the
quantitative phenomena in the natural and behavioral sciences, such as the problems
encountered in structural analysis and design. To determine the suitability of the developed
CFRP-confined model, all 208 experimental and predicted confined compressive strength
values were taken, and the results of the confined compressive strength ratio
40 60 80 100 120 140 160 180 200
40
60
80
100
120
140
160
Test data
R=0.9259

106
(predicted/experimental) were statistically analyzed with a 0.5% level of significance using
the SPSS software V.16.
The average and variance of all values were found to be 92% and 0.02, respectively.
Fig.5 shows the frequency histogram of the confined compressive strength ratio
(fcc)pred./(fcc)Exp population curve. The (fcc)pred./(fcc)Exp values are distributed around their mean
values. The magnitude of the frequency becomes smaller when the value moves away from
the mean central value. The probability of obtaining a ratio between 90% and 95% is 85%.
Fig. 5. Histogram and normal distribution curve for the second mathematical model
6. PARAMETRIC STUDY
One of the advantages of the ANN models is that parametric studies can be easily
conducted by simply varying one input parameter while all other input parameters are set to
constant values. Parametric studies can verify the performance of the model in simulating the
physical behavior of the CFRP-confined concrete due to the variation in certain parameter
values.
The second ANN model and Eq. 13 were used to complete this parametric study.
Figs. 6 and 7 show the relationship between the compressive strength (fୡ
′
ሻof the
unconfined concrete and that of the CFRP-confined concrete (fୡୡ) under different values of
tensile strength (f୊ୖ୔) of the CFRP and total thickness(t) of the CFRP, respectively. In
general and regardless of the other parameters, the compressive strength (fୡୡ) increases with
the increasing compressive strength (fୡ
′
ሻof the unconfined concrete.
Fig. 6 shows the effect of the tensile strength (f୊ୖ୔) of the CFRP on the compressive
strength of the CFRP-confined concrete (fୡୡ) under different values of compressive
strength(fୡ
′
) of the unconfined concrete with a constant total thickness (t) equal to 0.22mm.

107
The compressive strength of the confined concrete (fୡୡ) strongly affects the tensile strength
(f୊ୖ୔) of the CFRP, especially under high unconfined compressive strength and high tensile
strength of the CFRP. For the unconfined compressive strength of 25 MPa, changing the
tensile strength of the CFRP from 2,500 MPa to 4,000 MPa led to an increase in the confined
compressive strength from 55.18 MPa to 56.6 MPa (increasing by 2.57%), whereas for the
unconfined compressive strength of 65 MPa, the same change in the tensile strength (f୊ୖ୔) of
the CFRP led to an increase in the confined compressive strength (fୡୡ) from 75.64 MPa to
87.27 MPa (increasing by 15.37%).
Fig 6. Effect of the tensile strength of the CFRP on the compressive strength of the FRP-
confined concrete
Fig. 7 shows the effect of the total thickness(t) of the CFRP on the compressive
strength of the CFRP-confined concrete (fୡୡ) for different values of compressive strength
(fୡ
′
)of the unconfined concrete with a constant tensile strength (f୊ୖ୔) equal to 3,000 MPa. For
different thicknesses(t) of CFRP, the curves are parallel, that is, a low or high unconfined
compressive strength wields the same effect on the compressive strength of the CFRP-
confined concrete with different thicknesses.
7. CONCLUSIONS
Two mathematical models for predicting the confined compressive strength of an
CFRP circular concrete column have been developed using the ANN approach. The
importance study showed that the diameter and height of the specimen and the elastic
modulus of CFRP had little effect on predicting the confined compressive strength of the
CFRP circular concrete column; hence, they were excluded from building the second ANN
model, leaving only three input parameters. Both parametric and importance studies showed
that the tensile strength of CFRP had an effect on predicting the confined compressive
strength of the CFRP circular concrete column. Finally, the ANN approach was proven to be
good and efficient in developing the mathematical models.
20 25 30 35 40 45 50 55 60 65 70 75
30
40
50
60
70
80
90
100
ConfinedCompressiveStrength(MPa)
Compressive Strength (MPa)
f=1000MPa
f=2500MPa
f=4000MPa
t=0.22 mm

108
Fig 7. Effect of the total thickness of the CFRP on the compressive strength of the CFRP-
confined concrete
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