A Neural Network Approach to the Modelling, BACK-PROPAGATION AND NEURAL NETWORK THEORY A neural network…

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  • ELSEVIER

    J. Construct. Steel Res. Vol. 44, Nos. 1-2, pp. 91-105, 1997 1997 Elsevier Science Ltd. All rights reserved

    Printed in Great Britain PII: S0143-974X(97)00039-4 0143-974X/97 $17.00 + 0.00

    A Neural Network Approach to the Modelling, Calculation and Identification of Semi-Rigid Connections

    in Steel Structures

    G. E. Stavroulakis, a A. V. Avdelas, b* K. M. Abdalla c & P. D. Panagiotopoulos b,d

    aInstitute for Applied Mechanics, Department of Civil Engineering, Technical University Braunschweig, D-38106 Braunschweig, Germany

    blnstitute of Steel Structures, Department of Civil Engineering, Aristotle University, GR- 54006 Thessaloniki, Greece

    CDepartment of Civil Engineering, Jordan University of Science and Technology, Irbid, Jordan

    dFaculty of Mathematics and Physics, RWTH Aachen, D-52062 Aachen, Germany

    ABSTRACT

    A two-stage neural network approach is proposed for the elastoplastic analy- sis of steel structures with semi-rigid connections. At the first stage, the moment-rotation law of the connection is obtained from experimental results by the use of a neural network based on the perceptron model. At the second stage, the elastoplastic analysis problem is formulated for the given moment- rotation law as a Quadratic Programming Problem and solved by a neural network based on the Hopfield model 1997 Elsevier Science Ltd.

    1 INTRODUCTION

    The highly nonlinear effects that have to be considered in the detailed model- ling of semi-rigid steel structure connections (e.g. unilateral contact and fric- tion 'prying effects' arising between adjacent parts of the connection [1-3], local plastification effects, etc.) require the use of time-consuming and compli- cated software not always available to the practising engineer. Another approach, permitted by modem design codes [4], is their treatment by means of simplified nonlinear relations. In the present paper, an alternative to the

    *To whom correspondence should be addressed.

    91

  • 92 G. E. Stavroulakis et al.

    above procedures is proposed: estimation of the mechanical behaviour of the steel structure connection by the use of learning algorithms in an appropriately defined neural network environment of experimental steel connections data. An estimated simplified law results which will be used as a moment-rotation constitutive law for the joints in the structural analysis and design of the steel structure.

    By the use of the error correcting back-propagation algorithm [5], a multi- layer feedforward neural network can be trained to recognize and generalize the experimental data. In our case, the experimentally measured moment- rotation curves for various design parameters of the steel structure connection are used as training paradigms. Next, the neural network reproduces the moment rotation law for a given set of design variables and thus it can be used in every design or structural analysis procedure.

    The neural network theory, which provides a solid basis for the construction of the model estimator, is considered highly suitable for the study of complex problems in mechanics and engineering. Some recent representative appli- cations are mentioned next, without any aim of completeness, in the area of structural analysis and design parameter identification problems [6-9], material modelling [10,11], structural analysis [8,12,13] and optimization problems [ 14].

    The experimental data considered here concern the case of single angle beam to column bolted steel structure connections. These experiments have been reported in [15] and are included in the steel connection databases [16,17]. The neural network analysis is given next (see [14,18,19]). The exper- imentally measured moment-rotation laws for steel structure connections are first preprocessed in a form suitable for neural network treatment. These data are next used for training a multilayer feed-forward back-propagation neural network. The trained network provides a model-free predictor, with satisfac- tory accuracy.

    It should be mentioned here that other artificial neural network models can be used as well, instead of the back-propagation neural method. It is also worth mentioning that the methodology which is proposed in this paper could be generalized to include more complicated effects (for instance dynamic behaviour, fatigue and creep effects).

    2 BACK-PROPAGATION AND NEURAL NETWORK THEORY

    A neural network is defined by its node characteristics, the learning rules and the network topology. The learning rules control the improvement of the net- work performance through appropriate adaptive changes of the weights of the links. Furthermore, one of the most remarkable properties of neural networks

  • Neural network analysis of semi-rigid connections 93

    is their considerable fault tolerance due to the increased numbers of locally connected processing nodes.

    Supervised learning is the case in which the required output data, with respect to a given set of input data, are known. Then, the whole set of input- output learning paradigms can be used to adjust the values of the connection weights or some variables of the activation functions, such as being able to reconstruct the implicit highly nonlinear mapping between input and output variables. It must be noted that no specific model has been assumed for the mapping between input and output variables; thus, the trained neural network provides us with a model-free estimator which simulates, for instance, the mechanical behaviour of a structural component [9,10] or a structure [10,20].

    The supervised learning procedure of a back-propagation neural network can be formulated in the following way, using the same notation as in [2 I]. Take one training example p f rom the set of available training examples T = { 1 .... ,t} with input-output data vectors denoted by [Xp,yp], respectively.

    Algorithm 1.

    = ( X ( I ) y(1) aT to the input nodes of layer (1). 1. Apply input vector xp , v . . . . . . p,,l: 2. Execute feedforward signal processing

    zj- =fj(r)) , rj = ~ x i w o. (1) i

    A sigmoidal activation function of the form f~(rj) = I/(1 + e - % ) can be used here.

    3. Calculate the error terms 8(p,'7 ) for the processing units of the output layer (last layer).

    4. Back-propagate the error and calculate for each previous layer j = (m - 1) . . . . . 1 the error terms 6pq) ) in all units of the j th hidden layer, i = 1 ... . . n:.

    5. Update the weights of various layers.

    Note that while in step 5, the network is assumed to be fully connected, i.e. all nodes of layer j - 1 are connected to each one of the nodes in layer j ; the generalization to more flexible interconnection schemes is obvious.

    One pass through all available learning examples (i.e. execution of the algorithm for all p E 7) is called a learning epoch. The error at this epoch is given by

    = - x;; , ) . (2) p ~ T i = 1 , . . . , n m

  • 94 G. E. Stavroulakis et al.

    Learning should continue until a reasonably small error is obtained. Since back-propagation of the error is performed for each individual learn-

    ing example [xp,yp] sequentially, this variant of the back-propagation learning algorithm is known as the on-line or per example or pattern learning version. Another variant of the learning algorithm is the off-line or batch mode training where weight correction is performed once per training epoch only after all changes due to error back-propagation have been accumulated for all learning examples p, p s T--{1, . . . , t} (see e.g. [22], p. 111; [14], pp. 129, 139). In this case the iteration step 5 of Algorithm 1 is given by

    w~)(t + 1) = w~)(t) + r l ~ aQpi)Xpl "1" o l m ~ l ( t - - 1) pet

    (3)

    where 0 -< a -< 1 (a value a = 0.9 is proposed in [14], p. 133) and Apw~)(t - 1) is the adaptation of w~ ) performed in the previous iteration step. A momentum (inertial) term has been added here in the weight adjustment step in order to enhance the speed of convergence in the backpropagation algorithm. This more refined version has been used in the computer implemen- tation. Note here that learning algorithms are actually optimization algorithms and can easily be adapted to the solution of minimum problems (cf. e.g. [8,14,22-24]).

    Using again the same notation as in [21], the production mode (generalization) after the training of the network takes the i~ollowing steps.

    Algorithm 2.

    1. Apply input vector xi,, = (x~ 1) ..... x~11)) T to the input nodes of layer (1). 2. Feed-forward the signal in layers 2 ..... m and in each processing

    element compute ~ and apply the activation (transfer) function x~, ). 3. Vector xout (xtm),. ~m) X = ..,X,, m) is the response of the neural network to

    the input xin.

    The algorithm starts from an initial value of w 0. which is produced by taking variables in a reasonably small range (e.g. [ - 1, + 1] according to [7], or even [ - 0.3, + 0.3]). Nevertheless, it must be noted here that learning in a neural network is algorithmically a hard problem (in general it is proved to be a nonpolynomial difficult NP problem, see [25]), and therefore the size of both the network and the training examples must be kept to the minimum necessary. For this reason, engineering experience and a set of good, represen- tative experimental data must be used as learning examples for the neural net- work.

    It is well known that neural computers are not yet commercially available.

  • Neural network analysis of semi-rigid connections 95

    Neural network computations are best suited to a parallel computer environ- ment, which even permits fairly good hardware implementation [14]. Here, all the computer si

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