Abstract

The estimation of structural reliability is a process that requires a large number of computational hours when statistical data are not available since it is necessary to perform a large amount of analysis or numerical simulations to estimate parameters related to the reliability. A methodology is proposed for estimating the structural reliability index, as well as the demand and structural capacity factors inherent to the structure, given the fundamental vibration period and the height of the structure, by using artificial neural networks (ANN) and, alternatively, the response surface method (RSM). Both approaches are applied to steel wind turbine towers. For the cases studied, ANN allow evaluating the reliability index and both the demand and structural capacity factors with greater accuracy than when using RSM.

1. Introduction

The methodology proposed by SAC-FEMA [1] for the evaluation of the demand and capacity factors based on structural reliability is a basic procedure that engineers usually use; however, its application implies carrying out a large number of analyses or simulations to estimate the safety factors and the reliability index implicit in the structural designs. Due to this limitation, numerical simulation methods have been applied to obtain those factors; for example, FORM and SORM methods, together with simplified probabilistic models, consider a given limit state [2, 3], or considering material fatigue [4, 5]; however, these procedures require great numerical processing time.

In the literature, methods have been proposed to obtain the probability of failure of structures that optimize computational resources. An example is the response surface method (RSM) [6], which can be as simple as a second-order equation, up to exponential or high-order response surfaces [7, 8], and even modified techniques where heavy polynomial calibrations are applied [9], as well as the use of theoretical evidence and control points to create the response surface [10].

In recent decades, computational processes have been used to solve complex engineering problems [11, 12], such as optimization design [13, 14], structural response [15], and others. Artificial neural networks (ANN) for structural reliability analysis have been used since 1989 when Hornik et al. [1618] proved that multilayer networks can approximate any function with a high degree of accuracy. Some other applications are as follows: Chen [1921] uses genetic algorithms to calculate the reliability index of bridges, trusses, and flat frames; Bojorquez et al. [22] propose a methodology based on optimization to find live, dead, and earthquake load factors for design of buildings located on soft soil; Dai and Cao [23] use hybrid methods (ANN and wavelet neural network) to obtain the probability of failure of various structures; Santana Gomes [18] calculates the probability of failure of several structures using adaptive ANN and compares its results obtained with these networks with those from Monte Carlo simulations; Wen et al. [24] optimize ANN to obtain the reliability of gas lines and compare their results with an unoptimized network and with Monte Carlo simulations; however, there are very few studies focused on the use of artificial intelligence for the calculation of structural safety factors; for example, Koopialipoor [25] uses the ant colony optimization algorithm to maximize the safety factor of retaining walls; Zeroual [26] calculates safety factors for dams, using a feedforward backpropagation network; Gordan et al. [27] obtain safety factors for retaining walls using ANN and “artificial bee colony”; and Zhang et al. [28] propose a Bayesian network for calculating pile resistance factors.

Response surface and ANN methods have focused mainly on obtaining the probability of failure of the structures; however, no efforts have been made to evaluate the structural reliability index, nor the demand and structural capacity factors implicit in wind turbine towers, which is a research gap that is addressed in the present study.

Based on the above, the main objective of this research is to propose a methodology to estimate the structural reliability index, together with the structural demand and structural capacity factors; the estimations are through ANN and alternative through RSM. The objective of the experimental analysis is to compare the performance of ANN and RSM based on accuracy. For this study, both approaches are applied, as part of the proposed methodology, only to steel wind turbine towers installed in open terrain, specifically in La Ventosa region of Oaxaca, in Mexico.

2. Reliability Analysis

Cornell et al. [1] propose a probabilistic methodology to obtain structural reliability, which is expressed in terms of the annual structural failure probability or the structural reliability index . The objective is to obtain the probability of exceeding a structural parameter level (for example, the maximum expected displacement associated with a given intensity).

Structural reliability can be described in terms of the mean annual exceedance rate , as shown in (1), which represents the expected number of times that environmental actions d, exceed the structural capacity C, evaluated for an intensity y, according to the site environmental hazard curve [29, 30].

Cornell et al. [1] propose closed equations to solve (1), and their method is based on characterizing the natural hazard and the median structural demand . In (2), is a function of the intensity y and the shape parameters k and r. In (3), a and b are the shape parameters.

Considering the previous aspects, (1) can be rewritten in (2) and (3) [31, 32].where is the median of the structural capacity corresponding to a certain limit state; and correspond to the variances of the natural logarithm (random uncertainties) of capacity and structural demand, respectively; and represent the epistemic uncertainties of capacity and structural demand, respectively.

Once the expected number of failures per year is calculated, the probability of failure can be obtained using (5) (assuming that it is a nonhomogeneous stochastic Poisson process). From this, the reliability index can be obtained (see (6)).

To achieve the appropriate level of reliability, it is established that the median factored capacity associated with a limit state must be greater than or equal to the factored demand associated with a desired annual failure rate [1].where is the structural capacity factor, and is the structural demand factor. is calculated with (8), where is a desirable or target annual failure probability, a and b are parameters that characterize the demand, and k characterizes the hazard (i.e., seismic or wind).

The capacity and demand factors are, respectively, as follows:where represents the total uncertainties associated with the structural capacity, and represents the total uncertainties corresponding to the structural demand.

The relation between factored capacity and factored demand is called the confidence factor . This factor indicates whether the structural design will achieve the desired performance for the limit state being reviewed. If is greater than or equal to 1, the structure's average annual failure rate is on the safety side [1].

3. Response Surface Method

A response surface refers to a polynomial that relates a mechanical model to its numerical response. Basically, it is an equation that relates the variables of a system with results obtained either from experimentation, numerical simulation, or theoretical data [6]. The particularity of the method is to obtain a function simple enough, in grade and in the number of variables, and that, at the same time, allows solving the problem.

The second-order equation is the most used way to solve response surfaces; however, for problems that present some degree of nonlinearity, the second-order polynomial equation can be applied with the mixed terms of the variables that intervene in the function [7].where is the number of variables, that is, height, diameter, weight, forces, and , , , and are unknown coefficients, which are obtained through regression analysis, such as using the least-squares method [33], or some other adjustment model that adapts to the problem.

To validate the results obtained from the response surface, the coefficient of determination is used (see (12)), which indicates the precision of the approximation between the real data and those predicted by the response surface [7].where is the total number of points, and the mean value of function is as follows:

In the present study, second-order response surfaces are obtained from structural reliability analysis, corresponding to 54 different models of wind turbine steel towers.

4. Artificial Neural Networks

Artificial neural networks (ANN) are algorithms designed to solve a specific problem. They imitate the human learning process, based on the configuration of the neurons of the nervous system, which are capable of learning through experience, generalizing, extending, or expanding a situation, and abstracting the characteristics of the object of study [34, 35]. McCulloch and Pitts [36] propose the first model of an artificial neural network; many ANN forms have emerged since this appearance. ANN can be classified in different ways, including the following: according to the number of layers, in single layer and multilayer [37]; by the architecture learning, in supervised [38], unsupervised [39], hybrid [40, 41], and reinforced [42, 43]; and by the data flow for the learning algorithm, in feedback and unidirectional [35].

A perceptron is the basic computational unit of an ANN. The perceptron introduction by Rosenblatt [44] was a radical innovation to this area. A study by Minsky and Papert [45] revealed that perceptrons could only learn linearly separable functions, limiting their research. In the 1980s, the works were reactivated by overcoming this limitation with the combination of perceptron layers (also called multilayer perceptron or MLP) [46]. To understand MLP, Taud [47] provides a brief introduction to a single-layer perceptron with one neuron. This perceptron works from input data , provided externally or by another perceptron; then, these data are transformed through linear coefficients called “weights” and a bias value . From this point, the transfer function transforms these weights to output data towards another neuron or final responses as represented in (14) and Figure 1. The MLP is a unidirectional layered feedforward ANN where the information flows from the input layer to the output layer, passing through the hidden layers [48]. Each neuron connection has its own weight. For the same layer, perceptrons have the same activation function that generally is a sigmoid [47].

The training architecture, or training algorithm, is responsible for updating the weights of the network. There are different training architectures. The regularization techniques, like Levenberg-Marquardt (LM) and Bayesian regularization (BR), allow obtaining lower errors than other algorithms applied in approximation problems [49]. Bui et al. [50] describe LM and BR techniques.

The way data are entered into networks affects their training. An extensive database increases the number of patterns required for training and, therefore, the network’s complexity. There are several recommendations to achieve a better trained ANN [51, 52], For example, reducing the sample size using statistical analysis and combining the number of variables to optimize the number of input and output variables [53], starting with a greater number of input variables and fewer neurons [54], or the scaling of data in a specific interval. Rafiq [55] proposed scaling in the interval [0, 1] or [−1, +1] by methods like linear normalization, as shown in equation (15). The data normalization into the range [−1, 1] would raise network training convergence because this range is the most sensitive for the variables’ sigmoid functions [55, 56].where S is the normalized value of the variable , and and are the minimum and maximum values of the variable, respectively.

Once the ANN are trained, it is necessary to validate the results. The most usual way to do this is by measuring the error, such as the mean absolute error, the root-mean-square error, or the mean square error (MSE) [55].where is the number of components of the output vector, is the vector of desired output values, and is the vector of output values predicted by the ANN.

In the present study, multilayer perceptron-based ANN are used, with unidirectional supervised learning, using a Sigmoid activation and transfer function and a linear output function. For the training, a database with unmodified (real) values and another database with normalized values are used in the training architectures Levenberg-Marquardt (LM) and Bayesian regularization (BR) using the Matlab tool [57]. The ANN are applied to obtain both demand and structural capacity factors and the structural reliability index implicit in steel wind turbine towers.

5. Case Study

Response surfaces were generated to obtain the structural reliability index, as well as the structural demand and structural capacity factors of wind turbine towers from the analysis results of 54 different steel towers' models. Likewise, ANN were trained to predict the same parameters.

The steel support towers were considered conical of variable diameter and thickness, based on S355 structural steel, with a modified density equal to 8500 kg/m3, which considers the contribution of accessories and screws. The rotor characteristics were considered the same for all tower models with a total weight of 836 kN and a diameter of 82.26 m.

It is considered the structure is located in an open terrain. Therefore, to simulate the wind forces on the tower, the auto-regressive moving average (ARMA) method [58] was applied at each meter of the tower’s height. Concerning the force on the rotor, the blade element momentum (BEM) method was applied for the rotor in operation, while the simplified BEM method was applied for the standstill condition [59].

Figure 2 shows the steps followed, and later sections (5.1 to 5.4) detail each step.

5.1. Parametric Analysis of Stability and Buckling

As a first step, a parametric analysis of stability and buckling is performed to determine the support towers’ acceptable geometry; it is carried out based on what is established by DNV-Riso [60] and Nicholson [61]. Different support tower geometries were tested [59] with heights of 70, 75, 80, and 85 m, lower and upper diameters of 3.0 to 4.5 m, and thicknesses of 2.5 to 4.5 cm.

The calculation of the displacement at the top of each tower model, the tower fundamental vibration frequency, the analysis of stability, and the tower's buckling at each meter was according to Section 7.4.7 of DNV-Risø [60]. For this purpose, the authors developed a program in Matlab [57].

The following conditions were verified to take as acceptable the configuration of the support tower:(a)Maximum diameter allowed, 4.5 m(b)Maximum thickness allowed, 0.040 m(b)Diameter at top of the tower > width of the nacelle(c)Tower natural frequency > rotor frequency(d)Critical compressive stress > acting forces(e)Displacement for a speed corresponding to a 50-year return period <3% of the tower’s height, corresponding to the collapse limit state [62]

There were obtained 88,320 different models of steel support towers. From them, there were selected 54 models that covered heights from 70 to 85 m, fundamental periods between 1.5 s and 3.0 s, and values of acceptable diameters and thicknesses. Table 11 shows the geometric characteristics of the 54 models, the fundamental vibration period, and the lateral force associated with the collapse design velocity, equal to 50 m/s.

5.2. Reliability Analysis

For each model, a structural reliability analysis was performed, solving equations (2) to (10) to obtain the structural capacity factor, the structural demand factor, the structural reliability index, and the confidence factor. The structural capacity was determined from the capacity curves obtained from 15 nonlinear incremental dynamic analyses (IDAs). The process was repeated 10 times, resulting in 810 structural reliability analyses.

The median of the structural capacity, the structural demand, and their respective uncertainties were obtained using capacity curves represented by tower’s top displacement versus mean wind speed. An expected annual probability of failure, equal to 0.0036, was selected [63]. The parameters and , obtained for the 54 models, were between 12 and 14 and between 6 and 8, respectively.

The results of the reliability analysis are represented with black dots in Figures 37. Also, they were used to form a database of 810 x11 elements, where each row corresponded to the results of the reliability analysis of a specific model, and the columns corresponded to the height, diameter, thickness, structural vibration period, lateral force, structural capacity factor, structural demand factor, reliability index, and confidence factor. The database was used to create the response surfaces, as well as to train the ANN.

5.3. Response Surface Method (RSM)

The main objective of response surface is that from simple variables, such as height, structural period, or geometry, complex parameters such as reliability index and the demand and structural capacity factors can be estimated as a function of a preset probability of annual failure. Table 1 lists the variables in the database with their corresponding notation.

To obtain the response surface, two different variables were related in polynomials up to the second degree. The coefficients of the equations were obtained using the least-squares method. Finally, the combination of variables that showed the highest coefficient of determination R2 was chosen, which indicates the model's quality to replicate the results and their variations.

Tables 24 show the correlation coefficients obtained from three response surfaces for the structural capacity factor, the structural demand factor, and the structural reliability index, respectively. Remarkably, the combination of the fundamental structural vibration period and the tower height yields a higher coefficient of determination with a second-degree equation to obtain the factors of capacity and of structural demand; on the other hand, to obtain the reliability index, the variables that yield a higher coefficient of determination are the fundamental structural vibration period and the lateral force. For higher-degree equations, the coefficient of determination tends to be higher; however, the equation becomes less comfortable for its application from an engineering point of view.

5.4. Artificial Neural Networks (ANN)

Alternatively, ANN were trained to predict the structural demand and capacity factors, as well as the structural reliability index. Analogously to what was done for the response surface, the variables that allowed the ANN to be trained with a higher coefficient of determination and a lower mean square error were chosen.

The results from Levenberg-Marquardt (LM) and Bayesian regularization (B-R) training architectures were compared. Two databases were used: one with exact data obtained from the reliability analysis and the other with normalized data using (15). Finally, it was concluded that the databases with exact numbers and the Levenberg-Marquardt training architecture resulted in values closer to the median value of the structural capacity and demand factors and also closer to the verification models' reliability index. Tables 5 and 6 show the coefficient of determination and the mean square error of the trained ANN.

6. Comparison of Methods

6.1. Response Surface Method

The main objective of the response surfaces proposed in this section is to evaluate the reliability index, as well as the demand and capacity factors, of an installed structure, by means of simple parameters such as the structural vibration period (x) and its height (y). The reliability index obtained from the 54 models is represented in equation (17) and in Figure 3.

Once the reliability index was obtained, the structural capacity factor and the structural demand factor could be determined as a function of the reliability index and the structural vibration period , as shown in (18) and (19) and in Figures 4 and 5. The main objective of these response surfaces is to obtain the capacity and demand factor required to achieve the desired reliability index, given the fundamental structural vibration period of the steel support tower.

Besides, response surfaces were obtained to determine the capacity factor and the structural demand factor necessary to achieve a desirable structural reliability index equal to β = 3.4 [63], as a function of the tower’s height ( and the structural vibration period (. This is shown in equations (20) and (21) and in Figures 6 and 7.

6.2. Artificial Neural Networks

Similar to the response surfaces, the main objective of the ANN is to obtain the reliability index of the existing structure, starting from simple parameters and avoiding excessive computing hours or complicated structural analyses.

The ANN were trained here using the database of the 54 models selected and using ten hidden layers and random samples for training, validation, and testing with a ratio of 70%, 15%, and 15% of the total database, respectively. After the testing, the results were verified with three different models whose characteristics are shown in Table 7 and were not found among the 54 models.

To predict the structural reliability index β, the following input parameters were used: tower’s height, structural fundamental vibration period, lateral force associated with a 50-year return period, and the desirable confidence factor should be greater than 1.0. Figure 8 shows results of the reliability index calculated from 10 reliability analyses (Equations (2) to (10)) for the three verification models, as well as the ANN training results.

The following input parameters were used to predict the structural capacity factor: tower’s height, structural vibration period, lateral force associated with a 50-year return period, structural reliability index, confidence factor, and displacement at the top of the tower, corresponding to the collapse limit state. Figure 9 shows the three verification models' reliability analysis results for the capacity factor and those predicted with ANN.

The following input parameters were used to predict the structural demand factor: the tower’s height, the structural vibration period, and the structural capacity factor. Figure 10 shows the three verification models' reliability analysis results and those predicted with ANN.

Table 8 compares the median value of the structural capacity factor, the structural demand factor, and the reliability index obtained from 10 reliability analyses (Equations (2) to (10)) for the three verification models, with the value obtained from the proposed equations and with ANN, respectively. The first column 1 in Table 8 corresponds to the tower’s height, column 2 is an identifying model number, column 3 indicates the fundamental structural vibration period, and columns 4, 5, 6, and 7 show results obtained from the reliability analyses (column 4), from the equations proposed in this study (columns 5–6), and from ANN predictions (column 7). The coefficient of determination (R2) and the mean square error (MSE) of the equations and ANN are also shown, compared to the reliability analyses results. Tables 9 and 10 show similar results for the structural demand factor and the structural reliability index.

Table 8 shows that, for the three verification models, ANN give place to greater accuracy for the capacity factor (R2 = 0.96), with an MSE six times smaller than that corresponding to equation (18) and 30% smaller than that to equation (20). Similarly, the reliability index obtained with ANN presents a higher R2 value than that corresponding to equation (17) (0.93 vs. 0.45), and an MSE value is eight times smaller (see Table 10). Even though for the structural demand factor obtained with ANN, the value of R2 is slightly smaller (1%) than that obtained with equation (19), and an MSE value is five times smaller. Therefore, for the cases studied here, it is concluded that ANN results lead to greater accuracy than those obtained with the RSM (see Table 9).

Figures 1113 show the results of Tables 810, respectively; it is clear that the results predicted by the ANN are more accurate than the results obtained by the response surfaces. This is because the training of ANN involves a greater number of variables for predicting the reliability parameters.

In Figure 11, it can be observed that for the 70–14 model, the ANN result does not present any difference for the capacity factor value obtained from the reliability analysis, while those calculated with (18) and with (20) present relative errors of 1.7% and 0.25%, respectively. For the 75–16 model, (20) leads to a smaller relative error (0.28%) than that corresponding to equation (18) (0.45%) and to ANN (1.14%). For the 80–16 model, the ANN present a smaller error (1.04%) than those corresponding to equations (18) and (20) (3.39% and 1.73%, respectively). In conclusion, for the cases corresponding to the structural capacity factor evaluation, ANN allow obtaining values with smaller errors than RSM.

In Figure 12, it can be seen that for 70–14 model, the ANN present a difference of 0.13% with respect to the structural demand factor value obtained from the reliability analysis, while (19) and (21) give place to relative errors of 0.22% and 0.27%, respectively. For the 75–16 model, the ANN present a smaller relative error (0.09%) than that obtained from equations (19) and (21) (both 0.23%). For the 80–16 model, ANN present an error (0.05%) which is smaller than that obtained from equations (19) and (21) (0.29% and 0.15%, respectively). Thus, in general, for the structural demand factor evaluation, ANN give place to values with minor errors than those corresponding to RSM.

Figure 13 shows that for 70–14 model, the ANN present a difference with respect to the structural reliability index value obtained from the reliability analysis equal to 0.47%, while (17) gives place to a relative error of 0.43%. For the 75–16 and 80–16 models, the ANN present smaller relative errors (1.32% and 0.05%, respectively) than those corresponding to equation (17) (1.80% and 3.57%, respectively). It is concluded that for the structural reliability index evaluation, ANN allow obtaining values with minor errors than those obtained with RSM.

7. Conclusions

Response surface and ANN methods allow estimating the structural reliability index as well as the structural demand and capacity factors of wind turbine towers, using basic variables. The results obtained from the present study are valid for support towers with similar characteristics to those described in Table 11 and installed inland in open terrain; however, the proposed methodology may apply to another type of structures.

With the ANN trained in this study, it is possible to evaluate the structural reliability index and the demand and structural capacity factors inherent in steel wind turbine towers like those located in La Ventosa, Mexico.

The response surfaces proposed in this study allow obtaining the reliability index inherent to the structure as a function of its height and fundamental structural vibration period, as well as obtaining the demand and structural capacity factors required to reach the desired reliability level, given the fundamental structural vibration period and the tower’s height.

For the wind turbine towers analyzed here, the ANN, trained with the Levenberg-Marquardt algorithm and a database with the exact values obtained from the reliability analysis, it was possible to predict more accurately the structural reliability index, as well as the demand and structural capacity factors, than when the Response Surface method was used. This result is because ANN recognize patterns of behavior and involve a larger number of variables in solving a problem.

To achieve a good training of ANN, it is necessary to have a wide database in which extreme and intermediate data of the variables involved are considered, as well as the choice of the appropriate number of input variables for the training of the network, since a large number of variables will tend to over-training or slow down the learning process. Likewise, uniformity must be maintained in the magnitude of the data, that is, although the input data units are different, they must have the same order of magnitude.

This study is a part of an ongoing research. Future works intend to expand the number of variables such as rotor area, type of terrain, and wind potential, as well as to consider different heights of support towers. In addition, ANN will be used in future research studies to predict the structural performance as well as to optimize the structural design of wind turbine towers, taking into account variables such as structural reliability index, the total cost of the structure, and wind potential.

Appendix

Table 11 shows the geometric characteristics of the 54 models that conform to the database, such as the tower’s height (column 1), an identification number for the model (column 2), diameter at the base and the top of the tower (column 3 and 4, respectively), thickness at the base and the top of the tower (column 5 and 6, respectively), and other characteristics as the fundamental vibration period (column 7) and the lateral force associated with the collapse design velocity, equal to 50 m/s (column 8).

Data Availability

The data supporting the results can be obtained under request to the first author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was done with the support of DGAPA-UNAM under project PAPIIT-IN100320. Thanks are due to PASPA-DGAPA-UNAM for the support given to the second author during her academic stay at the Instituto Tecnológico de Ciudad Madero.