An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network

doi:10.1007/s11709-017-0440-8

Front. Struct. Civ. Eng.

2017, Vol. 11

Issue (4) : 470-479 https://doi.org/10.1007/s11709-017-0440-8

RESEARCH ARTICLE

An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network

P. ZAKIAN(

)

Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran 14115–397, Iran

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Abstract

Since a lot of engineering problems are along with uncertain parameters, stochastic methods are of great importance for incorporating random nature of a system property or random nature of a system input. In this study, the stochastic dynamic analysis of soil mass is performed by finite element method in the frequency domain. Two methods are used for stochastic analysis of soil media which are spectral decomposition and Monte Carlo methods. Shear modulus of soil is considered as a random field and the seismic excitation is also imposed as a random process. In this research, artificial neural network is proposed and added to Monte Carlo method for sake of reducing computational effort of the random analysis. Then, the effects of the proposed artificial neural network are illustrated on decreasing computational time of Monte Carlo simulations in comparison with standard Monte Carlo and spectral decomposition methods. Numerical verifications are provided to indicate capabilities, accuracy and efficiency of the proposed strategy compared to the other techniques.

Keywords stochastic analysis random seismic excitation finite element method artificial neural network frequency domain analysis Monte Carlo simulation

Corresponding Author(s): P. ZAKIAN

Online First Date: 25 July 2017 Issue Date: 10 November 2017

Cite this article:

P. ZAKIAN. An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network[J]. Front. Struct. Civ. Eng., 2017, 11(4): 470-479.

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http://journal.hep.com.cn/fsce/EN/10.1007/s11709-017-0440-8
http://journal.hep.com.cn/fsce/EN/Y2017/V11/I4/470

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Fig.1 Radial basis function neural network.

Fig.2 Spectral density function of the seismic excitation.

Fig.3 Schematic of the soil Medium.

Fig.4 The shear modulus distribution over soil depth.

Fig.5 Mean spectral density function of soil response with SFEM.

Fig.6 Standard deviation spectral density function of soil response with SFEM.

Fig.7 Mean spectral density function of soil response with MC and MC-ANN procedures.

Fig.8 Standard deviation spectral density function of soil response with MC and MC-ANN procedures

Fig.9 Error of mean of spectral density function of soil response

Fig.10 Error of standard deviation of spectral density function of soil response

Fig.11 Convergence curve of radial basis ANN training.

Tab.1 Elapsed Time for the three procedures.

a	Coefficients of polynomial chaos
$b$	Correlation length
$b ‾$	Bias
$c$	Coefficient of polynomial chaos in compact form
$c ‾$	Central input vector for ANN
$C$	Response vector in frequency domain
diag [.]	Diagonal terms of a matrix
$d$	Euclidean distance
$E [.]$	Expected value
F₀	Scale factor
$F$	Force vector in frequency domain
f(s)	Radial basis function
$G$	Shear modulus of soil
$G *$	Complex shear modulus
$H$	Transfer function
$K ‾ e$	Element stiffness matrix: mean part
$K e$	Element stiffness matrix
$K ‾$	stiffness matrix: mean part
$K$	stiffness matrix in frequency domain
$l e$	Length of an element
$m$	Number of KL terms
M_e	Element mass matrix
M	Mass matrix
m_I	Influence vector
$N$	Number of elements
$P$	Number of PC terms
$R α α$	Autocorrelation function
R	Epicentral distance
r	polynomial chaos in compact form
s	Earthquake duration
$S$	Spectral density function
t_l	Strong motion duration
T_a	Mean period of the earthquake excitation
u	Horizontal displacement
$u ¨ g ? max ?$	Maximum ground acceleration
$w$	Weights of the second layer for ANN
x	Input vector for ANN

$φ$	Eigenfunction
$λ$	Eigenvalue
$Δ u ¨ g$	Correlation factor
$δ G$	Scale of fluctuation
$σ G$	Standard deviation of shear modulus
$ρ$	Soil mass density
$ψ 0$	Shape function for non-stationary effect
$Γ$	Polynomial chaos
$ω ‾$	Solution to the transcendental equations
$ω$	Component frequency
$ω g$	Frequency of the ground
$ξ g$	Damping ratio of ground
$ξ$	Random variable
$θ$	Member of random space
$α G$	Fluctuation part of shear modulus
$β$	Frictional damping
$〈 . 〉$	Mean value

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