**1. Introduction**

470 Fuzzy Inference System – Theory and Applications

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Soft computing is an approximate solution to a precisely formulated problem or more typically, an approximate solution to an imprecisely formulated problem (Zadeh, 1993). It is a new field appearing in the recent past to solve some problems such as decision-making, modeling and control problems. Soft computing is an emerging approach to computing which parallels the remarkable ability of the human mind to reason and learn in an environment of uncertainty and imprecision (Jang el at., 1997). It consists of many complementary tools such as artificial neural network (ANN), fuzzy logic (FL), and adaptive neuro-fuzzy inference system (ANFIS).

Artificial neural network (ANN) model is a system of interconnected computational neurons arranged in an organized fashion to carry out an extensive computing to perform a mathematical mapping (Rafiq et al., 2001). The first interest in neural network (or parallel distributed processing) emerged after the introduction of simplified neurons by McCulloch & Pitts, (1943). These neurons were presented as models of biological neurons and as conceptual components for circuits that could perform computational works. ANN can be most adequately characterized as a computational model with particular properties such as the ability to adapt or learn, to generalize, or to cluster or organize data in which the operation is based on parallel processing.

ANN has a large number of highly interconnected processing elements (nodes or units) that usually operate in parallel and are configured in regular architectures. The collective behavior of an ANN, like a human brain, demonstrates the ability to learn, recall, and generalize from training patterns or data. ANN is inspired by modeling networks of biological neurons in the brain. Hence, the processing elements in ANN are also called artificial neurons (Rafiq et al., 2001). Artificial neural network described in this chapter is mostly applied to solve many civil engineering applications such as structural analysis and design (Cladera & Mar, 2004a, 2004b; Hajela & Berke, 1991; Sanad & Saka, 2001), structural damage assessment (Feng & Bahng, 1999; Mukherjee et al., 1996), structural dynamics and control (Chen et al., 1995; Feng & Kim, 1998) and pavement condition-rating modeling (Eldin & Senouuci, 1995).

Neural Network and Adaptive Neuro-Fuzzy

Fig. 2. Natural (biological) neurons

**2.1 Learning process** 

neural network.

Inference System Applied to Civil Engineering Problems 473

The complexity of real neurons is highly abstracted when modeling artificial neurons. These basically consist of *inputs*(like synapses), which are multiplied by *weights* (strength of the respective signals), and then computed by a mathematical function which determines the *activation* of the neuron. Another function (which may be the identity) computes the *output* of the artificial neuron (sometimes independent on a certain *threshold*). ANN combines

Compared to conventional digital computing techniques, neural networks are advantageous because of their special features, such as the massively parallel processing, distributed storing of information, low sensitivity to error, their very robust operation after training,

An artificial neuron is composed of five main parts: inputs, weights, sum function, activation function and outputs. Inputs are information that enters the cell from other cells of from external world. Weights are values that express the effect of an input set or another process element in the previous layer on this process element. Sum function is a function that calculates the effect of inputs and weights totally on this process element. This function

The information is propagated through the neural network layer by layer, always in the same direction. Besides the input and output layers there can be other intermediate layers of neurons, which are usually called hidden layers. Fig. 3 shows the structure of a typical

The inputs to the jth node are represented as an input factor, a, with component ai (i=1 to n), and the output by bj. The values w1j, w2j, …, and wnj are weight factors associated with each input to the node. This is something like the varying synaptic strengths of biological neurons. Weights are adaptive coefficients within the network that determine the intensity of the input signal. Every input (a1, a2, …, an) is multiplied by its corresponding weight factor (w1j, w2j, …, wnj), and the node uses this weighted input (w1j a1, w2j a2, …, wnj an) to perform further calculations. If the weight factor is positive, (wijai) tends to excite the node. If the weight factor is negative, (wijai) inhibits the node. In the initial setup of a neural

calculates the net input that comes to a cell (Topcu & Sardemir, 2007).

artificial neurons in order to process information (Gershenson, 2003).

generalization and adaptability to new information (Waszczyszyn, 1998).

The adaptive neuro-fuzzy inference system (ANFIS), first proposed by Jang, 1993, is one of the examples of neuro-fuzzy systems in which a fuzzy system is implemented in the framework of adaptive networks. ANFIS constructs an input-output mapping based both on human knowledge (in the form of fuzzy if-then rules) and on generated input-output data pairs by using a hybrid algorithm that is the combination of the gradient descent and least squares estimates. Readers are referred to References (Jang, 1993; Mashrei, 2010) for more details on the ANFIS. After generated input-output by training, the ANFIS can be used to recognize data that is similar to any of the examples shown during the training phase .The adaptive neuro-fuzzy inference system has been used in the area of civil engineering to solve many problems (Abdulkadir et al., 2006; Akbulut et al., 2004; Fonseca el at., 2007; Tesfamariam & Najjaran, 2007).

Most of the problems solved in civil and structural engineering using ANFIS and ANN are prediction of behavior based on given experimental results that are used for training and testing data. The matter of modeling is to solve a problem by predicting which is obtained by mapping a set of variables in input space to a set of response variables in output space through a model as represented in Fig. 1. In the box representing a model in this figure, conventionally a mathematical model is used. However, the conventional modeling of the underlying systems often tends to become quite intractable and very difficult. Recently an alternative approach to modeling has emerged under the rubric of soft computing with neural network and fuzzy logic as its main constituents. The development of these models, however, requires a set of data. Fortunately, for many problems of civil engineering such data are available.

The purpose of this chapter is to investigate the accuracy of an adaptive neuro-fuzzy inference system and neural network to solve civil engineering problems: The ANN and ANFIS are used to predict the shear strength of concrete beams reinforced with fiber reinforced polymer (FRP) bars and shear strength of ferrocement members. The performance of the ANFIS and ANN models are compared with experimental values and with those of the other methods to assess the efficiency of these models. The study is based on the available databases.

Fig. 1. An input-output mapping

### **2. Artificial neural network**

One type of network sees the nodes as artificial neurons. These are called artificial neural network (ANN). An artificial neuron is a computational model inspired in by natural neurons. Natural neurons receive signals through *synapses* located on the dendrites or membrane of the neuron. When the signals received are strong enough (surpass a certain *threshold*), the neuron is *activated* and emits a signal through the *axon*. This signal might be sent to another synapse, and might activate other neurons (Gershenson, 2003). Fig. 2 shows a natural neuron.

The adaptive neuro-fuzzy inference system (ANFIS), first proposed by Jang, 1993, is one of the examples of neuro-fuzzy systems in which a fuzzy system is implemented in the framework of adaptive networks. ANFIS constructs an input-output mapping based both on human knowledge (in the form of fuzzy if-then rules) and on generated input-output data pairs by using a hybrid algorithm that is the combination of the gradient descent and least squares estimates. Readers are referred to References (Jang, 1993; Mashrei, 2010) for more details on the ANFIS. After generated input-output by training, the ANFIS can be used to recognize data that is similar to any of the examples shown during the training phase .The adaptive neuro-fuzzy inference system has been used in the area of civil engineering to solve many problems (Abdulkadir et al., 2006; Akbulut et al., 2004; Fonseca el at., 2007;

Most of the problems solved in civil and structural engineering using ANFIS and ANN are prediction of behavior based on given experimental results that are used for training and testing data. The matter of modeling is to solve a problem by predicting which is obtained by mapping a set of variables in input space to a set of response variables in output space through a model as represented in Fig. 1. In the box representing a model in this figure, conventionally a mathematical model is used. However, the conventional modeling of the underlying systems often tends to become quite intractable and very difficult. Recently an alternative approach to modeling has emerged under the rubric of soft computing with neural network and fuzzy logic as its main constituents. The development of these models, however, requires a set of data. Fortunately, for many problems of civil engineering such

The purpose of this chapter is to investigate the accuracy of an adaptive neuro-fuzzy inference system and neural network to solve civil engineering problems: The ANN and ANFIS are used to predict the shear strength of concrete beams reinforced with fiber reinforced polymer (FRP) bars and shear strength of ferrocement members. The performance of the ANFIS and ANN models are compared with experimental values and with those of the other methods to assess the efficiency of these models. The study is based

One type of network sees the nodes as artificial neurons. These are called artificial neural network (ANN). An artificial neuron is a computational model inspired in by natural neurons. Natural neurons receive signals through *synapses* located on the dendrites or membrane of the neuron. When the signals received are strong enough (surpass a certain *threshold*), the neuron is *activated* and emits a signal through the *axon*. This signal might be sent to another synapse, and might activate other neurons (Gershenson, 2003). Fig. 2 shows

*Input Model Output*

Tesfamariam & Najjaran, 2007).

data are available.

on the available databases.

Fig. 1. An input-output mapping

**2. Artificial neural network** 

a natural neuron.

The complexity of real neurons is highly abstracted when modeling artificial neurons. These basically consist of *inputs*(like synapses), which are multiplied by *weights* (strength of the respective signals), and then computed by a mathematical function which determines the *activation* of the neuron. Another function (which may be the identity) computes the *output* of the artificial neuron (sometimes independent on a certain *threshold*). ANN combines artificial neurons in order to process information (Gershenson, 2003).

Compared to conventional digital computing techniques, neural networks are advantageous because of their special features, such as the massively parallel processing, distributed storing of information, low sensitivity to error, their very robust operation after training, generalization and adaptability to new information (Waszczyszyn, 1998).

Fig. 2. Natural (biological) neurons
