Keywords: neural network, non-linear behaviour, restoring force, earthquake, dynamic response analysis.

This paper describes multiple layered neural network to simulate the non-linear hysteretic behaviour like Ramberg-Osgood model, modified bilinear model and Takeda model. In this study, based on the pattern recognition ability of neural network, non-linear hysteretic behaviour was modeled by the network directly without replacing it with a mathematical model. The effectiveness and applicability of the network in numerical analysis were evaluated. Neuron is the nervous cell system of creature. Neuron model can simulates the information processing of this brain nervous system industrially. Multiple layered neural network was adopted. Information is transmitted toward the output layer from the input layer in one-way direction on the network, Multiple layered neural network is simple to handle and easy to apply. Neural network transmits the signals from the input layer to the output layer by way of the hidden layers when the input signals is received, and output the output signals finally. When an error between these output signals and the data for learning occurred during, the weight of connection is modified to reduce this error.

This modification process is equivalent to learning by neural network. Modification proceeds to minimize the error toward the input layer from the output layer in learning algorithm. The parameters that can evaluate hysteretic behaviour like Ramberg-Osgood model, modified bilinear model and Takeda model need to be selected adequately for the input layer of neural network. It is to be desired that selected parameters can be taken easily from loading tests as the data for learning of the network. The input layer were selected as follows referring the past studies [1,2,3,4]. Maximum displacement, maximum load, minimum displacement, minimum load, displacement and load at the latest turning point, increases of previous displacement, increases of previous load, previous tangential stiffness, previous displacement, previous load were selected. Above eleven units and present displacement were defined as inputs for the input layer. The output layer has one unit of present tangential stiffness. Structure of the network was three layers network with one hidden layer.

Forced displacement of increased and decreased sinusoidal wave was given to single degree of freedom system with non-linear spring of Ramberg-Osgood model, modified bilinear model and Takeda model. The response was adopted as a data for learning of neural network. Forced displacement was formed with continuously and gradually increased and decreased sinusoidal wave. Square of error in the data for learning was converged in three types of hysteretic model. Tangential stiffness and hysteretic curve were compared. The estimated value by the network after learning and the three types of hysteretic curves were approximately agreed. Error Back Propagation Method (BP method in following) is used as learning algorithm developed by Rumelhart in this study. BP method is the most general learning algorithm for neural network. In this study, average coefficient a and learning velocity coefficient ? were introduced to control the adjustment of the weight.

Applicability of learned neural network as a subroutine in dynamic response analysis was investigated. Modified data of Japan Meteorological Agency Kobe in HyogokenNanbu earthquake 1995 was used as input acceleration for dynamic response analysis. Dynamic response analyses using single degree of freedom system considering mathematical model like Ramberg-Osgood model, modified bilinear model and Takeda model and single degree of freedom system considering neural network model were conducted. The time histories of displacement and restoring force between estimated by the network after learning and calculated by mathematical model in three types of hysteretic model were compared. In Ramberg- Osgood model, the time histories of displacement and restoring force approximately agreed each other. In modified bilinear model, maximum responses of both models were agreed each other. But in the first half of the time history of displacement, the error is relatively large. In Takeda model, the phase of responses of both models was not agreed in the latter half of the time histories.

Generalized multiple layered neural network to evaluate non-linear hysteretic curve was constructed. The network can recognize well the three types of hysteretic curve. The network is available as a subroutine of non-linear spring in dynamic response analysis.

1

J. Ghaboussi, J.H. Garrett Jr and X. Wu, "Knowledge-Based Modeling of Material Behavior with Neural Networks", ASCE Vol.177, No.1, 132-153, 1991. doi:10.1061/(ASCE)0733-9399(1991)117:1(132)

2

K. Yamamoto, "Modeling of Hysteretic Behavior with Neural Networks and its Application to Non-Linear Dynamic Analysis" Journal of Structural Engineering JSCE Vol.38A, 85-94, 1993.

3

T. Mazda, Y. Kabayama, T. Irie and T. Takayama, "A Study of Application of Neural Network to Non-Linear Dynamic Analysis", Journal of Structural Engineering,JSCE, Vol.42A, 635-644, 1996.

4

T. Mazda, H. Otsuka and W. Yabuki, "Recognition of Hysteretic Behaviors of High Damping Rubber Bearing using Neural Network", Seismic Engineering 2000, ASME PVP-Vol.402-1, Volume 1, 245-250, 2000.

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