Prediction of Inelastic Response Spectra Using Artificial Neural Networks

Several studies have been oriented to develop methodologies for estimating inelastic response of structures; however, the estimation of inelastic seismic response spectra requires complex analyses, in such a way that traditional methods can hardly get an acceptable error. In this paper, an Artificial Neural Network ANN model is presented as an alternative to estimate inelastic response spectra for earthquake ground motion records. The moment magnitude MW , fault mechanism FM , Joyner-Boore distance dJB , shear-wave velocity Vs30 , fundamental period of the structure T1 , and themaximumductility μu were selected as inputs of the ANNmodel. Fifty earthquake ground motions taken from the NGA database and recorded at sites with different types of soils are used during the training phase of the Feedforward Multilayer Perceptron model. The Backpropagation algorithm was selected to train the network. The ANN results present an acceptable concordance with the real seismic response spectra preserving the spectral shape between the actual and the estimated spectra.


Introduction
The response spectrum is one of the most useful tools in earthquake engineering.In fact, earthquake-resistant design is based on the use of response spectra to estimate the seismic demand on structures.The response spectrum concept was used for the first time by Benioff 1 , and subsequently by Housner 2 and Biot 3 .Nowadays, seismic design codes around the world use response spectra to establish the design forces.Hence, several studies have tried to establish methodologies to estimate elastic and inelastic response spectra; in particular, the estimation of inelastic seismic response spectra requires complex numerical

Seismic Response Spectrum
The seismic response spectrum can be graphically displayed as an element that integrates the maximum response of a single-degree-of-freedom system with some level of damping ratio ξ and ductility μ when subjected to ground acceleration üg t .The seismic response may be defined by different parameters displacements, velocities, accelerations, etc. .In this study, pseudo-acceleration will be used to estimate the seismic response spectra, and consequently it will be the parameter of the output network, which is directly related with the resistance required by an earthquake-resistant structure.Moreover, the spectral shape in terms of the pseudo-acceleration spectrum is a very important parameter related with the ground motion potential of an earthquake Boj órquez and Iervolino 14 .
The idealized force-displacement relation of a structure subjected to an increasing load is shown in Figure 1.In this figure, the elastic and the inelastic behaviors are illustrated.The main difference between the elastic and the inelastic behavior is the capacity to dissipate energy when the system presents inelastic behavior.In Figure 1, the inelastic system behavior is represented by the well-known elasto-plastic curve which is commonly used in several seismic design codes around the world.
For earthquake engineering purposes, it is desirable to estimate the peak displacement of an inelastic system u m due to the action of the earthquake ground motion and to compare this displacement with the peak displacement caused by the same excitation acting in the corresponding linear system u o .The elastic system is assumed to have the same stiffness as that of the inelastic system during the initial loading.Both systems have the same mass and damping.
The normalized yield strength f y of an inelastic system is defined as follows: where f y and u y are the force and deformation at yielding.If the normalized yield strength of a system is less than unity, the system will yield and deform into the inelastic range e.g., f y 0.5 implies that the yield strength of the system is one half of the minimum strength required for the system to remain elastic during the ground motion .
The procedure to obtain the seismic response spectrum for inelastic systems corresponding to specified levels of ductility is summarized as follows.
1 Numerically define the earthquake ground motion üg t .
2 Select and fix the damping ratio ξ for which the spectrum is to be plotted.
3 Select a value for the natural period of vibration of the system T 1 .
4 Estimate the response at each time instant u t of the linear system with T 1 and ξ equal to the values previously selected.From u t compute the peak deformation u o and the peak force where k is the stiffness of the system.Note that for the estimation of the elastic response spectra an iterative procedure is not required as in the case of the inelastic response spectra, where it is necessary to calculate the strength required to have a specific target ductility value, while in the elastic case the ductility is one.
5 Determine the response u t of an inelastic system with the same T 1 and ξ and yield force f y f y .fo , for a selected f y < 1.From u t determine the peak deformation u m and the associated ductility factor μ u u m /u y .Repeat the analysis for several values of f y to develop data points f y , μ u , until the target ductility is obtained.
6 Estimate the spectral ordinates corresponding to the value of f y computed in step 5 associated to the target ductility.The pseudo-spectral acceleration is obtained as Mathematical Problems in Engineering Sa 2 • π /T 1 2 • u m , corresponding to the ordinate of one point on the spectrum with abscissa equal to T 1 .
7 Repeat steps 3 to 6 for a wide range of natural vibration periods.
For further information about seismic response spectra see Chopra 15 .
The response spectrum for inelastic system developed according to this procedure can be used to establish the seismic design forces for ductile structures.The design forces of an inelastic system are smaller than the design forces of its corresponding elastic system, implying smaller sections of the structural members.In other words, elastic seismic designs are more expensive than inelastic seismic designs.The knowledge of the inelastic seismic response of structures is important in earthquake engineering due to the economical optimization of earthquake-resistant structures.

Artificial Neural Networks
The theory of Artificial Neural Networks arises from the need of solving complex problems, not only as a sequence of steps, but also as the evolution of computational systems inspired by the human brain and therefore endowed with certain "intelligence."An ANN is a mathematical or a computational model inspired by the structure and functional aspects of biological neural networks.The structure of a neural network is as follows: neurons are the main processing element, these are connected to other neurons via a signal weight synapse , the entries are the dendrites, and the result is the axon Figure 2 .As in the case of neural networks, the ANN needs a process of learning to establish relationships between the variables that define a specific phenomenon.The processing power of an ANN is due to its distributed parallel structure, and the ability to learn from some examples, obtaining acceptable results from patterns never shown to it.
There are many topologies established by different authors to define the structure of the ANN; nevertheless, in this work the Feedforward Multilayer Perceptron FMP was selected Shepherd, 16 .Figure 3 illustrates the architecture of the FMP.The architecture begins with an input layer which is connected to a hidden layer; this can be connected to another hidden layer or directly to the output layer.Because the information flow is always from the input layer to the output layer, the output of a layer is always the input of the next layer.
The training of the ANN used in this study was carried out using the "Backpropagation" algorithm proposed by Rumelhart and McClelland 17 .The procedure is as follows: 1 A set of patterns consisting of pairs of inputs and outputs is applied to the network. 2 The input data information is entered through the first layer.This information is propagated over the network according to a propagation rule: the entries are multiplied by the weights of the connection between layers, and the output is transformed by a nonlinear function and transferred to the next layer there are several transfer functions, e.g., step, linear, mixed, Gaussian, hyperbolic tangent, secant hyperbolic, and sigmoid .Many investigations have used a sigmoid transfer function in the hidden layers and linear function in the output layer getting satisfactory results.For this reason herein the sigmoid function was selected as propagation rule.The same procedure is applied to the following layers until the output of the network is obtained.The distribution of the records in terms of moment magnitude and d JB distances is provided in Figure 4.In this figure, it can be observed that the records were obtained for different distances and from different events as moment magnitudes indicate; the wide range of the selected distances and magnitudes is very important for the training of the Artificial Neural Networks.
Tables 1, 2, 3, 4, and 5 summarize the main characteristics of the seismic records used in this study corresponding to all types of soil zones in accordance with the shear-wave velocity .Soil type A corresponds to rock and soil type E to soft soil.In the tables, the first column refers to the record number, the second column corresponds to the moment magnitude of the event, the fourth column refers to the Joyner and Boore distance, and the last column is the shear wave velocity above site depth of 30 m.In the next subsections, the definition of the failure type column 3 and the soil type are given.

Failure Mechanisms
The selected ground motion records were generated by four different failure mechanisms: Longitudinal 0 , Normal 1 , Reverse 2 , and Oblique 3 .It is observed that for the selection of the record accelerations there were considered an average of 12 records for each type of failure, so that ANN could generalize the estimation of spectra coming from motions corresponding to different type of failure mechanism.The use of different record features is very helpful to reduce the error in the estimation of inelastic response spectra.Figure 5 shows the number of events corresponding to each type of failure mechanism.

Types of Soil
As stated earlier five soil types are used in this study.They are expressed as a function of shear-wave velocity.Table 6 illustrates the soil type as a function of shear-wave velocity, as classified by NEHRP.Note that soil type A represents a very hard soil, while soil types E corresponds to a soft soil.Ten records were selected for each soil type.From Figures 4 and 5, and Tables 1, 2, 3, 4, and 5, it can clearly be seen that the selected ground motion records cover a wide range of earthquake sources and site characteristics.

Artificial Neural Network Model
The aim of this part of the study is to implement Artificial Neural Networks to estimate inelastic response spectra using the Multilayer Perceptron Backpropagation .To achieve this objective, in the first place, appropriate parameters must be selected for the training phase, and in the second places, it is necessary to find the optimal network architecture.The most important limitation of ANN is that its efficiency depends on the training algorithm and network architecture.Unfortunately, there are no rules for determining both characteristics of the network.Only by using the trial and error procedure it is possible to find the optimal network.In order to obtain satisfactory results with the Backpropagation algorithm it is necessary that the values of the training vector represent the entire domain of the output.

Estimation of Input and Output Parameters
An appropriate selection of the parameters that represent an ANN is required for the correct working of the network.The parameters required to estimate the seismic response spectra in terms of pseudo-acceleration are moment magnitude, mechanism of failure, Joyner-Boore distance, shear-waves velocity, fundamental period of vibration of the system and maximum ductility.According to the above, inputs to the network are defined here by the values of M w , F M , d JB , V s30 , T 1 , and μ u .The output node is represented by the pseudo-acceleration at first mode of vibration Sa T 1 .The values of the input parameters are shown in Tables 1 and  6, and the input range values are presented in Table 7.Note that the input parameters are usually selected to propose ground motion attenuation prediction equations, and the chosen ductility levels are commonly used for the seismic design of buildings.

Architectures Used for the ANN Model
A large number of architectures were tested in order to obtain the best ANN model.The format for the architecture arrangements shown in Table 8 is described by where I represents the number of neurons in the input layer; H i represents the number of neurons in the ith hidden layer; O represents the number of neurons in the output layer.Table 8 shows some of the models that were used.In this table we can see that models with more than one hidden layer have a lower total error in the training phase compared with one hidden layer models.Nevertheless, these models in the testing phase generated a much higher total error than models with one hidden layer.This is because the model is overtrained and the network estimates the spectra in the training phase with very good accuracy; however, the same is not valid for the testing phase, which implies that the network could not be generalized.It turned out that the optimal architecture for this problem is a multilayer neural network with feedforward.The selection of the optimal architecture of an ANN is not an easy task as it is necessary to test a large number of architectures to achieve the best one.In this paper, under the authors' judgment, it was assumed that the optimal model architecture consists of an input layer, one hidden layer, and one output layer, with the topology 6×100×1 which means that 6, 100, and 1 neurons were used in the input, hidden and output layers, respectively see Table 8 .

ANN Model, Training, and Testing Phases
As it was described before, the Backpropagation algorithm was selected for the training phase.The transfer functions used in the network layers were sigmoid for the hidden layer and linear function for the output layer.During training it was found that the ANN had trouble in estimating the inelastic response spectra, so it was necessary to test a large number of architectures.To train and test the ANN models, a computer program was developed that includes routines for MATLAB Neural Network Tool Box Demuth et al. 19 .The selection of training parameters was such that the ANN had a predictive capability for events not used during training.Figure 6 shows the regression results in the networks training, which compares the value estimated by the network and the actual value.While Figure 6   to case b was much larger than that corresponding to case a.This is due to the "overlearning" problem, which has been observed in all the networks with two or more hidden layers.

Test Results
After the training gave adequate results, in the test phase some random records were chosen from the NGA database in order to test the network these records were never shown to the ANN model .The test phase results are shown in Figures 8 a -  The results of the ANN model to estimate inelastic response spectrum corresponding to different accelerations records are acceptable.It is observed that the error in the estimation increases as ductility gets larger.When estimating response spectra for different soil types, the network presents more difficulties in estimating spectra for soil type E than for soil type A; hence the estimation, of the spectra for soft soils generates a higher error than for intermediate or stiff soils.This is because there is more uncertainty when soft soil records are considered to estimate nonlinear structural response.The most important observation is that the ANN model retains all the characteristics of the spectral shape of the earthquake response spectra, which is a crucial aspect related to the ground motion potential of an earthquake.

Conclusions
An Artificial Neural Network model to estimate inelastic seismic response spectra for different types of soils and seismicity zones is proposed.The moment magnitude of the event, mechanism of failure, Joyner and Boore distance, shear-wave velocity, fundamental period of the structure, and ductility were selected as the inputs for the ANN model.The output was represented by the spectral pseudo-acceleration response.It is important to say that the input parameters selected are usually used to propose ground motion attenuation prediction equations.
It is concluded that the ANN model proposed to estimate inelastic seismic response spectra gives an acceptable concordance when compared to that of the real spectra.The main conclusion is that the ANN represents a good accuracy for estimating the elastic response spectra, since it gives the lowest error.In the case of the inelastic spectra, the error increases as the ductility increases; however, the error in the estimation of inelastic spectra using the ANN model is acceptable; more importantly, it preserves the spectral shape of the real response spectra.
It is suggested that for successful operation of the ANN model, it is essential to select properly the architecture, method and learning algorithms as well as the main parameters that represent the specific problem under consideration.

Figure 1 :
Figure1: Force-displacement relationship of an inelastic system and its corresponding linear system.

Figure 5 :
Figure 5: Distribution of selected failure mechanisms.

a shows the 6 ×b
100 × 1 Model, Figure 6 b illustrates the 6 × 100 × 100 × 100 × 1 Architecture.The regression line indicates the precision of the ANN in the estimation of the real value of the spectral acceleration; the error in the training phase is represented by the slope of the line an ideal ANN model should have a 1 : 1 slope .It is observed that for both cases the slope of the regression line is almost 1, which implies that the model predicts with good accuracy the spectral acceleration.Although for Case b the error estimated in the training is slightly lower than the error for Case a, in the test phase the error produced by the network corresponding Architecture 6 × 100 × 100 × 100 × 1

Figure 6 :
Figure 6: Comparison of the regression results of the ANN corresponding to the training phase.

Figures 7 a
Figures 7 a -7 d compare the inelastic seismic response spectra for different ductility levels and soil types obtained during the training phase dashed line with the actual spectra computed with the real records solid line .The results of the training stage show an acceptable relation between the predicted and the real spectra.The maximum error between the actual response spectra and that obtained with ANN was 12%.It is observed that the spectral shape presents a clear definition throughout the range of periods.This is very important because the spectral shape of the pseudo-acceleration spectrum is an indicator of the ground motion potential of an earthquake Boj órquez and Iervolino 14 , Bojórquez et al. 20 ; moreover, ground motion intensity measures based on spectral shape are very useful to estimate the seismic fragility of buildings in terms of peak and energy demands Boj órquez et al.21 , Bojórquez et al. 22 .The control of seismic energy demands is very important on structures subjected to long duration ground motion records Teran-Gilmore and Jirsa 23 , Bojorquez et al. 24 .Note that the shape of the spectra shown in Figures 7 a -7 d is very similar for different nonlinear levels and types of soil, which suggests that the training of the ANN model was satisfactory.
8 d .Note that in the testing stage the ANN results are shown to be as good as those obtained during the training stage; in fact, the results shown in Figures 8 a -8 d indicate a very good relationship between the predicted spectra via ANN and the real spectra; this is valid for different types of soils and ductility levels.It is observed that the ANN model keeps its spectral shape in all the cases.

Figure 7 :
Figure 7: Comparison of earthquake response spectra obtained during the training phase ANN versus real .

Figure 8 :
Figure 8: Comparison of earthquake response spectra obtained during the testing phase ANN versus real .

Table 1 :
Ground motion records for soil type A.

Table 2 :
Ground motion records for soil type B.

Table 3 :
Ground motion records for soil type C.

Table 4 :
Ground motion records for soil type D.

Table 5 :
Ground motion records for soil type E.

Table 6 :
Soil types under consideration.

Table 7 :
Range of input parameters for the neural network.