Elsevier

Engineering Structures

Volume 29, Issue 3, March 2007, Pages 383-394
Engineering Structures

A soft computing based approach for the prediction of ultimate strength of metal plates in compression

https://doi.org/10.1016/j.engstruct.2006.05.005Get rights and content

Abstract

This paper presents two plate strength formulations applicable to metals with nonlinear stress–strain curves, such as aluminum and stainless steel alloys, obtained by soft computing techniques, namely Neural Networks (NN) and Genetic Programming (GP). The proposed soft computing formulations are based on well-defined FE results available in the literature. The proposed formulations enable determination of the buckling strength of rectangular plates in terms of Ramberg–Osgood parameters. The strength curves obtained by the proposed soft computing formulations show perfect agreement with FE results. The formulations are later compared with related codes and results are found to be quite satisfactory.

Introduction

In spite of the fact that studies on the buckling of columns go back to the end of 19th century, viable theoretical solutions for the plastic buckling of plates were only proposed in the late 1930s and 1940s [1]. Although the theory of plastic buckling of columns is well developed, several aspects of the theory of plastic buckling of plates are still controversial. Determination of the plastic buckling load of a plate is significantly more difficult than its elastic counterpart as the stress–strain relationship beyond the proportional limit is more complex. In the case of plastic buckling of columns the stresses are uniaxial, whereas in the case of plates the stresses are two or three dimensional which brings extra difficulties in the proper representation of the stress–strain relationship. Thus numerical methods are strongly recommended for stability analysis of plates in the plastic region [2]. This paper offers an alternative novel approach for the formulation of plate strength using soft computing techniques. Neural Networks (NNs) and Gene-Expression Programming (GEP) which is an extension of Genetic Programming (GP) are used for closed-form solution of plate strength applicable to metals with nonlinear stress–strain curves, such as aluminum and stainless steel alloys. The formulation is based on well-established FE results from the literature. The formulation is proposed in terms Ramberg–Osgood parameters. Results of the soft computing formulations agree well with FE results. The proposed NN model is seen to be more accurate than related codes. On the other hand the GEP formulation is quite short and practical for use compared to existing codes.

Section snippets

Inelastic buckling of plates

The phenomenon of buckling can be categorized (by plasticity) into three classes, namely elastic buckling, elastic–plastic buckling and plastic buckling where the last two are called inelastic buckling. Elastic buckling is only observed in the elastic regime. On the other hand elastic–plastic buckling occurs after a local region inside the plate deforms plastically. Plastic buckling refers to buckling that occurs in the regime of gross yielding, i.e., after the plate has yielded over large

Soft computing techniques

The definition of soft computing is not precise. Lotfi A. Zadeh, the inventor of the term soft computing, describes it as follows [10]:

“Soft computing is a collection of methodologies that aim to exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low solution cost. Its principal constituents are fuzzy logic, neurocomputing, and probabilistic reasoning. Soft computing is likely to play an increasingly important role in many application areas, including

Optimal NN model selection

The performance of a NN model mainly depends on the network architecture and parameter settings. One of the most difficult tasks in NN studies is to find this optimal network architecture which is based on determination of the numbers of optimal layers and neurons in the hidden layers by a trial and error approach. The assignment of initial weights and other related parameters may also influence the performance of the NN to a great extent. However, there is no well defined rule or procedure to

Numerical application

The main focus of this study is to obtain closed-form solutions of plate strength by means of NNs and GEP. Data needed for the training process are obtained from Ref. [5]. Bezkorovainy et al. [5] performed a wide range of FE analysis of plate strength in terms of e,n, and λ and have compared these FE results with the results of analytical equations they have derived given in Eqs. (6), (7), (8), (9). These values (S/χ) are presented in Table 1. Thus to obtain the FE results, Eqs. (6), (7), (8),

Explicit formulation of NN models

The explicit formulation for the proposed NN model is obtained by using the well trained NN parameters which are biases, and weights for the input and hidden layer and the normalization factors, both for inputs and outputs of the proposed NN model. Related weights in the derivations of NN based formulations are given in Table 6. Each input is multiplied by a connection weight. Thus the main focus is to obtain the explicit formulation as follows: σuσ0.2=f(λ,e,n). Revisiting Eq. (10)ui=j=1Hwijxj+

Explicit formulation of the GEP model

Fig. 11 shows the Expression Tree (ET) of the formulation which is actually σuσ0.2=(d(0)/(d(0)+d(0)))+(d(0)d(1))+(d(0)/((c1+(d(2)c2))/((d(0)/d(2))d(0)))) where d0λd1ed2n and c1 and c2 are constants which are c1=5.353;c2=2.322; and finally the formulation becomes: σuσ0.2=1/2+λe+λ/n5.353+2.322n.

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