**3. Adaptive neuro-fuzzy inference system (ANFIS)**

The fuzzy set theory developed by (Zadeh, 1965) provides as a mathematical framework to deal with vagueness associated with the description of a variable. The commonly used

Neural Network and Adaptive Neuro-Fuzzy

**3.1 Architecture of ANFIS** 

X1

Inference System Applied to Civil Engineering Problems 479

Fig. 7 shows the architecture of a typical ANFIS with two inputs X1 and X2, two rules and one output *f*, for the first order Sugeno fuzzy model, where each input is assumed to have two associated membership functions (MFs). For a first-order Sugeno fuzzy model a typical

The architecture of the proposed (ANFIS), it contains five layers where the node functions in the same layer are of the same function family. Inputs, outputs and implemented

w2 w2

N

w<sup>1</sup> w1

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

where *X* is the input to node *i*, μ*Ai* is the membership function (which can be triangular, trapezoidal, gaussian functions or other shapes) of the linguistic label *Ai* associated with this node and *Oi* is the degree of match to which the input *X* satisfies the quantifier *Ai*. In the

where { ��� ��} are the parameters of the MFs governing the Gaussian functions. The

**Layer 2:** Every node in this layer multiplies the incoming signals from layer 1 and sends the

� (����)� ��

� � ����(�) (8)

Σ

w1f1

w2f2

X1 X2

X1 X2

�� � ����(��) � ���(��)� � � ������� (10)

� � (9)

f

rule set with two fuzzy if–then rules can be expressed as (Jang, 1993):

where: *m1*, *n1*, *q1* and *m2*, *n2*, *q2* are the parameters of the output function.

mathematical models of the nodes of each layer are explained below. **Layer 1:** The node function of every node *i* in this layer take the form:

current study, the Gaussian shaped MFs defined below are utilized.

parameters in this layer are usually referred to as premise parameters.

where the output of this layer ( wi) represents the firing strength of a rule.

��

���(�) � ��� �� �

Rule (1): If *X1* is *A*1 and *X2* is *B*1, then *f1 = m1 X1 + n1X2 + q1*, Rule (2): If *X1* is *A*2 and *X2* is *B*2, then *f2 = m2 X1 + n2X2 + q2*.

Fig. 7. Structure of the proposed ANFIS model

A1

A2

B2

B1

X2 N

product out as follows,

fuzzy inference system (FIS) is the actual process of mapping from a given input to output using fuzzy logic.

Fuzzy logic is particularly useful in the development of expert systems. Expert systems are built by capturing the knowledge of humans: however, such knowledge is known to be qualitative and inexact. Experts may be only partially knowledgeable about the problem domain, or data may not be fully available, but decisions are still expected. In these situations, educated guesses need to be made to provide solutions to problems. This is where fuzzy logic can be employed as a tool to deal with imprecision and qualitative aspects that are associated with problem solving (Jang, 1993).

A fuzzy set is a set without clear or sharp boundaries or without binary membership characteristics. Unlike a conventional set where object either belongs or do not belong to the set, partial membership in a fuzzy set is possible. In other words, there is a softness associated with the membership of elements in a fuzzy set (Jang, 1993).A fuzzy set may be represented by a membership function. This function gives the grade (degree) of membership within the set. The membership function maps the elements of the universe on to numerical values in the interval [0, 1]. The membership functions most commonly used in control theory are triangular, trapezoidal, Gaussian, generalized bell, sigmoidal and difference sigmoidal membership functions (Jang et al., 1997; Matlab toolbox, 2009; Zaho & Bose, 2002).

As mentioned previously, the fuzzy inference system is the process of formulating the mapping from a given input to an output using fuzzy logic. The dynamic behavior of an FIS is characterized by a set of linguistic description rules based on expert knowledge.

The fuzzy system and neural networks are complementary technologies.The most important reason for combining fuzzy systems with neural networks is to use the learning capability of neural network. While the learning capability is an advantage from the view point of a fuzzy system, from the viewpoint of a neural network there are additional advantages to a combined system. Because a neuro-fuzzy system is based on linguistic rules, we can easily integrate prior knowledge in to the system, and this can substantially shorten the learning process. One of the popular integrated systems is an ANFIS, which is an integration of a fuzzy inference system with a back-propagation algorithm (Jang et al., 1997; Lin & Lee 1996).

There are two types of fuzzy inference systems that can be implemented: Mamdani-type and Sugeno-type (Mamdani & Assilian, 1975; Sugeno, 1985). Because the Sugeno system is more compact and computationally more efficient than a Mamdani system, it lends itself to the use of adaptive techniques for constructing the fuzzy models. These adaptive techniques can be used to customize the membership functions so that the fuzzy system best models the data. The fuzzy inference system based on neuro-adaptive learning techniques is termed adaptive neuro-fuzzy inference system (Hamidian & Seyedpoor, 2009).

In order for an FIS to be mature and well established so that it can work appropriately in prediction mode, its initial structure and parameters (linear and non-linear) need to be tuned or adapted through a learning process using a sufficient input-output pattern of data. One of the most commonly used learning systems for adapting the linear and nonlinear parameters of an FIS, particularly the first order Sugeno fuzzy model, is the ANFIS. ANFIS is a class of adaptive networks that are functionally equivalent to fuzzy inference systems (Jang, 1993).

### **3.1 Architecture of ANFIS**

478 Fuzzy Inference System – Theory and Applications

fuzzy inference system (FIS) is the actual process of mapping from a given input to output

Fuzzy logic is particularly useful in the development of expert systems. Expert systems are built by capturing the knowledge of humans: however, such knowledge is known to be qualitative and inexact. Experts may be only partially knowledgeable about the problem domain, or data may not be fully available, but decisions are still expected. In these situations, educated guesses need to be made to provide solutions to problems. This is where fuzzy logic can be employed as a tool to deal with imprecision and qualitative aspects

A fuzzy set is a set without clear or sharp boundaries or without binary membership characteristics. Unlike a conventional set where object either belongs or do not belong to the set, partial membership in a fuzzy set is possible. In other words, there is a softness associated with the membership of elements in a fuzzy set (Jang, 1993).A fuzzy set may be represented by a membership function. This function gives the grade (degree) of membership within the set. The membership function maps the elements of the universe on to numerical values in the interval [0, 1]. The membership functions most commonly used in control theory are triangular, trapezoidal, Gaussian, generalized bell, sigmoidal and difference sigmoidal membership functions (Jang et al., 1997; Matlab toolbox, 2009;

As mentioned previously, the fuzzy inference system is the process of formulating the mapping from a given input to an output using fuzzy logic. The dynamic behavior of an FIS

The fuzzy system and neural networks are complementary technologies.The most important reason for combining fuzzy systems with neural networks is to use the learning capability of neural network. While the learning capability is an advantage from the view point of a fuzzy system, from the viewpoint of a neural network there are additional advantages to a combined system. Because a neuro-fuzzy system is based on linguistic rules, we can easily integrate prior knowledge in to the system, and this can substantially shorten the learning process. One of the popular integrated systems is an ANFIS, which is an integration of a fuzzy inference system with a back-propagation algorithm (Jang et al., 1997; Lin & Lee 1996). There are two types of fuzzy inference systems that can be implemented: Mamdani-type and Sugeno-type (Mamdani & Assilian, 1975; Sugeno, 1985). Because the Sugeno system is more compact and computationally more efficient than a Mamdani system, it lends itself to the use of adaptive techniques for constructing the fuzzy models. These adaptive techniques can be used to customize the membership functions so that the fuzzy system best models the data. The fuzzy inference system based on neuro-adaptive learning techniques is termed

In order for an FIS to be mature and well established so that it can work appropriately in prediction mode, its initial structure and parameters (linear and non-linear) need to be tuned or adapted through a learning process using a sufficient input-output pattern of data. One of the most commonly used learning systems for adapting the linear and nonlinear parameters of an FIS, particularly the first order Sugeno fuzzy model, is the ANFIS. ANFIS is a class of adaptive networks that are functionally equivalent to fuzzy inference

is characterized by a set of linguistic description rules based on expert knowledge.

adaptive neuro-fuzzy inference system (Hamidian & Seyedpoor, 2009).

using fuzzy logic.

Zaho & Bose, 2002).

systems (Jang, 1993).

that are associated with problem solving (Jang, 1993).

Fig. 7 shows the architecture of a typical ANFIS with two inputs X1 and X2, two rules and one output *f*, for the first order Sugeno fuzzy model, where each input is assumed to have two associated membership functions (MFs). For a first-order Sugeno fuzzy model a typical rule set with two fuzzy if–then rules can be expressed as (Jang, 1993):

Rule (1): If *X1* is *A*1 and *X2* is *B*1, then *f1 = m1 X1 + n1X2 + q1*, Rule (2): If *X1* is *A*2 and *X2* is *B*2, then *f2 = m2 X1 + n2X2 + q2*.

where: *m1*, *n1*, *q1* and *m2*, *n2*, *q2* are the parameters of the output function.

Fig. 7. Structure of the proposed ANFIS model

The architecture of the proposed (ANFIS), it contains five layers where the node functions in the same layer are of the same function family. Inputs, outputs and implemented mathematical models of the nodes of each layer are explained below.

**Layer 1:** The node function of every node *i* in this layer take the form:

$$\mathcal{O}\_l^1 = \; \mathfrak{u} \mathcal{A}\_l(\mathcal{X}) \tag{8}$$

where *X* is the input to node *i*, μ*Ai* is the membership function (which can be triangular, trapezoidal, gaussian functions or other shapes) of the linguistic label *Ai* associated with this node and *Oi* is the degree of match to which the input *X* satisfies the quantifier *Ai*. In the current study, the Gaussian shaped MFs defined below are utilized.

$$\mu A\_l(X) = \exp\left\{-\frac{1}{2}\frac{(X - c\_l)^2}{\sigma\_l^2}\right\} \tag{9}$$

where { ��� ��} are the parameters of the MFs governing the Gaussian functions. The parameters in this layer are usually referred to as premise parameters.

**Layer 2:** Every node in this layer multiplies the incoming signals from layer 1 and sends the product out as follows,

$$\mu\_{l} = \mu A\_{l}(X\_{1}) \times \mu B\_{l}(X\_{2}), l = 1, 2 \tag{10}$$

where the output of this layer ( wi) represents the firing strength of a rule.

$$
\omega\_l^- = \frac{\mathbf{w}\_l}{\mathbf{w}\_1 + \mathbf{w}\_2} \tag{11}
$$

$$\mathcal{O}\_l^\phi = \left. w\_l^- f\_l = \left. w\_l^- (m\_l X\_1 + n\_l X\_2 + q\_l), l = 1, 2 \right. \tag{12}$$

$$\mathcal{O}\_l^5 = \Sigma\_l \,\boldsymbol{\omega}\_l^- \, f\_l = \frac{\Sigma\_l \,\boldsymbol{\omega}\_l f\_l}{\Sigma\_l \,\boldsymbol{\omega}\_l}, l = 1, 2 \tag{13}$$

$$f = \frac{\mathbf{w}\_1}{\mathbf{w}\_1 + \mathbf{w}\_2} f\_1 + \frac{\mathbf{w}\_2}{\mathbf{w}\_1 + \mathbf{w}\_2} f\_2 = \mathbf{w}\_1^- f\_1 + \mathbf{w}\_2^- f\_2 \tag{14}$$

$$\frac{V\_u}{p\_\cdot d} = \frac{\sqrt{f\_c'}}{a/d} \left\{ 0.0856 + 0.0028 \frac{v\_f f\_y}{\sqrt{f\_c'}} \right\} \tag{15}$$

Neural Network and Adaptive Neuro-Fuzzy

small number of epochs.

**4.1.2.3 Normalizing input and output data sets** 

**4.1.2.4 Number of hidden layers and nodes in each hidden layer** 

Inference System Applied to Civil Engineering Problems 483

section (d), the width of specimens cross section (b),yield tensile strength of wire mesh reinforcement (*fy*), cylinder compressive strength of mortar (*fc'*), total volume fraction of wire mesh (*vf*) and shear span to depth ratio (*a/d*). The shear strength of ferrocement member represents the target variable. Table 1 summarizes the ranges of each different variable.

Normalization (scaling down) of input and output data sets within a uniform range before they are applied to the neural network is essential to prevent larger numbers from overriding smaller ones, and to prevent premature saturation of hidden nodes, which impedes the learning process. The limitation of input and output values within a specified range are due to the large difference in the values of the data provided to the neural network. Besides, the activation function used in the back-propagation neural network is a hyperbolic tangent function, the lower and upper limits of this function are -1 and +1 respectively. In this study Eq. 5 mentioned above is used to normalize the input and output parameters. That equation gives the required results with a certain mean square error by a

The network is tested with an increasing number of nodes in hidden layer. It is found that one-hidden layer network with four nodes gives the optimal configurations with minimum mean square error (MSE). As an activation function, a hyperbolic tangent function is

In this study the initial weights are randomly chosen. The network has been trained

Fig. 8 shows the performance for training and generalization (testing) sets using resilient back-propagation training algorithm, the network is trained for 420 epochs to check if the performance (MSE) for either training or testing sets might diverge. The network performance with resilient back-propagation training algorithm have been tested for training and generalizing patterns, as shown in Fig. 9 (a) and (b). A good agreement has

selected for the hidden layer and a purelin function is used for the output layer.

continually through updating weights until the final error achieved is 8.48\*10-4.

been noted in the predicted values compared with the actual (targets) values.

Fig. 8. Convergence of the BPNN for training and testing sets

#### **4.1.2.1 Selection of the training and testing patterns**

The experimental data that are used to train the neural network are obtained from literature **(**Mansur & Ong, 1987; Mashrei, 2010; Rao et al., 2006**)** as shown in Table A in appendix. The data used to build the neural network model should be divided into two subsets: training set and testing set. The testing set contains approximately 13% from total database. The training phase is needed to produce a neural network that is both stable and convergent. Therefore, selecting what data to use for training a network is one of the most important steps in building a neural network model. The total numbers of 69 test specimens was utilized. The training set contained 60 specimens and the testing set was comprised of 9 specimens.

Neural networks interpolate data very well. Therefore, patterns chosen for training set must cover upper and lower boundaries and a sufficient number of samples representing particular features over the entire training domain (Rafiq et al., 2001).An important aspect of developing neural networks is determining how well the network performs once training is complete. The performance of a trained network is checked by involving two main criteria:


To effectively visualize how well a network performs recall and generalization steps, the learning curve is generated which represents the mean square error (MSE) for both the recall of training data sets and generalization of testing set with the number of iteration or epoch. The error between the training data sets and the generalization of testing sets should converge upon the same point corresponding to the best set of weight factors for the network.

#### **4.1.2.2 Input and output layers**

In the developed neural network model there is an input layer, where input data are presented to the network, and an output layer of one neuron representing the shear strength of ferrocement member. In this study the parameters which may be introduced as the components of the input vector consist of six inputs: the total depth of specimens cross


Table 1. Range of parameters in the database

The experimental data that are used to train the neural network are obtained from literature **(**Mansur & Ong, 1987; Mashrei, 2010; Rao et al., 2006**)** as shown in Table A in appendix. The data used to build the neural network model should be divided into two subsets: training set and testing set. The testing set contains approximately 13% from total database. The training phase is needed to produce a neural network that is both stable and convergent. Therefore, selecting what data to use for training a network is one of the most important steps in building a neural network model. The total numbers of 69 test specimens was utilized. The training set contained 60 specimens and the testing set was

Neural networks interpolate data very well. Therefore, patterns chosen for training set must cover upper and lower boundaries and a sufficient number of samples representing particular features over the entire training domain (Rafiq et al., 2001).An important aspect of developing neural networks is determining how well the network performs once training is complete. The performance of a trained network is checked by involving two main criteria: 1. How well the neural network recalls the predicted response from data sets used to train the network (called the recall step). A well trained network should be able to produce

2. How well the network predicts responses from data sets that were not used in the training (called the generalization step). Generalization is affected by three factors: the size and the efficiency of the training data set, the architecture of the network, and the physical complexity of the problem. A well generalized network should be able to sense

To effectively visualize how well a network performs recall and generalization steps, the learning curve is generated which represents the mean square error (MSE) for both the recall of training data sets and generalization of testing set with the number of iteration or epoch. The error between the training data sets and the generalization of testing sets should converge

In the developed neural network model there is an input layer, where input data are presented to the network, and an output layer of one neuron representing the shear strength of ferrocement member. In this study the parameters which may be introduced as the components of the input vector consist of six inputs: the total depth of specimens cross

> Width of specimens (b) (mm) 100-200 Total depth of specimens (d) (mm) 25-50 Shear span to depth ratio (*a/d*) 1-7

> yield strength of wire mesh (*fy*) (MPa) 380-410 Volume fraction of wire mesh (*vf*) % 0-5.7

Compressive strength of mortar (*fc*

Table 1. Range of parameters in the database

Parameters Range

*'*

) 26.5-44.1

upon the same point corresponding to the best set of weight factors for the network.

**4.1.2.1 Selection of the training and testing patterns** 

an output that deviates very little from desired value.

comprised of 9 specimens.

the new input patterns.

**4.1.2.2 Input and output layers** 

section (d), the width of specimens cross section (b),yield tensile strength of wire mesh reinforcement (*fy*), cylinder compressive strength of mortar (*fc'*), total volume fraction of wire mesh (*vf*) and shear span to depth ratio (*a/d*). The shear strength of ferrocement member represents the target variable. Table 1 summarizes the ranges of each different variable.

#### **4.1.2.3 Normalizing input and output data sets**

Normalization (scaling down) of input and output data sets within a uniform range before they are applied to the neural network is essential to prevent larger numbers from overriding smaller ones, and to prevent premature saturation of hidden nodes, which impedes the learning process. The limitation of input and output values within a specified range are due to the large difference in the values of the data provided to the neural network. Besides, the activation function used in the back-propagation neural network is a hyperbolic tangent function, the lower and upper limits of this function are -1 and +1 respectively. In this study Eq. 5 mentioned above is used to normalize the input and output parameters. That equation gives the required results with a certain mean square error by a small number of epochs.

#### **4.1.2.4 Number of hidden layers and nodes in each hidden layer**

The network is tested with an increasing number of nodes in hidden layer. It is found that one-hidden layer network with four nodes gives the optimal configurations with minimum mean square error (MSE). As an activation function, a hyperbolic tangent function is selected for the hidden layer and a purelin function is used for the output layer.

In this study the initial weights are randomly chosen. The network has been trained continually through updating weights until the final error achieved is 8.48\*10-4.

Fig. 8 shows the performance for training and generalization (testing) sets using resilient back-propagation training algorithm, the network is trained for 420 epochs to check if the performance (MSE) for either training or testing sets might diverge. The network performance with resilient back-propagation training algorithm have been tested for training and generalizing patterns, as shown in Fig. 9 (a) and (b). A good agreement has been noted in the predicted values compared with the actual (targets) values.

Fig. 8. Convergence of the BPNN for training and testing sets

Neural Network and Adaptive Neuro-Fuzzy

(Jang, 1997; Jonic'

**4.1.3.1 Database**

decided when testing data set.

**4.1.3.2 Modeling and results** 

Fig. 10. Structure of the proposed ANFIS model

Inference System Applied to Civil Engineering Problems 485

technique introduced by (Chiu, 1994)with (genfis2) function was used. Given separate sets of input and output data, the genfis2 uses a subtractive clustering method to generate a Fuzzy Inference System (FIS). When there is only one output, genfis2 may be used to generate an initial FIS for ANFIS training by first implementing subtractive clustering on the data. The genfis2 function uses the subclust function to estimate the antecedent membership functions and a set of rules. This function returns an FIS structure that contains a set of fuzzy rules to cover the feature space (Fuzzy Logic Toolbox, 2009). For a given set of data, subtractive clustering method was used for estimating the number of clusters and the cluster centers in a set of data. It assumes each data point is a potential cluster center and calculates a measure of the potential for each data point based on the density of surrounding data points. The algorithm selects the data point with the highest potential as the first cluster center and then delimits the potential of data points near the first cluster center. The algorithm then selects the data point with the highest remaining potential as the next cluster center and delimits the potential of data points near this new cluster center. This process of acquiring a new cluster center and delimiting the potential of surrounding data points repeats until the potential of all data points falls below a threshold. The range of influence of a cluster center in each of the data dimensions is called cluster radius. A small cluster radius will lead to finding many small clusters in the data (resulting in many rules) and vice versa

, 1999). Membership functions (MFs) and numbers are appropriately

The adaptive neuro-fuzzy inference system model is developed to predict the shear strength of ferrocement members. The same database of (69) specimens as in the previous BPNN model is used for the development of this model. The total data is divided at random into two groups (training data set, and testing data set), as shown in Table A in Appendix.

The ANFIS model is developed to predict shear strength of ferrocement specimens with MFs of type (gussmf) for all input variables and linear for the output. The number of MFs assigned to each input variable is chosen by trial and error. After training and testing, the

Fig. 9. Comparison between BPNN results and target results

#### **4.1.2.5 Relative importance**

Once the artificial neural network has been trained, a relative importance is used to investigate the influence of the various parameters on the shear strength. The effect of each parameter on the shear strength of ferrocement is clear in Table 2. After training all the data sets with the final model, the relative importance of each input variable is evaluated. The methodology suggested by Garson, (1991) is used. The relative importance of the various input factors can be assessed by examining input-hidden-output layer connection weights. This is carried out by partitioning the hidden-output connection weights into components connected with each input neuron. Table 2 lists the relative importance of the input variables in the BPNN model. It can be observed that for shear strength of the ferrocement member, the shear span to depth ratio (a d� ) is the most important factor among the input variables and volume fraction of wire mesh is the second most important factor comparing with the others. Therefore, it can be concluded that (a d� ) ratio has the most influence on the shear strength of ferrocement.


Table 2. Relative importance (RI) (%) of BPNN model

#### **4.1.3 Adaptive neural fuzzy inference system (ANFIS) model**

In the developed ANFIS, six variables consisting of width (b) and depth (d) of the specimens, yield tensile strength of wire mesh reinforcement (��), cylinder compressive strength of mortar (��′), total volume fraction of wire mesh (��) and shear span to depth ratio (� �� ) are selected as input variables to predict the shear strength of ferrocement members, which is the target variable. In this investigation the subtractive clustering technique introduced by (Chiu, 1994)with (genfis2) function was used. Given separate sets of input and output data, the genfis2 uses a subtractive clustering method to generate a Fuzzy Inference System (FIS). When there is only one output, genfis2 may be used to generate an initial FIS for ANFIS training by first implementing subtractive clustering on the data. The genfis2 function uses the subclust function to estimate the antecedent membership functions and a set of rules. This function returns an FIS structure that contains a set of fuzzy rules to cover the feature space (Fuzzy Logic Toolbox, 2009). For a given set of data, subtractive clustering method was used for estimating the number of clusters and the cluster centers in a set of data. It assumes each data point is a potential cluster center and calculates a measure of the potential for each data point based on the density of surrounding data points. The algorithm selects the data point with the highest potential as the first cluster center and then delimits the potential of data points near the first cluster center. The algorithm then selects the data point with the highest remaining potential as the next cluster center and delimits the potential of data points near this new cluster center. This process of acquiring a new cluster center and delimiting the potential of surrounding data points repeats until the potential of all data points falls below a threshold. The range of influence of a cluster center in each of the data dimensions is called cluster radius. A small cluster radius will lead to finding many small clusters in the data (resulting in many rules) and vice versa (Jang, 1997; Jonic' , 1999). Membership functions (MFs) and numbers are appropriately decided when testing data set.

#### **4.1.3.1 Database**

484 Fuzzy Inference System – Theory and Applications

(a) For training pattern (b) For testing pattern

Once the artificial neural network has been trained, a relative importance is used to investigate the influence of the various parameters on the shear strength. The effect of each parameter on the shear strength of ferrocement is clear in Table 2. After training all the data sets with the final model, the relative importance of each input variable is evaluated. The methodology suggested by Garson, (1991) is used. The relative importance of the various input factors can be assessed by examining input-hidden-output layer connection weights. This is carried out by partitioning the hidden-output connection weights into components connected with each input neuron. Table 2 lists the relative importance of the input variables in the BPNN model. It can be observed that for shear strength of the ferrocement

variables and volume fraction of wire mesh is the second most important factor comparing

��′ (MPa)

RI (%) 7.11 20.0 8.89 5.28 38.32 20.4

In the developed ANFIS, six variables consisting of width (b) and depth (d) of the specimens, yield tensile strength of wire mesh reinforcement (��), cylinder compressive strength of mortar (��′), total volume fraction of wire mesh (��) and shear span to depth

�� ) are selected as input variables to predict the shear strength of ferrocement members, which is the target variable. In this investigation the subtractive clustering

d� ) is the most important factor among the input

�� (MPa)

d� ) ratio has the most influence on the

�

�� ��

(%)

Fig. 9. Comparison between BPNN results and target results

**4.1.2.5 Relative importance** 

member, the shear span to depth ratio (a

b (mm)

shear strength of ferrocement.

Input variables

ratio (�

with the others. Therefore, it can be concluded that (a

Table 2. Relative importance (RI) (%) of BPNN model

d (mm)

**4.1.3 Adaptive neural fuzzy inference system (ANFIS) model** 

The adaptive neuro-fuzzy inference system model is developed to predict the shear strength of ferrocement members. The same database of (69) specimens as in the previous BPNN model is used for the development of this model. The total data is divided at random into two groups (training data set, and testing data set), as shown in Table A in Appendix.

#### **4.1.3.2 Modeling and results**

The ANFIS model is developed to predict shear strength of ferrocement specimens with MFs of type (gussmf) for all input variables and linear for the output. The number of MFs assigned to each input variable is chosen by trial and error. After training and testing, the

Fig. 10. Structure of the proposed ANFIS model

Neural Network and Adaptive Neuro-Fuzzy

Inference System Applied to Civil Engineering Problems 487

models are better than those of the empirical method. Also in Table 3 the correlation coefficient R of predicted shear strength by BPNN, ANFIS, and the empirical method are summarized. As shown in Table 4, both ANFIS and BPNN produce a higher correlation coefficient R as compared with the empirical method. Therefore, the BPNN as well as ANFIS can serve as

reliable and simple tools for the prediction of shear strength of ferrocement.

(a) For training pattern (b) For training pattern

V� V�

Fig. 13. Comparison experimental and predicted values for training data Set

0

5

10

Vu (Predicted) (kN)

15

20

Specimens No. Average of of Ve / Vi STDEV of Ve / Vi

V� V�

Training set 60 1.01 1.01 1.21 0.14 0.13 0.27 Testing set 9 1.03 1.03 1.23 0.09 0.08 0.31 Table 3. Comparison between experimental and predicted results for training and testing sets

BPNN ANFIS

empirical expression 100% agreement line

0 5 10 15 20

Vu (Target) kN

V� V� V� V� V� V� V� V�

Fig. 12. Comparison between ANFIS results and target results

number of MFs is fixed as six MFs for each input variable. This is chosen when the ANFIS model reaches an acceptable satisfactory level. The structure of ANFIS model is developed as shown in Fig. 10. The basic flow diagram of computations in ANFIS is illustrated in Fig. 11. A comparison between the prediction from ANFIS and target value for each of training and testing data set is shown in Fig. 12(a) and (b) respectively. The predictions appear to be quite good with correlation coefficient R approaches one.

#### **4.1.4 Comparison between experimental and theoretical results**

The predictions of shear strength of ferrocement members as obtained from BPNN, ANFIS, and the empirical available method (Eq.15) (Rao et al., 2006) are compared with the experimental results and shown for both training and testing sets in Figs.13 and 14 and Table 3 . In Table 3 the ratios of experimental (Ve) to theoretical predictions of the shear strength (Vi) of the ferrocement specimens are calculated. The theoretical predications include those obtained by BPNN (V1), ANFIS (V2), and empirical method (Eq.15) (V3). The average and the standard deviation of the ratios Ve/Vi are given in this table for both training and testing set. It can be seen that BPNN and ANFIS models give average values of Ve/V1 and Ve/V2 of 1.01 and standard deviations of 0.14 and 0.13, respectively for training set and the average values of Ve/V1 and Ve/V2 of 1.03 and standard deviations of 0.09 and 0.08, respectively for testing set , which are better than the values obtained for the empirical method. Figs. 13 and 14 confirm the same conclusion the predictions of BPNN and ANFIS

Fig. 11. The basic flow diagram of computations in ANFIS

number of MFs is fixed as six MFs for each input variable. This is chosen when the ANFIS model reaches an acceptable satisfactory level. The structure of ANFIS model is developed as shown in Fig. 10. The basic flow diagram of computations in ANFIS is illustrated in Fig. 11. A comparison between the prediction from ANFIS and target value for each of training and testing data set is shown in Fig. 12(a) and (b) respectively. The predictions appear to be

The predictions of shear strength of ferrocement members as obtained from BPNN, ANFIS, and the empirical available method (Eq.15) (Rao et al., 2006) are compared with the experimental results and shown for both training and testing sets in Figs.13 and 14 and Table 3 . In Table 3 the ratios of experimental (Ve) to theoretical predictions of the shear strength (Vi) of the ferrocement specimens are calculated. The theoretical predications include those obtained by BPNN (V1), ANFIS (V2), and empirical method (Eq.15) (V3). The average and the standard deviation of the ratios Ve/Vi are given in this table for both training and testing set. It can be seen that BPNN and ANFIS models give average values of Ve/V1 and Ve/V2 of 1.01 and standard deviations of 0.14 and 0.13, respectively for training set and the average values of Ve/V1 and Ve/V2 of 1.03 and standard deviations of 0.09 and 0.08, respectively for testing set , which are better than the values obtained for the empirical method. Figs. 13 and 14 confirm the same conclusion the predictions of BPNN and ANFIS

> Initialize the fuzzy system Use *genfis2* command

Give other parameter for learning Important are: Number of iterations Tolerance (error)

Start learning process Use command *anfis*  Stop when tolerance is achieved

> Validate with Independent data

quite good with correlation coefficient R approaches one.

Fig. 11. The basic flow diagram of computations in ANFIS

**4.1.4 Comparison between experimental and theoretical results** 

models are better than those of the empirical method. Also in Table 3 the correlation coefficient R of predicted shear strength by BPNN, ANFIS, and the empirical method are summarized. As shown in Table 4, both ANFIS and BPNN produce a higher correlation coefficient R as compared with the empirical method. Therefore, the BPNN as well as ANFIS can serve as reliable and simple tools for the prediction of shear strength of ferrocement.

(a) For training pattern (b) For training pattern

Fig. 12. Comparison between ANFIS results and target results


Table 3. Comparison between experimental and predicted results for training and testing sets

Fig. 13. Comparison experimental and predicted values for training data Set

Neural Network and Adaptive Neuro-Fuzzy

member (Wu & Bailey, 2005).

amounts of steel reinforcement.

03), can be expressed as follows:

reinforced beams as follows:

proposed equation by El-Sayed et al. (2005a).

**4.2.1.2 Tureyen and Frosch equation (2003)** 

**4.2.1.3 El-Sayed et al. equation (2005a)** 

modification for proposing the Eq. below:

**4.2.1.1 American Concrete Institute (ACI 440.1R-03)** 

��� <sup>=</sup> ���� ������

Inference System Applied to Civil Engineering Problems 489

2005a) is the use of FRP for reinforcement in concrete structures. The use of FRP in concrete structures include: (a) the internal reinforcing (rod or bar) which is used instead of the steel wire (rod) equivalent; and (b) the external bonded reinforcement, which is typically used to repair/strengthen the structure by plating or wrapping FRP tape, sheet or fabric around the

There are fundamental differences between the steel and FRP reinforcements: the latter has a lower modulus of elasticity, the modulus of elasticity for commercially available glass and aramid FRP bars is 20 to 25 % that of steel compared to 60 to 75 % for carbon FRP bars (Bank, 2006) linear stress–strain diagram up to rupture with no discernible yield point and different bond strength according to the type of FRP product. These characteristics affect the shear capacity of FRP reinforced concrete members. Due to the relatively low modulus of elasticity of FRP bars, concrete members reinforced longitudinally with FRP bars experience reduced shear strength compared to the shear strength of those reinforced with the same

Some of empirical equations have been developed to estimate shear strength of concrete beams reinforced with FRP. Most of the shear design provisions incorporated in these codes and guides are based on the design formulas of members reinforced with conventional steel considering some modifications to account for the substantial differences between FRP and steel reinforcement. These provisions use the well-known Vc + Vs method of shear design, which is based on the truss analogy. This section reviews the concrete shear strength of members longitudinally reinforced with FRP bars, ���, as recommended by the American Concrete Institute (ACI 440.1R-03, 2003), Tureyen and Frosch Equation (2003), and the

The equation for shear strength proposed by the American Concrete Institute (ACI 440.1R-

This equation was developed by Tureyen and Frosch, 2003. It was developed from a model that calculates the concrete contribution to shear strength of reinforced concrete beams. The equation was simplified to provide a design formula applicable FRP

> � ( ��� � �

They applied the same procedure in ACI 440.1R-03 to derive Eq. 1 above, with some

��� ) � ���

� �

��� (16)

���) (17)

� = ������ � (����)� � ���� (18)

� ( ��� � �

��� <sup>=</sup> �

where: � = ��= cracked transformed section neutral axis depth ( mm).

Fig. 14. Comparison experimental and predicted values for testing data Set


Table 4. Comparison summary of correlation R
