Engineering Structures 29 (2007) 383–394www.elsevier.com/locate/engstruct

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

Abdulkadir Cevika,∗, Ibrahim H. Guzelbeyb,∗∗

a Department of Civil Engineering, University Of Gaziantep, 27310, Turkeyb Department of Mechanical Engineering, University Of Gaziantep, 27310, Turkey

Received 6 January 2006; received in revised form 11 May 2006; accepted 12 May 2006Available online 30 June 2006

Abstract

This paper presents two plate strength formulations applicable to metals with nonlinear stress–strain curves, such as aluminum and stainlesssteel alloys, obtained by soft computing techniques, namely Neural Networks (NN) and Genetic Programming (GP). The proposed soft computingformulations are based on well-defined FE results available in the literature. The proposed formulations enable determination of the bucklingstrength of rectangular plates in terms of Ramberg–Osgood parameters. The strength curves obtained by the proposed soft computing formulationsshow perfect agreement with FE results. The formulations are later compared with related codes and results are found to be quite satisfactory.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Soft computing; Neural networks; Genetic programming; Buckling; Plates

1. Introduction

In spite of the fact that studies on the buckling of columnsgo back to the end of 19th century, viable theoretical solutionsfor the plastic buckling of plates were only proposed in thelate 1930s and 1940s [1]. Although the theory of plasticbuckling of columns is well developed, several aspects ofthe theory of plastic buckling of plates are still controversial.Determination of the plastic buckling load of a plate issignificantly more difficult than its elastic counterpart as thestress–strain relationship beyond the proportional limit is morecomplex. In the case of plastic buckling of columns the stressesare uniaxial, whereas in the case of plates the stresses aretwo or three dimensional which brings extra difficulties in theproper representation of the stress–strain relationship. Thusnumerical methods are strongly recommended for stabilityanalysis of plates in the plastic region [2]. This paper offersan alternative novel approach for the formulation of plate

∗ Corresponding author. Tel.: +90 342 3601200x2409; fax: +90 342 3601107.∗∗ Corresponding author. Tel.: +90 342 3601200x2513; fax: +90 342 360

1104.E-mail addresses: akcevik@gantep.edu.tr (A. Cevik),

guzelbeyih@gantep.edu.tr (I.H. Guzelbey).

0141-0296/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2006.05.005

strength using soft computing techniques. Neural Networks(NNs) and Gene-Expression Programming (GEP) which is anextension of Genetic Programming (GP) are used for closed-form solution of plate strength applicable to metals withnonlinear stress–strain curves, such as aluminum and stainlesssteel alloys. The formulation is based on well-established FEresults from the literature. The formulation is proposed in termsRamberg–Osgood parameters. Results of the soft computingformulations agree well with FE results. The proposed NNmodel is seen to be more accurate than related codes. On theother hand the GEP formulation is quite short and practical foruse compared to existing codes.

2. Inelastic buckling of plates

The phenomenon of buckling can be categorized (by plastic-ity) into three classes, namely elastic buckling, elastic–plasticbuckling and plastic buckling where the last two are called in-elastic buckling. Elastic buckling is only observed in the elasticregime. On the other hand elastic–plastic buckling occurs af-ter a local region inside the plate deforms plastically. Plasticbuckling refers to buckling that occurs in the regime of grossyielding, i.e., after the plate has yielded over large areas [3].

Elastic buckling analysis of plates may be based on threeclasses, namely the incremental (flow) theory of plasticity,

384 A. Cevik, I.H. Guzelbey / Engineering Structures 29 (2007) 383–394

deformation theory and slip theory. The successes of thesemethods are still controversial. For example the deformationtheory gives a better prediction of critical buckling loads forlong simply supported plates and the incremental theory givesbetter results for cylinders under compression and torsion. Onthe other hand hybrid methods are also available, being basedon both the deformation theory and the incremental theory [4].

2.1. Ultimate strength of metal plates with nonlinear mechani-cal properties

Metals with nonlinear mechanical properties, such asstainless steel, aluminum or brass, do not have a yieldplateau as do ordinary hot-rolled carbon structural steels,which for compression design purposes can be modeled aselastic–perfectly-plastic. However, the mechanical response ofmany modern cold-formed structural shapes of carbon steel,including tubulars, is becoming increasingly nonlinear becauseof many factors like residual stresses introduced during cold-forming, material properties, fabrication related imperfections,loading conditions and boundary conditions. The strength of thecomponent plates of such sections can also be determined usingthe strength curve formulation developed by Bezkorovainyet al. [5].

In the absence of a yield plateau, it is common practice todefine an equivalent yield stress for nonlinear metals. This isusually chosen as the 0.2% proof stress, defined as the stresswith a plastic strain of 0.2%. It is also common practice torepresent the stress–strain curve by a Ramberg–Osgood curve,which is defined in terms of the initial elastic modulus (Eo), aproof stress, and a parameter n that defines the sharpness of theknee of the stress–strain curve. If the proof stress is chosen asthe 0.2% proof stress (σ0.2), the Ramberg–Osgood curve takesthe form [6]

ε =σ

E0+ 0.002

(σ

σ0.2

)n

. (1)

The proportionality stress (σp) for such metals is defined as the0.01% proof stress (σ0.01). Thus the n-parameter can be definedas:

n =ln(20)

ln(σ0.2/σ0.1). (2)

Thus it is possible to define the strength of a platewhich depends on the stress–strain curve in terms of theRamberg–Osgood parameters (E0, n, σ0.2) as the stress–straincurve can be defined in terms of the Ramberg–Osgoodparameters. Based on this approach Raamussen and Rondal [7]developed equations for the strength of metal columns, whichwere later modified to design curves for stainless steel [8] andaluminum columns [9].

Ref. [5] is based on a similar study of Rasmussen andRondal [7] for uniformly compressed metal plates simplysupported along four edges. Bezkorovainy et al. [5] firstperformed FE analysis to obtain strength curves for a largenumber of Ramberg–Osgood parameters. The FE modelproduced agreement with tests on stainless steel plates to

within a few percent. The FE model can therefore be expectedto be accurate. The width of the plate was 100 mm in allanalyses, while the thickness was varied to produce a set ofpredetermined plate slenderness values,

λ =

√σ0.2

σcr(3)

where σcr is the elastic critical stress

σcr =4π2 E0τ

12(1 − ν2)

(tb

)2

. (4)

Strength curves were determined for all (n, e)-permutations ofthe n- and e-values given in Table 1where e is defined as:

e =σ0.2

E0. (5)

For each (n, e)-combination, the plate strength was determinedfor the slenderness values λ = 0.5, 0.75, 1, 1.25, 1.5, 2, 2.5, 3given in Table 1.

The plate strength (s) is given as the average ultimate stress(σu) nondimensionalized with respect to the 0.2% proof stress(σ0.2), where the ultimate stress is the maximum load obtainedin the FE analysis divided by the cross-sectional area. S isexpressed as:

S = σu/σ0.2. (6)

Having obtained plate strength curves for a wide range ofn- and e-values, Bezkorovainy et al. [5] derived analyticalapproximations to these curves, i.e., a nondimensional platestrength (χ), by adopting a generalized Winter curve given inthe following form:

χ = (α/λ) − (β/λ2) (7)

where λ is the plate slenderness and α and β are calculatedfunctions of the Ramberg–Osgood parameters derived byBezkorovainy et al. [5] given as follows:

α =

0.92 + 0.07 tanh(

n − 32.1

)− (0.026 exp[−0.55(n − 3)] + 0.019)(6 − 2000e)

3 ≤ n ≤ 10 (a)

α10 + (1 − α10)n − 10

9010 < n ≤ 100 (b)

(8)

β =

0.18 + 0.045 tanh(

n − 32.5

)− (0.01 exp[−1.6(n − 3)] + 0.005)(6 − 2000e)

3 ≤ n ≤ 10 (a)

β10 + (0.22 − β10)n − 10

9010 ≤ n ≤ 100 (b).

(9)

3. Soft computing techniques

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

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29(2007)383–394

385

Table 1Values of (S/χ) for a complete set of finite element ana

λ e = 0.001 e = 0. e = 0.0025 e = 0.003n = 3 n = 5 n = 10 n = 100 n = 3 5 n = 10 n = 100 n = 3 n = 5 n = 10 n = 100 n = 3 n = 5 n = 10 n = 100

0.5 0.97 1.01 0.98 0.98 1.00 1.02 0.99 0.98 0.99 1.02 1.00 0.95 0.97 1.02 1.010.75 0.98 0.97 0.98 1.00 1.04 1.00 1.01 1.04 1.00 1.00 1.01 1.03 1.00 1.00 1.021 0.94 0.93 0.99 1.04 1.00 0.99 1.03 1.00 0.96 0.99 1.04 0.98 0.96 0.98 1.041.25 0.96 0.909 0.98 1.00 1.00 0.96 1.00 0.98 0.95 0.96 1.01 0.98 0.95 0.95 1.011.5 0.98 0.913 0.97 0.99 1.02 0.97 0.99 1.01 0.97 0.97 0.99 0.98 0.95 0.96 1.012 1.03 0.94 1.00 1.01 1.06 1.01 1.00 1.05 0.99 1.00 1.03 1.02 0.99 1.00 1.042.5 1.07 0.97 1.02 1.02 1.12 1.03 1.06 1.13 1.06 1.04 1.03 1.09 1.05 1.03 1.063 1.08 1.01 1.06 1.06 1.13 1.05 1.00 1.09 1.04 1.03 1.08 1.06 1.02 1.04 1.06

lyses

0015 e = 0.002n = 5 n = 10 n = 100 n = 3 n =

1.04 1.00 0.99 0.99 1.011.01 1.00 1.00 1.05 1.010.95 0.99 1.04 1.00 0.960.95 0.97 0.99 1.01 0.950.95 0.97 0.99 1.02 0.960.99 1.01 1.01 1.06 1.001.04 1.02 1.03 1.09 1.031.04 1.06 1.06 1.07 1.03

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“Soft computing is a collection of methodologies that aim toexploit the tolerance for imprecision and uncertainty to achievetractability, robustness, and low solution cost. Its principalconstituents are fuzzy logic, neurocomputing, and probabilisticreasoning. Soft computing is likely to play an increasinglyimportant role in many application areas, including softwareengineering. The role model for soft computing is the humanmind”.

Soft computing can be seen as an attempt to collecttechniques that mimic living organisms: plants, animals,human beings, which are soft, flexible, adaptive and clever.It can be described as a family of problem-solving methodsthat have analogy with biological reasoning and problemsolving. It includes basic methods such as Fuzzy Logic (FL),Neural Networks (NN), Genetic Algorithms (GA) and geneticprogramming—methods which do not derive from classicaltheories. Soft computing can also be seen as a foundation forthe growing field of Computational Intelligence (CI) as analternative to traditional Artificial Intelligence (AI) which isbased on hard computing [11].

In many ways, soft computing represents a significantparadigm shift in the aims of computing—a shift which reflectsthe fact that the human mind, unlike present day computers,possesses a remarkable ability to store and process informationwhich is pervasively imprecise, uncertain and lacking incategorization [12].

Two soft computing approaches based on neural networksand genetic programming are within the scope of this study andwill be described in this section.

3.1. Background on neural networks

A neural network is a ‘machine’ that is designed to model theway in which the brain performs a particular task or function ofinterest; the network is usually implemented using electroniccomponents or simulated in software on a digital computer.Neural networks are an information processing technique builton processing elements, called neurons, that are connected toeach other [13].

An artificial neuron (Fig. 1) is the basic element of a neuralnetwork which consists of three main components, namelyweights, bias and an activation function

where ui =

H∑j=1

wi j x j + bi . (10)

The summation ui is transformed as the output usinga scalar-to-scalar function called an “activation or transferfunction” as follows:

O = f (ui ). (11)

Neural networks are commonly classified by their networktopology, (i.e., feedback, feed forward) and learning or trainingalgorithms (i.e., Supervised, Unsupervised). For example, amultilayer feed forward neural network with back propagationindicates the architecture and learning algorithm of the neuralnetwork. A back propagation algorithm is used in this study

Fig. 1. Basic elements of an artificial neuron.

and is the most widely used supervised training method fortraining multilayer neural networks due to its simplicity andapplicability. It is based on the generalized delta rule and waspopularized by Rumelhart et al. [14].

3.2. Background on genetic programming (GP)

GP is an extension to genetic algorithms proposed byKoza [15]. Koza, the early pioneer, defines GP as a domain-independent problem-solving approach in which computer pro-grams are evolved to solve, or approximately solve, problemsbased on the Darwinian principle of reproduction and survivalof the fittest and analogs of naturally occurring genetic opera-tions such as crossover, (sexual recombination) and mutation.

GP reproduces computer programs to solve problems byexecuting the following steps [Fig. 2]:

(1) Generate an initial population of random compositions ofthe functions and terminals of the problem (computerprograms).

(2) Execute each program in the population and assign it afitness value according to how well it solves the problem.

(3) Create a new population of computer programs.(i) Copy the best existing programs (reproduction)

(ii) Create new computer programs by mutation.(iii) Create new computer programs by crossover (sexual

reproduction).(iv) Select an architecture-altering operation from the

programs stored so far.(4) The best computer program that appeared in any

generation, the best-so-far solution, is designated as theresult of genetic programming [15].

3.2.1. Brief overview of GEPGene-Expression Programming (GEP) software which is

used in this study is an extension to GP that evolves computerprograms of different sizes and shapes encoded in linearchromosomes of fixed length. The chromosomes are composedof multiple genes, each gene encoding a smaller sub-program.Furthermore, the structural and functional organization of thelinear chromosomes allows the unconstrained operation ofimportant genetic operators such as mutation, transposition andrecombination. One strength of the GEP approach is that thecreation of genetic diversity is extremely simplified as geneticoperators work at the chromosome level. Another strength of

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Fig. 2. Genetic programming flowchart [15].

GEP consists of its unique, multigenic nature which allowsthe evolution of more complex programs composed of severalsub-programs. As a result GEP surpasses the old GP systemby 100–10,000 times [16–18]. APS 3.0 [19], GEP softwaredeveloped by Candida Ferreira is used in this study.

The fundamental difference between GA, GP and GEP isdue to the nature of the individuals: in GAs the individualsare linear strings of fixed length (chromosomes); in GP the in-dividuals are nonlinear entities of different sizes and shapes(parse trees); and in GEP the individuals are encoded as lin-ear strings of fixed length (the genome or chromosomes) whichare afterwards expressed as nonlinear entities of different sizesand shapes (i.e., simple diagram representations or expressiontrees). Thus the two main parameters in GEP are the chromo-somes and Expression Trees (ETs). The process of informationdecoding (from the chromosomes to the ETs) is called trans-lation which is based on a set of rules. The genetic code isvery simple; there exists a one-to-one relationship between thesymbols of the chromosome and the functions or terminals theyrepresent. The rules, which are also very simple, determine thespatial organization of the functions and terminals in the ETsand the type of interaction between sub-ETs [16–18].

That’s why two languages are utilized in GEP: the languageof the genes and the language of ETs. A significant advantageof GEP is that it enables us to infer exactly the phenotype given

Fig. 3. Expression tree (ET).

the sequence of a gene and vice versa; this is called Karvalanguage. Consider, for example, the algebraic expression (d4∗√

(d3 − d0 + d3 ∗ d4) − d4); this can be represented by adiagram which is the expression tree shown in Fig. 3.

4. Optimal NN model selection

The performance of a NN model mainly depends on thenetwork architecture and parameter settings. One of the mostdifficult tasks in NN studies is to find this optimal networkarchitecture which is based on determination of the numbersof optimal layers and neurons in the hidden layers by a trialand error approach. The assignment of initial weights andother related parameters may also influence the performanceof the NN to a great extent. However, there is no well defined

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Fig. 4. Optimal NN selection process.

Table 2Back propagation training algorithms used in NN training

MATLAB function name Algorithm

trainbfg BFGS quasi-Newton backpropagation

traincgf Fletcher–Powell conjugategradient back propagation

traincgp Polak–Ribiere conjugategradient back propagation

traingd Gradient descent backpropagation

traingda Gradient descent with adaptivelr back propagation

traingdx Gradient descent w/momentum& adaptive linear backpropagation

trainlm Levenberg–Marquardt backpropagation

trainoss One step secant backpropagation

trainrp Resilient back propagation(Rprop)

trainscg Scaled conjugate gradient backpropagation

rule or procedure to have optimal network architecture andparameter settings where the trial and error method remainsvalid. This process is very time consuming. The determinationof an optimal NN model depends on the proper selection of thefollowing NN parameters [20]:

number of hidden neuronslearning-rate parametersmomentum constantslearning algorithmsnumber of epochs.

There is a dangerous common misconception concerningiterative training. It says that neural networks can beovertrained. It says that there is an optimal amount of training,and that continuing past that point will improve performanceon the training set but degrade performance in the generalpopulation. Another important task to be considered should bethe risk of overfitting, which occurs when too many hidden

neurons are used. Rather than learning only the general patternsneeded to produce the correct decision, the network focusesexcessively on individual samples. Since it takes a considerabletime to learn these individual samples, while the importantpatterns are more likely to be at least partially learned quickly,the quality of the network often reaches a peak as trainingprogresses and then deteriorates [21].

In this study Matlab NN toolbox is used for NN applications.Various back propagation training algorithms are used as givenin Table 2. The Matlab NN toolbox randomly assigns the initialweights for each run each time which considerably changesthe performance of the trained NN even if all parametersand NN architectures are kept constant. This leads to extradifficulties in the selection of optimal network architecture andparameter settings. To overcome this difficulty a program hasbeen developed in Matlab based on the procedures given inSimon [20] and Masters [21] which handles the trial and errorprocess automatically. The program tries various numbers oflayers and neurons in the hidden layers for both the first andsecond hidden layers for a constant epoch several times andselects the best NN architecture with the minimum MAPE(Mean Absolute % Error) or RMSE (Root Mean Squared Error)of the testing set, as the training of the testing set is morecritical. Measures of errors used in this study are given in theform of formulas in Appendix. For instance NN architecturewith 1 hidden layer with 7 nodes is tested 10 times and thebest NN is stored; in the second cycle the number of hiddennodes is increased up to 8 and the process is repeated. The bestNN for cycle 8 is compared with cycle 7 and the best one isstored as the best NN. This process is repeated N times whereN denotes the number of hidden nodes for the first hidden layer.This whole process is repeated for a changing number of nodesin the second hidden layer. Moreover this selection process isperformed for different back propagation training algorithmssuch as trainlm, trainscg and trainbfg given in Table 2. Theprogram begins with the simplest NN architecture, i.e. a NNwith 1 hidden node for the first and second hidden layers, andends up with optimal NN architecture as shown in Fig. 4. Aflowchart is of the whole process is shown in Fig. 5.

A. Cevik, I.H. Guzelbey / Engineering Structures 29 (2007) 383–394 389

Fig. 5. Flowchart of optimal NN selection.

5. Numerical application

The main focus of this study is to obtain closed-formsolutions of plate strength by means of NNs and GEP. Dataneeded for the training process are obtained from Ref. [5].Bezkorovainy et al. [5] performed a wide range of FE analysisof plate strength in terms of e, n, and λ and have compared these

FE results with the results of analytical equations they havederived given in Eqs. (6)–(9). These values (S/χ) are presentedin Table 1. Thus to obtain the FE results, Eqs. (6)–(9) have beenused to find the nondimensional χ values and in return the FEresults have been obtained from Table 1. This study is basedon the soft computing modeling of these FE results given inTable 3. Among the FE results a number of cases have been

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.Guzelbey

/Engineering

Structures29

(2007)383–394

Table 3FE results for (

σuσ0.2

)

λ e = 0.001 e = 0 2 e = 0.0025 e = 0.003n = 3 n = 5 n = 10 n = 100 n = n = 5 n = 10 n = 100 n = 3 n = 5 n = 10 n = 100 n = 3 n = 5 n = 10 n = 100

0.5 0.9700 1.1184 1.0825 0.9757 0.970 1.0408 1.0811 0.9496 0.8991 0.9860 1.0596 0.9419 0.8482 0.9347 1.0390 0.93450.75 1.2672 1.2370 1.2136 1.1743 1.289 1.2043 1.1878 1.1401 1.1908 1.1548 1.1641 1.1184 1.1360 1.1196 1.1412 1.10841 1.5161 1.4168 1.4513 1.4712 1.538 1.3697 1.3920 1.3995 1.4085 1.3276 1.3640 1.3857 1.3243 1.2879 1.3237 1.35931.25 1.8634 1.6294 1.6826 1.6715 1.846 1.5961 1.5808 1.6047 1.6487 1.5475 1.5490 1.5891 1.5786 1.5019 1.5027 1.55861.5 2.2273 1.8898 1.9180 1.9158 2.201 1.8634 1.8393 1.8389 1.9804 1.8260 1.8023 1.8027 1.8375 1.7360 1.7486 1.80362 3.0294 2.4775 2.5098 2.4961 2.954 2.4731 2.4309 2.3715 2.6498 2.3750 2.3584 2.3942 2.4578 2.3060 2.3119 2.37052.5 3.8656 3.1118 3.1105 3.0733 3.830 3.1013 3.0120 3.0642 3.4920 3.0966 2.9800 2.9183 3.2134 2.9787 2.8932 2.94473 4.6286 3.8211 3.8079 3.7706 4.581 3.6583 3.6170 3.4125 3.9878 3.5841 3.4767 3.6120 3.6977 3.4138 3.4412 3.4758

.0015 e = 0.003 n = 5 n = 10 n = 100 n = 3

9 1.1102 1.0818 0.9673 0.93403 1.2448 1.2126 1.1511 1.25005 1.3999 1.4210 1.4415 1.47064 1.6477 1.6306 1.6211 1.77824 1.9032 1.8778 1.8765 2.09597 2.5263 2.4818 2.4446 2.80794 3.2306 3.0453 3.0392 3.53901 3.8104 3.7280 3.6923 4.1154

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Fig. 6. Proposed NN model for the prediction of σu .

Fig. 7. Performance of the NN model for the test set.

randomly selected as test and training sets. As generalizationcapability is the most important issue in NN applications, thetest set was kept as big as possible, consisting of 50 datasets,the training set consisting of 110 datasets.

5.1. Results of NN and GEP models

The optimal NN architecture in this part was found to bea 3-5-1 NN architecture with a hyperbolic tangent sigmoidtransfer function (tansig). The training algorithm was quasi-Newton back propagation (BFGS). The optimum NN modelis given in Fig. 6. Related parameters of the GEP models forboth GEP trainings are given in Table 4. Statistical parametersof learning and training sets of the NN model are presentedin Table 5. Prediction of the NN and actual values of learningand testing sets and their corresponding correlation are givenin Figs. 7 and 8. Prediction of the GEP and actual values oflearning and testing sets and their corresponding correlation aregiven in Figs. 9 and 10. The overall accuracy and correlationof NN and GEP models are quite satisfactory compared to FEresults.

Table 4Parameters of the GEP models

P1 Function set +, −, ∗, /,√

, exp(x)

P2 Chromosomes 200P3 Head size: 6, 8, 10P4 Number of genes: 1P5 Linking function: Addition, MultiplicationP6 Fitness function error type: MAEP7 Mutation rate: 0,044P8 Inversion rate: 0,1P9 One-point recombination rate: 0,3P10 Two-point recombination rate: 0,3P11 Gene recombination rate: 0,1P12 Gene transposition rate: 0,1

Table 5Statistical parameters of the proposed NN and GEP models

MSE RMSE SSE MAPE (%)

NN training set 0.0018 0.042 0.193 2.028NN test set 0.008 0.09 0.402 3.025GEP training set 0.026 0.161 1.298 6.104GEP test set 0.023 0.152 2.5466 6.179

Fig. 8. Performance of the NN model for the training set.

6. Explicit formulation of NN models

The explicit formulation for the proposed NN model isobtained by using the well trained NN parameters whichare biases, and weights for the input and hidden layer andthe normalization factors, both for inputs and outputs of theproposed NN model. Related weights in the derivations ofNN based formulations are given in Table 6. Each input ismultiplied by a connection weight. Thus the main focus is toobtain the explicit formulation as follows:

σu

σ0.2= f (λ, e, n). (12)

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Fig. 9. Performance of the GEP model for the test set.

Fig. 10. Performance of the GEP model for the training set.

Table 6Weights between inputs and the hidden layer

Weights Number of hidden layer neurons (i)1 2 3 4 5

w1i −2.0846 3.6183 −4.8607 −0.33863 −2.0238w2i 0.349 −0.8433 0.14596 0.020628 0.23784w3i 19.573 −19.117 −9.4351 0.007103 13.224

Revisiting Eq. (10)

ui =

H∑j=1

wi j x j + bi

where each input is multiplied by its corresponding weight andits bias is added. wi j is the weight matrix of the first hiddenlayer in Table 6 and X j is the corresponding parameter vectorgiven as X = [λ, e, n] where bi is the bias matrix to the first

hidden layer given as

b = [3.5011 −1.3388 −5.9156 1.7062 2.8130].

The summation ui is transformed using a scalar-to-scalarfunction called an “activation or transfer function”, F(ui )

yielding a value called the unit’s “activation”.

Yi = f (ui ).

Following the steps above leads to:

u1 = (−2.081 ∗ λ) + (0.349 ∗ e) + (19.57 ∗ n) + 3.5u2 = (3.62 ∗ λ) + (−0.84 ∗ e) + (−19.12 ∗ n) − 1.339u3 = (−4.86 ∗ λ) + (0.146 ∗ e) + (−9.44 ∗ n) − 5.916u4 = (−0.34 ∗ λ) + (−0.02 ∗ e) + (0.007 ∗ n) + 1.70u5 = (−2.02 ∗ λ) + (0.238 ∗ e) + (13.22 ∗ n) + 2.81.

The activation function used in this study is a hyperbolictangent sigmoid transfer function (tansig) f (ui ) = ( 2

1+e−2(ui ))

performed for each hidden node in the first hidden layerThus the output is

O = 2/(1 + exp −2 ∗ (Σ (w2i ∗ f (ui )) + b2))

where w2i is the weight vector to the output layer given as

w2i = [−42.567 0.1263 −0.88153 −11.098 8.6415]

and b2 is the bias added which is

b2 = 43.7.

It should be noted that the inputs entering the network werenormalized before the training as

λ∗= λ/3 e∗

= e/0.003 n∗= n/100

and the output before the training was normalized by 5, thus thefinal output is

R = O ∗ 5.

The steps given so far may seem to be too complex,particularly for those who do not have a neural networkbackground. The same steps can be given in a simpler formas follows:σu

σ0.2= f (λ, e, n)

σu

σ0.2= 5 ∗

(2

1 + e−2W

)where W = (−42.57) ∗

(2

1 + e−2U1

)+ (0.126)

∗

(2

1 + e−2U2

)+ (−0.88) ∗

(2

1 + e−2U3

)+ (−11.01) ∗

(2

1 + e−2U4

)+ (8.64) ∗

(2

1 + e−2U5

)+ 43.7.

u1 = (−0.695 ∗ λ) + (116.3 ∗ e) + (0.1957 ∗ n) + 3.5u2 = (1.20 ∗ λ) + (−281 ∗ e) + (−0.1912 ∗ n) − 1.339

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Table 7Range of variables

λ e n

Minimum value 0.5 0.001 3Maximum value 3.0 0.003 100

u3 = (−1.62 ∗ λ) + (48.65 ∗ e) + (−0.0944 ∗ n) − 5.916u4 = (−0.113 ∗ λ) + (6.876 ∗ e) + (0.00007 ∗ n) + 1.70u5 = (−0.675 ∗ λ) + (79.28 ∗ e) + (0.1322 ∗ n) + 2.81.

It should be noted that the proposed explicit formulation of theNN GEP models presented above are valid only for the rangesof the training set given in Table 7.

7. Explicit formulation of the GEP model

Fig. 11 shows the Expression Tree (ET) of the formulationwhich is actuallyσu

σ0.2= (d(0)/(d(0) + d(0))) + (d(0) − d(1))

+ (d(0)/((c1 + (d(2) ∗ c2))/((d(0)/d(2)) ∗ d(0))))

where d0 → λ

d1 → e

d2 → n

and c1 and c2 are constants which are

c1 = 5.353; c2 = 2.322;

and finally the formulation becomes:

σu

σ0.2= 1/2 + λ − e +

λ/n5.353 + 2.322 ∗ n

.

7.1. Conclusion

This paper presents an alternative soft computing basedapproach for the prediction of the nondimensional ultimatestrength of metal plates in compression with nonlinearmechanical properties. NNs and GEP are like soft computingtechniques for numerical application. The proposed NNand GEP models are based on well-defined FE resultsfor nondimensional ultimate strength of metal plates incompression with nonlinear mechanical properties from theliterature. The ultimate plate strength is obtained in termsof λ, e, n being the plate slenderness ratio, σ0.2

E0and the

Ramberg–Osgood parameter. The closed-form solutions arealso presented for the NN and GEP models. The results of theproposed NN and GEP models are quite satisfactory comparedto FE results. A very high accuracy and a perfect correlation(R = 0.999; MAPE = 2.34%), which is better than existinganalytical formulations, has been obtained for the NN modelsince a new optimum NN selection algorithm was used for theselection of the optimum NN architecture. On the other handthe explicit formulation obtained by GEP (R = 0.96; MAPE =

6.1%) is quite short and practical for use compared to existing

Fig. 11. Expression tree of the GEP model.

analytic expressions. This study verifies the robustness of softcomputing techniques for the explicit formulation and analysisof various engineering problems where it is difficult to obtain ananalytic expression from experimental and numerical results.

Appendix

Mean Squared Error

MSE =

∑(X − Y )2

n.

Root Mean Squared Error

RMSE =

[∑(X − Y )2

n

]1/2

.

Mean Absolute Percentage Error

MAPE =

∣∣ X−YY

∣∣ × 100n

.

Sum Squared Error

SSE =

∑(X − Y )2

where n = total number of pairs X and Y values. X is the targetvalue and Y is the output value.

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