The approximation of the probability density function (PDF) of the structural response is importance in structural reliability analysis. The recently developed cubic normal distribution can be used to represent the PDF of structural response due to its high flexibility and its large applicable range, whose parameters are evaluated by its first four moments. Therefore, efficient estimation of the moments is of great importance. Although some methods have been developed, evaluating the moments of structural response from the sight of balancing accuracy and efficiency remains a challenge, especially when the structural response is implicit and high-dimensional. In this paper, based on the artificial neural network (ANN), a new method is developed to efficiently estimate the statistical moments of structural response. The main procedure of the proposed method includes two steps: the structural response is approximated by the ANN and then the moments of structural response can be easily obtained. A RC frame structure with non-linear behavior is used to demonstrate the efficiency, accuracy, and applicability of the proposed method. The results show that the proposed method is of high accuracy and efficiency and provides a robust tool for representing the PDF of structural response.The study is partially supported by the National Natural Science Foundation of China (Grant Nos.: 51820105014, 51738001). The support is gratefully acknowledged

Cai, Chao-Huang

Li, Chun-Qing

Lu, Zhao-Hui

Zhao, Yan-Gang

SNU Open Repository and Archive

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  1 Estimating Distribution of Structural Responses based on Cubic Normal Distribution and Artificial Neural Network  Chao-Huang Cai Graduate Student, School of Civil Engineering, Central South University, Changsha, China Zhao-Hui Lu Professor, Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China (Corresponding author).  Yan-Gang Zhao Professor, Department of Architecture, Kanagawa University, Yokohama, Japan Chun-Qing Li  Professor, School of Engineering, RMIT University, Melbourne, VIC 3000, Australia ABSTRACT: The approximation of the probability density function (PDF) of the structural response is importance in structural reliability analysis. The recently developed cubic normal distribution can be used to represent the PDF of structural response due to its high flexibility and its large applicable range, whose parameters are evaluated by its first four moments. Therefore, efficient estimation of the moments is of great importance. Although some methods have been developed, evaluating the moments of structural response from the sight of balancing accuracy and efficiency remains a challenge, especially when the structural response is implicit and high-dimensional. In this paper, based on the artificial neural network (ANN), a new method is developed to efficiently estimate the statistical moments of structural response. The main procedure of the proposed method includes two steps: the structural response is approximated by the ANN and then the moments of structural response can be easily obtained. A RC frame structure with non-linear behavior is used to demonstrate the efficiency, accuracy, and applicability of the proposed method. The results show that the proposed method is of high accuracy and efficiency and provides a robust tool for representing the PDF of structural response.   1. INTRODUCTION Uncertainty is ubiquitous in civil engineering, such as loads, material properties, geometric dimensions, and boundary conditions. In this regard, the outputs of structural response are essentially random variables. For the design more rational, it is necessary to determine the statistical distribution of structural responses, which are functions of uncertain structural parameters described as random variables. The probability density function (PDF) of structural responses can, theoretically, be derived from the probability distributions of the input random variables. However, the derivation of the PDF of structural response is generally difficult because of the nonlinear, complicated, and implicit of structural responses.  Generally, the distribution of structural response can be determined through fitting a candidate distribution to the histogram and then performing goodness-of-fit tests. The candidate distribution is generally one- or two-parameter distributions, whose parameters evaluate from the mean and standard deviation of structural response. However, those one- or two-parameters distributions may be not flexible enough. The recently developed cubic normal distribution (Zhao et al. 2018a) can be adopted to represent the distributions of structural response 13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  2 duo to its high flexibility and its large applicable range. The parameter of cubic normal distribution is determined by the first four moments of random variable, i.e., mean, standard deviation, skewness, and kurtosis. In order to use the cubic normal distribution to represent the PDF of structural response, computing the first four moments of structural responses with efficiency and accuracy is of paramount importance.  Many approaches can be used to numerically calculate the first four moments of structural response, such as Taylor expansion based method (Ibrahim 1987; Singh and Lee 1993), original space two- or three-point estimate method (Rosenbluenth 1975; Gorman 1980), univariate dimension reduction method (UDRM) (Zhao and Ono 2000), bivariate dimension-reduction method (BDRM) (Zhao and Lu 2011; Xu and Rahman 2004; Fan et al. 2016; Cai et al. 2018), sparse grid stochastic collocation method (SGSCM) (He et al. 2014), cubature formulae (Xu and Lu 2017), etc.. Unfortunately, these methods may not be able to achieve a good tradeoff of accuracy and efficiency to estimate the high-order central moments when the dimensions of input random variables are large.  The objective of this paper is to represent the PDF of structural response using the cubic normal distribution, in which a high efficient method for determining the first four moments of structural response is developed.  2. THE CUBIC NORMAL DISTRIBUTION Without loss of generality, the structural response can be commonly expressed by a function Z = G(X), in which X = [X1, X2, …, Xn]T is an n-dimensional vector of random variables representing uncertain quantities such as loads, material properties, geometric dimensions, and boundary conditions. According to the fourth-moment transformation, the standardized form of Z can be approximated by a third-order polynomial of standard normal random variable U as following (Zhao and Lu 2007; Zhao et al. 2018b):  2 3( )Zs uZZZ S U a bU cU dU        (1) where Z and Z are the mean and standard deviation of Z, respectively. Su(U) is the third-order polynomial of U; and a, b, c, and d are the polynomial coefficients that can be determined by making the first four moments of Su(U) equal to those of Zs.  Based on the normal transformation theory, the CDF of Z should be equal to that of U, i.e., F(z) = (u). Taking the derivate of the left and right sides with respect to z, and using the relationship between Z and U, the PDF of Z, i.e., f(z), can be formulated as (Zhao et al. 2018a):  2( )( )(3 2 )Zuf zdu cu b   (2) where (u) is the PDF of standard normal random variable U. The complete monotonic expression expressed the relationship between u and z has been given by Zhao et al. (2018b).  3. STATISTICAL MOMENTS ESTIMATION To represent the PDF of structural response by the cubic normal distribution, computing the first four moments with efficiency and accuracy is of paramount importance. Many approximate methods (e.g., UDRM, BDRM, and SGSCM) can be used to numerically determine the first four moments of structural response. However, those methods may become not feasible especially when the implicit structural response with high dimension of input random variables because the evaluation of G(X) requires a time-consuming numerical calculation by mean of finite element analysis.  Since the artificial neural networks (ANNs) have the ability of accurately and effectively approximating the structural response. Based on ANN and Monte Carlo simulation (MCS) method, the present paper develops a new method for evaluating the first four moments of structural response.  The most widely used ANN is the multi-layer feed-forward back-propagation network, 13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  3 i.e., BP network. A typical BP network is shown in Figure 1, whose architecture is composed of a number of neurons situated on three or more layers, i.e., one input layer, one or more hidden layer and one output layer.   Figure 1. A three-layer feed-forward BP network with n inputs, k hidden neurons and one output The neuron is the basic processing element with several inputs and one output, and its output can be expressed as:  ,1nk k i i kia f u b       (3) where ui are the inputs; ak is the output; k,i and bk are the weights and bias of the kth neuron; and f() is the activation function or transfer function. The activation functions used in the present paper are the logistic function for hidden layers and the pure linear function for output layer, respectively. The logistic function is formulated as:  1( )1 exp( )f aa   (4) The number of neurons in input layer is the dimension of basic random variables, and that of output layer is the dimension of structural responses. However, the selection of the number of neurons in hidden layers is not a simple task and there is no general rule. In this paper, we first train several networks with different neurons and then select the best one.  The training of a network is an iterative process to update their weights and biases using optimization methods to minimize the mean square error between the predicted and the target values. Details of the ANN training algorithm can be found in Hagan et al. (1996). To overcome over-fitting during network training, the Bayesian regularization is introduced into the back-propagation algorithm. Because Bayesian regularization works best when the training data are confined within the range of [-1, 1], the inputs and outputs should be normalized. In this paper, the inputs and outputs are first normalized into [-1, 1] before presented to network by the following formulae (Lu et al. 2012):  ,min,max ,min2( )1i ii iu uuu u    (5) where ui are the original inputs; ui'  are the normalized inputs; and ui, min and ui, max are the original minimum and maximum of ui, respectively.  The training data must be prepared before training network. In this paper, based on Latin hypercube sample method, the samples in independent standard normal random variables (i.e., the inputs of network) are generated. With the aid of the inverse normal transformation (e.g., Liu and Der Kiureghian 1986), the original samples can be determined.  4. NUMERICAL EXAMPLE To illustrate the application of the proposed method, a 7-story RC frame structure with three bays is selected, as shown in Fig. 2. A non-linear structural analysis is performed. The total height of the building is 21.5 m. The height of the first story is 3.5 m, and that of upper stories is 3.0 m. The length of each bay is 6.0 m. The sizes of columns are 500500 mm for the first four stories and 450450 mm for 5th through 7th stories, and the sections of the beams are 200400 mm. More detailed information about the arrangement of the reinforcements of the column and beam members is shown in Figure 2.  1u2unu1u2unu1f1f1f2f ( )g u( )g u11,11,k n11b12b1kb21b21,121,221,k Input Layer Hidden Layer Output LayerNeuron13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  4  Figure 2. RC frame structure considered in example The hysteretic model developed by Mohd Yassin (1994) and perfect elastic plastic are used to simulate the nonlinear behavior of concrete material and reinforcing steel, respectively. The parameters and their distributions of materials are listed in Table 1, in which fpc, c, fpcu, cu, , ft, and Etc denote the concrete compressive strength, strain at maximum strength, crushing strength, strain at crushing strength, ratio between unloading slope at cu and initial slope, tensile strength, and tension softening stiffness, respectively; fy and Es represent the steel yield strength and Young’s modulus, respectively. The RC frame structure subjected to lateral loads, which is shown in Figure 2, is considered. The lateral loads (F1-F7) are assumed independent Gumbel distributed random variables, and their distribution parameters are listed in Table 1. In this example, we discuss the PDF of the horizontal displacement ux at Point1. The PDF of ux cannot be derived from the probability distributions of the input random variables because of the implicit structural response. To establish the network, 500 samples of independent standard normal random variables are generated by LHS, and 50 validation data are randomly selected. Several networks are trained that composed of one input layer, one output layer, and three hidden layers with 10, 20, and 30 neurons, and the best one was selected according to the mean squared error of validation data. The first four moments of horizontal displacement ux are determined as 0.06995, 0.02424, 0.94569 and 4.53626, respectively. Using the cubic normal distribution, the PDF of Z = ux can be given as:  2( )( )(3 2 )Zuf zdu cu b   (6a) in which Z = 0.02424, b = 0.9374, c = 0.14791, and  d = 0.013312. And u expressed as:  1/3 1/32 2 3q q cud                   (6b) where 2928 0.25q   ; 170.3 ( ) /Zq z d     Table 1. Distribution of Random Variables Variable Distribution Mean COV fpc Lognormal 48 Mpa 0.25 c Lognormal 2.310-3 0.25 fpcu Lognormal 40.8 Mpa 0.25 cu Lognormal 3.510-3 0.25  Lognormal 0.5 0.25 ft Lognormal 3.5 MPa 0.25 Ets Lognormal 4.0 GPa 0.25 fy Lognormal 400 MPa 0.05 Es Lognormal 200 GPa 0.05 F1 Gumbel 60 kN 0.40 F2 Gumbel 80 kN 0.40 F3 Gumbel 60 kN 0.40 F4 Gumbel 40 kN 0.40 F5 Gumbel 30 kN 0.40 F6 Gumbel 15 kN 0.40 F7 Gumbel 15 kN 0.40 Note: COV denotes coefficient of variation The PDF of the horizontal displacement ux at Point1 are pictured in Figure 3. It is seen that the PDF of the structural response function, determined by the proposed method, is in close agreement with the histogram obtained by MCS. It should be emphasized that the evaluation of the entire distribution is just based on the 500 times of repeated deterministic finite element analysis, which is highly efficient. 5. CONCLUSIONS Based on the cubic normal distribution, this paper proposed a method for representing the 6m 6m 6m3.5m3m3m3m3m3m3m0.5m0.4m0.4m0.5m0.4m0.2mF7F6F5F4F3F2F1C1 C1 C1 C1C1 C1 C1 C1C1 C1 C1 C1C1 C1 C1 C1C2 C2 C2 C2C2 C2 C2 C2C2 C2 C2 C2B1 B1 B1B1 B1 B1B1 B1 B1B1 B1 B1B1 B1 B1B1 B1 B1B1 B1 B1C1C2B161616181618Point113th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  5 PDF of structural response, in which the first four moments of structural response are obtained by combining the ANN and MCS. A nonlinear RC frame structure is used to validate the efficiency, accuracy, and applicability of the proposed method. The proposed method is highly efficient that only requires several hundred deterministic finite element analyses.   Figure 3. PDF comparisons for numerical example 6. ACKNOWLEDGMENTS The study is partially supported by the National Natural Science Foundation of China (Grant Nos.: 51820105014, 51738001). The support is gratefully acknowledged.  7. REFERENCES Cai, C. H., Lu, Z. H., Xu, J., and Zhao, Y. G. (2019). “An efficient algorithm for the evaluation of the statistical moments of performance functions.” Journal of Engineering Mechanics, 145(1): 06018007.  Fan, W. L., Wei, J. H., Ang, A. H-S., and Li, Z. L. (2016). “Adaptive estimation of statistical moments of the responses of random systems.” Probabilistic Engineering Mechanics, 43, 50–67. Gorman, M. R. (1980). ‘‘Reliability of structural systems.’’ PhD thesis, Case Western Reserve University, Cleveland, Ohio, 320–332.  Hagan, M. T., Demuth, H. B., and Beale, M. H., (1996). Neural Network Design. PWS, Boston. He J., Gao S.B., and Gong J. (2014). “A sparse grid stochastic collocation method for structural reliability analysis.” Structural Safety,51,29–34. Ibrahim, R. A. 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Estimating Distribution of Structural Responses based on Cubic Normal Distribution and Artificial Neural Network

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Estimating Distribution of Structural Responses based on Cubic Normal Distribution and Artificial Neural Network

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