Analytical prediction of available rotation capacity of cold-formedrectangular and square hollow section beams

Mario D'Aniello a, Esra Mete Güneyisi b,n, Raffaele Landolfo a, Kasım Mermerdaş c

a Department of Structures for Engineering and Architecture, University of Naples “Federico II”, Naples, Italyb Department of Civil Engineering, Gaziantep University, Gaziantep, Turkeyc Department of Civil Engineering, Hasan Kalyoncu University, Gaziantep, Turkey

a r t i c l e i n f o

Article history:Received 12 April 2013Accepted 14 September 2013Available online 18 October 2013

Keywords:Soft-computing methodsAnalytical formulationRotation capacitySteel beamsCold-formed hollow sections

a b s t r a c t

In this paper, a soft-computing based study aimed to estimate the available rotation capacity of cold-formed rectangular and square hollow section (RHS-SHS) steel beams is described and novel mathema-tical models based on neural network (NN) and genetic expression programming (GEP) are proposed. Inorder to develop the proposed formulations, a wide experimental database obtained from availablestudies in the literature has been considered. The data used in the NN and GEP models are arranged in aformat of eight input parameters covering both geometrical and mechanical properties such as width,depth and wall thickness of cross section, inside corner radius, yield stress, ratio of modulus of elasticityto hardening modulus, ratio of the strain under initial hardening to yield strain and shear length. Theaccuracy of the proposed formulations is verified against the experimental data and the rates ofefficiency and performance are compared with those provided by analytical semi-empirical formulationdeveloped by some of the Authors in a previous study. The proposed prediction models proved that theNN and GEP methods have strong potential for predicting available rotation capacity of cold-formed RHS-SHS steel beams.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Steel hollow sections are becoming even more used in a widerange of structural applications, thanks to both static and esthetic orfunctional advantages. Indeed, steel beams made of cold-formedhollow sections are generally torsionally stiffer than wide flangesections, thus requiring less bracing to restrain lateral-torsionalbuckling phenomena. Moreover, the use of such a kind of profilesallows reducing both finishing (e.g. they need less amount of paintthan open profiles) and maintenance costs (e.g. in those cases wherethe structural members are not covered, such as industrial ware-houses, rural constructions and garages, the water cannot be accu-mulated over the flanges and the birds cannot nest on the closedsections). Hollow profiles may be produced by three main differenttechnological processes, namely by welding plates or channel sectionstogether, hot-rolling and cold-forming. The latter method is veryeffective because using thin plates to built-up the sections leads tolight weight members having high strength-to-weight ratio, thusreducing both transportation and erection costs [1].

Although cold formed hollow beams are generally made of thinplates, it is quite common to design profiles with stocky walls inorder to fulfill requirements for plastic design as presented in

current codes, e.g. selecting Class 1 (Plastic) or Class 2 (Compact)sections according to EN 1993:1-1 [2] and EN 1998-1 [3]. Indeed,the codified slenderness limits for flange and web are calibrated inorder to guarantee that plastic hinges can provide the minimumductility to allow the plastic redistribution of bending momentsand the formation of a ductile mechanism [4,5]. The measure ofthe ductility provided by the plastic hinge is generally indicated asrotation capacity (R) [4].

Several experimental studies [1,6] demonstrated that somesections, although classifiable as plastic or compact according tocurrent steel design codes, can experience insufficient plasticrotations than those strictly necessary for plastic design. In addition,Wilkinson [1] clearly showed that considering independently flangeand web slenderness limits, as stated in current codes, may resultinappropriate.

These considerations highlight the need to develop effectiveand accurate formulations to predict the rotation capacity of coldformed hollow beams in order to overcome the fallacies of codifiedcross section classification, which is based only on the limitation oflocal slenderness for flange and web. Indeed, accurate predictionequations of rotation capacity allow to verify if the selected beamscan be suitable for the design purpose providing the expectedductility, especially for seismic applications [4,5,7].

In the literature, several studies have been conducted toobtain explicit formulations for estimating the rotation capacityof steel members using empirical methods, namely based on

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Thin-Walled Structures

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tws.2013.09.015

n Corresponding author. Tel.: þ90 342 3172423.E-mail address: eguneyisi@gantep.edu.tr (E.M. Güneyisi).

Thin-Walled Structures 77 (2014) 141–152

post-processing of experimental and/or numerical data [7–16].Especially, some of them were focused on utilizing computationaltools such as finite element analysis or developing computationalaids. For instance, Wilkinson and Hancock [16] carried out finiteelement analysis on cold-formed rectangular hollow section (RHS)beams to predict the rotation capacity of Class 1 (Plastic) and Class2 (Compact) beams. On the other hand, Gioncu et al. [10] andAnastasiadis et al. [11] developed a specific computer program(e.g. DUCTROT-M) to determine both flexural strength and avail-able rotation capacity of wide-flange beams, only.

Recently, novel approaches based on soft-computing techni-ques have been implemented to analyze and design steel struc-tures [17–22] and are becoming very popular thanks to theeffectiveness and versatility of the obtained results. These meth-ods were derived from artificial intelligence philosophy and themost commonly utilized soft-computing tools are based on geneticprogramming and artificial neural networks. For example, Lagarosand Papadrakakis [17] used soft-computing techniques to optimizethe structural design. Indeed, they proposed a neural networkbased prediction scheme of the structural response required toassess the quality of candidate design during the optimizationprocedure. In the study of Saka [18], a genetic algorithm basedoptimum design is reported for pitched roof steel portal frameswith haunches provided to the rafters at the eaves. Gholizadehet al. [19] used finite element (FE) and soft-computing techniquesnamed back-propagation (BP) neural network and adaptive neuro-fuzzy inference system (ANFIS) methods to propose models forestimation the web post-buckling load in castellated steel beams.Fonseca et al. [20] utilized neural networks to forecast steel beampatch load resistance, comparing the results with precedingmodels and existing design formulae. They concluded that thenetworks' percentage errors with the experimental results con-firms the possibility of using the unified methodology to generatenew trustworthy data. In the study of Gandomi et al. [21], analternative approach for predicting the flexural resistance andinitial rotational stiffness of semi-rigid joints in steel structuresusing linear genetic programming was suggested. Finally, Güneyisiet al. [22] proposed a new formulation of the flexural overstrengthfactor for steel beams with a wide range of cross-section typolo-gies using gene expression programming.

Nowadays, at the Authors' knowledge there has not yet beendeveloped a scientific study reporting soft-computing based estima-tion models for predicting the rotation capacity of cold-formed RHS-SHS steel beams. Therefore, at the light of the previous considera-tions, two different analytical prediction models have been devel-oped and the relevant main assumptions and predictive performanceare presented and discussed in the current study. The former modelhas been derived from gene expression programming (GEP) and thelatter from neural network (NN). Moreover, in order to test theadvantages of the proposed methods the performances of presentedformulations are compared with those provided by the more recentexisting empirical model which was proposed by D'Aniello et al. [7]and was developed using the same input parameters adopted for thenovel formulations presented hereinafter.

2. Rotation capacity of steel beams

2.1. Definition

Rotation capacity (R) is the measure of the flexural ductility ofthe beam. As shown in Fig. 1, it can be evaluated by means of themoment–rotation relationship [23–25] as follows:

R¼ θuθp

�1 ð1Þ

being θp the rotation corresponding to flexural yielding (Mp) and θuthe ultimate beam rotation, which corresponds to the bendingmoment dropping below Mp. The rotation capacity can be dividedinto two parts: the stable part of rotation capacity developed up tothe occurrence of local buckling and the unstable part due to post-buckling behavior [5].

In the past several studies have been carried out to evaluate therotation capacity required to achieve plastic collapse in steelstructures [1,24,26–34]. Kemp [26] suggested two minimum limitsin South African Code for the required rotation capacity (R45 andR43) to be utilized for plastic design at different performancelevels. Yura et al. [27] proposed as lower bound limit for plasticdesign R43 in the draft American LRFD Code. Currently, the NorthAmerican standard has supposed a rotation capacity of 3 to besufficient for the most civil engineering structures as stated inYura et al. [27] and AISC [28]. This value for R is based on limitingthe flange strain to four times the yield strain. Similarly, in thebackground document to the European standard, Bild et al. [29]and Sedlacek and Feldmann [24] studied and reported the rotationrequirements for three-span continuous beams and single bayframes under point loads. They concluded that a rotation capacityequal to 3 is adequate and the corresponding limiting width-to-thickness limit ratios for class 1 sections were developed.These limits are also adopted in the current Italian code NTC2008 [30]. Numerous researches were also addressed for cold-formed structural hollow sections. Korol and Hudoba [31] sug-gested a rotation capacity equal to 4 for plastic design. To satisfythis requirement some design rules for limiting width-to-thickness ratios were then developed. This rotation capacity wasfurther adopted by [1,32,34]. In Europe, Stranghoner et al. [34]conducted a parametric study to examine the rotation require-ments of square, rectangular, and circular hollow sections on athree-span continuous beam subjected to a point load in thecentral span. The investigated factors covered beam geometry,loading, cross-section, material, and serviceability requirements.Analysis of the results indicated that a rotation capacity of 3 issatisfactory to supply a full plastic behavior.

2.2. Existing models

The number of the studies regarding numerical modeling ofrotation capacity of cold-formed RHS-SHS steel beams is limited.In the present paper, the formulation proposed by D'Aniello et al.[7] was assumed as benchmark to compare the novel proposedmodels because (i) it was developed using the same inputparameters and (ii) it is the more recent empirical formulationfor cold-formed RHS-SHS steel beams presented in scientificliterature, providing also the better prediction of R at the currenttime [7].

The formulation given by [7] was obtained by a multiple linearregression analysis of experimental data presented in that studyand those reported also by [1,35]. This prediction formulation is

M/Mp

θ /θp

1

1 θu/θp

Mmax/Mp

R

s

Fig. 1. Generalized moment–rotation curve for a steel beam [7].

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152142

presented by Eq. (2), as follows:

R¼ �32:93þ0:049λ2f

þ0:676λ2w

þ20:528bf h

L2v�466:972

bf tfhLv

�1:904Af

Atotþ0:322

LmLv

þ32:475Mpl

Muð2Þ

where λf and λw are the flange and web slendernesses given inEqs. (3) and (4), respectively; bf is the flange width, h is the beamdepth, tf is the flange thickness, Lv is the shear length, Af/Atot is theratio between the area of the flange over the total area of the crosssection; Lm is the length of the plastic hinge given by Eq. (6) andcalculated using Eqs. (7) for c and (8) for β, Eh is the hardeningmodulus, εh is the strain corresponding to the beginning ofhardening, and εy the first yielding strain.

λf ¼bf2tf

ffiffiffiffiffif yE

sð3Þ

λw ¼ dw;e

tw

ffiffiffiffiffif yE

sð4Þ

where

dw;e ¼ 12

1þ AAw

ρ

� �dw ð5Þ

Lm ¼ 2cβ ð6Þwhere

c¼ bf �2ðtwþriÞ ð7Þ

β¼ 0:6dc

� �1=4

ð8Þ

3. Models based on soft-computing techniques

3.1. Generality

Zadeh [36] defines soft-computing as a collection of methodol-ogies that aim to exploit the tolerance for imprecision anduncertainty to achieve tractability, robustness, and low solutioncost. Its main components are fuzzy logic, neurocomputing, andprobabilistic reasoning. Soft-computing is likely to play an impor-tant role in wide variety of fields of application. The key model forsoft-computing is the human mind. The fuzzy logic, geneticalgorithm, genetic programming, and neural network can beaccepted as the main techniques of soft-computing. In the follow-ing neural network and genetic programming methods werealternatively used to derive two different prediction formulationof available rotation capacity of cold-formed RHS-SHS steel beams.

3.2. Gene expression programming (GEP)

Genetic programming (GP), proposed by Koza [37] is essentiallyan application of genetic algorithms to computer programs. GP hasbeen applied successfully to solve discrete, non-differentiable,combinatory, and general nonlinear engineering optimizationproblems [38]. It is an evolutionary algorithm based the metho-dology inspired by biological evolution to find computer thatperforms a task defined by a user. Therefore, it is a machinelearning technique used to construct a population of computerprograms according to a fitness landscape determined by aprogram's ability to perform a given computational task. Similarto genetic algorithm (GA), the GP needs only the problem to bedefined. Then, the program searches for a solution in a problem-independent manner [36,37].

Gene expression programming (GEP) was introduced by Ferreira[39] and it can be considered as a natural development of geneticalgorithms and genetic programming. GEP evolves computer pro-grams of different sizes and shapes encoded in linear chromosomesof fixed-length. GEP algorithm begins with the random generationof the fixed-length chromosomes of each individual for the initialpopulation. Then, the chromosomes are expressed and the fitness ofeach individual is evaluated based on the quality of the solution itrepresents.

To clarify the GEP basis it is convenient to draft the funda-mentals of GP. The GP reproduces computer programs to solveproblems by executing the following steps (as described in Fig. 2):

(1) Generate an initial population of random compositions of thefunctions and terminals of the problem (computer programs);

(2) Execute each program in the population and assign it a fitnessvalue according to how well it solves the problem;

(3) Create a new population of computer programs(i) Copy the best existing programs,(ii) Create new computer programs by mutation,(iii) Create new computer programs by crossover.

Differently from GP, the significant improvement of GEP is thatit makes it possible to infer exactly the phenotype given thesequence of a gene, and vice versa which is termed as Karvalanguage. For example, the following algebraic expression (Eq. (9))can be represented by a diagram which is the expression tree asfollows (Fig. 3).

Y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid3

sin d2þc1� ln c1

s� d1

vuut ð9Þ

3.3. Neural networks (NNs)

An artificial neural network (NN) is an information processingparadigm that is inspired by the way biological nervous systems,such as the brain, process information. The key element of thisparadigm is the novel structure of the information processingsystem. It is composed of a large number of highly interconnectedprocessing elements (neurons) working in unison to solve specificproblems. NNs, like people, learn by example. An NN is configuredfor a specific application, such as pattern recognition or dataclassification, through a learning process. Learning in biologicalsystems involves adjustments to the synaptic connections thatexist between the neurons.

The training of NNs by back propagation have three stages [40]:(i) the feed forward of the input training pattern, (ii) the calcula-tion and back propagation of the associated error, and (iii) theadjustment of the weights. This process can be used with anumber of different optimization strategies. The error betweenthe output of the network and the target value is propagatedbackward during the backward pass and used to update theweights of the previous layers as shown in Fig. 4 [41–43].

In this study, neural network fitting tool (nftool) provided as asoft-computing tool in MatlabV.R2012a was utilized to performneural network modeling. In fitting problems, a neural networkmay be used to map between a data set of numeric inputsand a set of numeric targets. The nftool helps create and train anetwork, and evaluate its performance using mean square errorand regression analysis.

A two-layer feed-forward network with sigmoid hidden neu-rons and linear output neurons, can fit multi-dimensional map-ping problems arbitrarily well, given consistent data and enoughneurons in its hidden layer. The network was trained withLevenberg–Marquardt back propagation algorithm.

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152 143

An artificial neuron consists of three main components namelyweights, bias, and an activation function. Each neuron receivesinputs I1, I2,…, In attached with a weight wi which shows the

connection strength for that input for each connection. Each inputis then multiplied by the corresponding weight of the neuronconnection. A bias can be defined as a type of connection weightwith a constant nonzero value added to the summation ofweighted inputs, as given in Eq. (10). Generalized algebraic matrixoperation was also given in Eq. (11). to clarify the mathematicaloperations in an artificial neuron.

Uk ¼ Biaskþ ∑n

j ¼ 1wj;k ð10Þ

Uk ¼

w11 w12 : : : w1n

w21 : :

:

:: :

: : :

: : :

wm1 : : : : wmn

2666664

3777775m�n

I1I2:

:

:

In

266664

377775n�1

þ

Bias1Bias2:

:

:

Biasm

266664

377775m�1

¼

U1

U2

:

:

:

Um

266664

377775m�1

ð11ÞSince nftool uses the normalized values in the range of [�1, 1], theinput parameters were normalized by means of Eq. (12) in order toget the prediction results after execution of the training process ofthe NN. Moreover, the obtained results are also in the normalizedform. Therefore, considering the Eq. (12) and the normalizationcoefficients a and b for outputs, de-normalization process isapplied and the results are monitored.

βnormalized ¼ aβþb ð12Þ

Fig. 3. A sample sub-expression tree for a mathematical operation.

Gen=0

Create initial random population

Termination criteron satisfied

YES Designate the result

End

NO

Evaluate fitness of each individial in the

population

Individiuals=0

Individiuals=M?Gen=Gen+1

Select genetic operation probabilistically

Select two individuals based on fitness

Crossover

Select one individual based on fitness

Select one individual based on fitness

Reproduction Mutation

Perform mutation Perform reproduction

Copy into new population

Perform crossover

Insert mutant into new population

Insert two offspring into new population

Individuals=Individual+1 Individuals=Individual+1 Individuals=Individual+2

YES

NO

Fig. 2. Flowchart for the genetic programming paradigm [37].

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152144

where β is the actual input parameter or output values given inTables 2 and 6, respectively. βnormalized is the normalized value ofinput parameters or outputs ranging between [�1,1]. a and b arenormalization coefficients given in the following equations (Eqs.(13) and (14)).

a¼ 2βmax�βmin

ð13Þ

b¼ �βmaxþβmin

βmax�βminð14Þ

where βmax and βmin are the maximum and minimum actual valuesof either inputs or outputs. The normalization coefficients for bothinput and output variables are given in Table 1.

4. Description of the database used for derivationof the models

The proposed formulations of R for cold-formed RHS-SHS steelbeams were derived using a set of 64 experimental data available

Start

Select the first NN algorithm

Train and test the NN with the first input variable

Reference=Test result of NN with the first input variable

i=1

Add one exogenous variable into the NN

Train and test the new NN

If test result >= Reference

No Yes

Variable does not stay in the model

Variable stays in the model

Reference=Test result of NN

i=i+1

If i >= no. of variables available

No

Train and test the NN with all the available variables

Choose another NN algorithm or architecture parameter

Identify the overall best model

End

Fig. 4. Forward strategy for selecting NN architecture and model [43].

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152 145

in the technical literature [1,6,35,44] for training and testing theproposed models.

Fig. 5 summarizes the structural configurations of the testedbeams, while the corresponding data sources are presented inTable 2. As it can be noted, the selected experimental data arethoroughly representative of a wide range of cross section typol-ogies as highlighted by the statistics of the main geometric andmechanical parameters reported in Table 3. In detail, the gener-ated models for RHS-SHS profiles covered the following inputparameters: b (width of section), d (height of section), t (wallthickness of section), r (inside corner radius), Lv (shear length), fy(yield stress), E/Eh (ratio of modulus of elasticity of steel tohardening modulus), and εh/εy (ratio of the strain correspondingto the beginning of hardening to yield strain).

All data samples were put in an order to establish a consistentsequence of the inputs to be used for derivation of the models. Theinput nodes cover the geometric properties of the section,mechanical properties of the material as well as the shear lengthof the steel beams. Thus, generally, eight inputs parameters wereutilized for development of prediction models.

The data set was randomly divided into two parts to obtaintraining and testing databases. For this, about one fourth of the

total data samples were considered to be the test database, whilethe rest was used as training database for Table 3. Therefore, 48data samples were used as training data while testing databasecontained 16 data samples. As seen in Table 4, the statistics of bothtraining and testing sets are in good agreement meaning bothrepresent almost the similar population.

The GeneXproTools.4.0 and MatlabV.R2012a softwares wereused for derivation of the GEP and NN based mathematicalmodels, respectively.

For clarity sake, in the next Sections, where it is discussed thecomparison between the experimental and predicted rotationcapacity, the effectiveness of the correlation is evaluated by meansof the correlation coefficient “r” (Eq. (15)), which describes the fitof the models’ output variable approximation curve to the actualtest data output variable curve. Higher r coefficients indicate amodel with better output approximation capability.

r¼ ∑ðmi�m′Þðpi�p′Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑ðmi�m′Þ2∑ðpi�p′Þ2

q ð15Þ

where m′ and p′ are mean values of measured (mi) and predicted(pi) values, respectively.

5. Proposed GEP model

The prediction model derived from GEP is presented in Eq. (16).The GEP parameters used for derivation of the mathematicalmodels are given in Table 5. As it can be seen from Table 5, inorder to provide an accurate model, various mathematical opera-tions were used.

The models developed by the software in its native language canbe automatically parsed into visually appealing expression trees,permitting a quicker and more complete comprehension of theirmathematical/logical intricacies. Fig. 6 demonstrates the expressiontree for the terms used in the formulation of the GEP model.

The performance of the proposed GEP prediction model inEq. (16) is graphically demonstrated in Fig. 7 for both training andtesting data sets. It seems that there is a close trend in the variationof the data between predicted and experimental data. Correlationcoefficients equal to 0.972 and 0.946 were calculated for trainingand testing databases, respectively, thus indicating strong correla-tion between actual and predicted values. Moreover, close values ofthe correlation coefficients may be considered as an evidence forthe consistency and good fitness of the proposed model.

R¼ R1þR2þR3þR4þR5þR6þR7þR8 ð16Þ

R1 ¼ e sin ð12:81311þd2 �d7 þ sin d3Þ ð16aÞ

R2 ¼ ½arctanðln d2Þ�5 �sin d2cos d5

ð16bÞ

R3 ¼ 102:9607277=ðd5 þd3 �d3�d2Þ ð16cÞ

R4 ¼ sin ð tan ð tan d0Þ�d0�d47Þ ð16dÞ

R5 ¼ arctan1

d6�d2

� �� �4� d0

4:940033

� �0:6

ð16eÞ

R6 ¼ d2�1

d1þd6�d7þd4�16:555786ð16fÞ

R7 ¼ d2�1

166:2496616�d33ð16gÞ

R8 ¼ arctanð tan ðð tan d23Þ0:2667ÞÞ ð16hÞ

Table 1Summary of experimental database for the steel beams with RHS-SHS sections.

No. Authors Testno.

Profiletype

Steel grade Testsetup

Loadinghistory

1 Wilkinson [1] 40 RHSþSHS MCSa 4PBTc Monotonic2 Zhou and Young [35] 15 RHSþSHS MCSþHSSb 4PBT Monotonic3 Gardner et al. [6] 3 RHSþSHS MCS 3PBTd Monotonic4 Landolfo et al. [44] 6 RHSþSHS MCS CBTe Monotonic

a MCS¼mild carbon steel.b HSS¼high strength steel.c 4PBT¼4 points bending test.d 3PBT = 3 points bending test.e CBT = cantilever bending test.

Fig. 5. Schematic view of test arrangement and geometry of cross-section variablesfor RHS-SHS steel beams.

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152146

6. Proposed NN model

A 8–9-1 NN architecture, as shown in Fig. 8, was adopted todevelop the NN model. That means there are 8 nodes in the input

layer, corresponding to 8 factors from I1 to I8, 9 nodes in thehidden layer, and one in the output layer corresponding to therotation capacity of the cold-formed RHS-SHS beam. It should benoted that all numeric variables were normalized to a range of

Table 2Experimental database for RHS and SHS profiles.

Ref. Exp no bf (mm) d (mm) t (mm) r (mm) fy (MPa) E/Eh εh/εy Lv (mm)

d0 d1 d2 d3 d4 d5 d6 d7

Wilkinson [1] 1 50.25 151.04 4.92 9.9 441 48.2 9.8 4502 50.41 150.92 4.9 10.7 441 48.2 9.8 4503 50.27 150.43 3.92 6.8 457 48.2 9.8 4504 50.4 150.44 3.87 7.3 457 48.2 9.8 4505 50.11 150.42 3.89 7.3 457 48.2 9.8 4506 50.16 150.21 3.89 5.4 423 48.2 9.8 4507 50.22 150.47 2.97 5.9 444 48.2 9.8 4508 50.01 150.79 2.95 5.8 444 48.2 9.8 4509 50.34 150.8 2.96 5.7 444 48.2 9.8 450

10 50.15 150.43 2.6 4.6 446 48.2 9.8 45011 50.41 150.39 2.57 4.6 446 48.2 9.8 45012 50.23 150.4 2.59 4.8 446 48.2 9.8 45013 50.4 150.31 2.64 5.3 440 48.2 9.8 45014 50.57 150.51 2.28 4.2 444 48.2 9.8 45015 50.7 100.45 2.06 3.8 449 48.2 9.8 45016 50.55 100.49 2.07 3.9 449 48.2 9.8 45017 50.24 100.46 2.04 4.7 449 48.2 9.8 45018 50.22 100.45 2.04 3.4 423 48.2 9.8 45019 50.1 75.48 1.94 4.4 411 48.2 9.8 40020 50.31 75.63 1.95 4.4 411 48.2 9.8 40021 25.28 75.31 1.98 3.7 457 48.2 9.8 40022 25.12 75.24 1.54 3.1 439 48.2 9.8 40023 25.2 74.9 1.54 3.4 439 48.2 9.8 40024 25.08 74.98 1.56 3.9 439 48.2 9.8 40025 25.12 75.27 1.55 3.4 422 48.2 9.8 40026 25.25 75.19 1.56 3.4 422 48.2 9.8 40027 50.13 150.46 3 6.2 370 48.2 9.8 45028 50.19 150.5 2.96 6.5 370 48.2 9.8 45029 50.51 150.45 3 6.8 382 48.2 9.8 45030 50.51 150.38 3 6.3 382 48.2 9.8 45031 50.43 100.91 2.06 3.6 400 48.2 9.8 45032 50.52 100.83 2.05 3.8 400 48.2 9.8 45033 75.84 125.56 2.92 6.6 397 48.2 9.8 45034 75.74 125.4 2.93 6.9 397 48.2 9.8 45035 75.56 125.4 2.91 7.1 397 48.2 9.8 45036 75.1 125.4 2.53 3.9 374 48.2 9.8 45037 100.27 100.43 2.88 5.2 445 48.2 9.8 45038 100.25 100.53 2.86 5.2 445 48.2 9.8 45039 50.21 150.32 3.9 7.9 349 48.2 9.8 45040 50.57 150.39 3.85 7.5 410 48.2 9.8 450

Zhou and Young [35] 41 40.1 40.1 1.96 2 447 48.2 9.8 480.742 40 40.1 3.88 4 565 48.2 9.8 480.343 80.5 80.4 1.91 4 398 48.2 9.8 480.744 79.9 79.8 4.77 7.5 448 48.2 9.8 48145 49.8 99.9 1.97 2 320 48.2 9.8 48046 49.6 99.7 3.88 4 378 48.2 9.8 479.747 59.9 120.2 1.84 2.5 361 48.2 9.8 480.748 59.7 120 3.89 5.5 392 48.2 9.8 480.749 40.2 40 1.94 2 707 48.2 9.8 414.350 50.1 50.3 1.54 1.5 622 48.2 9.8 41451 150.6 150.7 2.78 4.8 448 48.2 9.8 546.752 150.7 150.5 5.87 6 497 48.2 9.8 55053 80.5 140.3 3.09 6.5 486 48.2 9.8 48054 80.9 160.6 2.9 6 536 48.2 9.8 48055 109.1 197.7 4 8.5 503 48.2 9.8 548

Gardner et al. [6] 56 40.09 60.04 3.93 2.07 400 48.2 9.8 55057 40.42 40.31 3.7 3.1 410 48.2 9.8 55058 40.11 40.16 2.8 2.63 451 48.2 9.8 550

Landolfo et al. [44] 59 100 150 5 10 275 42.6 11 187560 80 160 4 8 275 42.6 11 187561 100 250 10 20 275 42.6 11 187562 160 160 6.3 12.6 355 48.2 9.8 187563 200 200 10 20 355 48.2 9.8 187564 250 250 8 16 355 48.2 9.8 1875

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152 147

[�1, 1] before being introduced to the NN. Therefore, one mustenter the normalized values in the mathematical operations givenfor NN model. It should also be noted that the final result obtainedfrom Eq. (17) is also in the normalized form which needs to bede-normalized according to Eq. (12) and normalization coefficientsgiven in Table 4.

R¼ Biasoutput layerþ ∑m

k ¼ 1Wkf ðUkÞ ð17Þ

where Biasoutput layer¼0.76679 and f(x) (Hyperbolic tangent) is theactivation function given in Eq. (18), Uk given in Eqs. (19) through(26), and Wk is the layer weight matrix given below.

f ðxÞ ¼ 21�e�2x�1 ð18Þ

U1 ¼ 1:994�0:44138X1þ0:038936X2�0:7512X3�0:85888X4

þ0:16204X5�0:97953X6�0:70885X7þ0:6166X8 ð19Þ

U2 ¼ �1:2141þ1:6453X1�0:5124X2�4:0952X3þ0:27556X4

þ1:57X5�0:42181X6þ1:2222X7�0:94306X8 ð20Þ

U3 ¼ �1:2932þ1:4389X1þ0:46072X2�1:5712X3þ0:22607X4

�0:97054X5�0:95105X6þ0:55178X7�2:0782X8 ð21Þ

U4 ¼ 0:025188þ0:66427X1�1:5455X2þ0:30345X3�0:97652X4

þ1:3317X5þ0:17963X6þ0:52345X7�0:23815X8 ð22Þ

U5 ¼ �1:0854�2:8601X1þ1:2491X2�0:99474X3þ0:35109X4

þ0:78051X5�1:5078X6þ0:56429X7�1:3615X8 ð23Þ

U6 ¼ �1:4512�2:1958X1þ0:40033X2þ2:9005X3�0:73111X4

�0:23174X5þ0:69238X6�0:11897X7�1:0793X8 ð24Þ

U7 ¼ �1:4316�0:28084X1�0:17944X2þ1:1617X3þ1:3321X4

�0:86308X5�1:3671X6þ0:42246X7þ0:70556X8 ð25Þ

Table 3Statistics of experimental data for RHS-SHS steel beams.

bf d t r fy E/Eh εh/εy Lv R

(a) Training dataNo. of data 48 48 48 48 48 48 48 48 48Mean 63.18 121.21 3.21 5.91 422.90 48.08 9.83 539.96 4.60Standard deviation 36.50 40.07 1.81 3.64 59.47 0.81 0.17 350.06 4.75COV 0.58 0.33 0.56 0.62 0.14 0.02 0.02 0.65 1.03Min. value 25.08 40 1.54 2 275 42.6 9.8 400 0.6Max. value 200 250 10 20 707 48.2 11 1875 21.91

(b) Test dataNo. of data 16 16 16 16 16 16 16 16 16Mean 72.67 123.83 3.47 5.93 432.38 47.50 9.95 741.17 5.75Standard deviation 52.25 63.57 1.54 3.71 95.03 1.91 0.41 563.92 4.79COV 0.72 0.51 0.44 0.63 0.22 0.04 0.04 0.76 0.83Min. value 40 40.1 1.54 1.5 275 42.6 9.8 414 1.1Max. value 250 250 8 16 622 48.2 11 1875 15.9

Table 4Normalization coefficients.

Variables Parameters βmax βmin a b

Input variables Width of section (mm) 250 25.08 0.008892 �1.22301Depth of section (mm) 250 40 0.009524 �1.38095Wall thickness (mm) 10 1.54 0.236407 �1.36407The inside corner radius (mm) 20 1.5 0.108108 �1.16216The yield stress (MPa) 707 275 0.00463 �2.27315The ratio of modulus of elasticity of

steel to hardening modulus48.2 42.6 0.357143 �16.2143

The ratio of the strain corresponding to thebeginning of hardening to yield strain

11 9.8 1.666667 �17.3333

The shear length (Lv) 1875 400 0.001356 �1.54237

Output variable Available rotation capacity (R) 21.9 0.6 0.093897 �1.05634

Table 5GEP parameters used for proposed models.

Parameters R for RHS-SHS profiles

P1 Function Set þ , � , n, /, √, ,̂ln, exp, sin, tan, inverse, Pow

P2 Number of generation 149072P3 Choromosomes 30P4 Head size 8P5 Linking function AdditionP6 Number of genes 8P7 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

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152148

U8 ¼ 2:1961þ0:24856X1þ0:09737X2þ1:5376X3þ1:3972X4

þ0:80282X5þ0:72682X6�0:54254X7�0:69853X8 ð26Þ

Wk ¼

�0:0488860:428581:7846�1:0958�1:2803�1:33321:79620:36989�1:0258

266666666666666664

377777777777777775

The obtained results from the NN model are also plotted in Fig. 7yielding 0.961 and 0.954 correlation coefficients for training andtesting data sets, respectively. Similar to the GEP model, theestimated results have close tendency to the experimental values.

7. Performance of the proposed models

To compare the prediction performances of the proposedmodels as well as D'Aniello et al. model [7], normalized resultscalculated by dividing predicted result by actual ones are given inTable 6. Additionally, to illustrate the tendency of the variation, thenormalized values are depicted in Fig. 9. The table revealed thatD'Aniello et al. model [7] failed in estimation of some R values byproducing either 0 or negative value. On the other hand, both ofthe proposed models appeared to be usable for the whole data set.

According to the normalized values (Rpredicted/Rexperimental,being the Rpredicted the calculated value of rotation capacity andRexperimental that experimentally measured), the perfect estimationperformance is equal to 1. Observing Fig. 9, it can be seen that theclosest trend in variation of the normalized values around 1 wasobserved for the NN model. Conversely, D'Aniello et al. model [7]revealed large fluctuations diverging from the actual experimentalvalue. The ratio of predicted to observed values for D'Aniello et al.

Fig. 6. Expression tree of GEP model for RHS-SHS steel beams (Where d0¼b (the width of section, expressed in mm); d1¼d (the depth of section, expressed in mm); d2¼t(the wall thickness of section, expressed in mm); d3¼r (the inside corner radius, expressed in mm); d4¼ fy (the yield stress, expressed in MPa); d5¼E/Eh (the ratio of modulusof elasticity of steel to hardening modulus); d6¼εh/εy (the ratio of the strain corresponding to the beginning of hardening to yield strain); d7¼Lv (the shear length, beingequal to (L1-L2)/2 for 4PTB test and L for CBT test, and L/2 for 3PTB where L, L1, L2 are described in Fig. 5 and expressed in mm)).

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152 149

model [7] were higher than those of the developed NN or GEPmodels, which generally either underestimated or overestimatedthe experimental values. For example, as recognizable in Table 6,for the experimental R values of 0.6 and 3.5, the most extremelower and upper normalized values of the data were observed as0.1 and 3.7, respectively. Critical observation of Table 6 alsoindicated that for R values being less than 3, the predictionperformance of the proposed models was not fairly good. On theother hand, for higher R values than 3 the proposed modelsprovided high prediction capability with certain accuracy. It is

worth noting that the higher levels of R values (R43) indicatehigher ductility and sufficient rotation capacity for plastic designaccording to Eurocode 3 [2].

Fig. 7. Evaluation of experimental and predicted rotation capacity for RHS-SHSsteel beams: (a) train set and (b) test set.

X1=bf

X2=d

X4=r

X5=fy

X3=t

X8=Lv

X7= h/ y

X6=E/Eh

R

Fig. 8. Architecture of neural network model used to predict the rotation capacity.

Table 6The performances of prediction model.

Ref. Exp no Rexp Normalized data (Rpredicted/Rexperimental)

GEP model NN model D'Aniello et al.model [7]

Wilkinson [1] 1 13.0 0.9 1.1 0.62 9.0 1.1 1.5 0.73 6.6 1.3 1.1 1.44 7.7 1.1 0.9 0.85 7.2 1.1 1.0 1.16 9.5 0.7 1.0 0.67 2.7 1.2 0.9 1.78 2.3 1.6 1.1 2.19 2.9 1.3 0.9 1.4

10 1.4 1.9 1.1 0.311 1.2 2.1 1.2 NAn

12 2.2 1.3 0.7 1.013 1.1 2.1 1.2 3.114 0.6 3.9 2.0 0.115 0.8 2.5 1.8 1.716 0.8 1.9 1.6 NA17 1.3 1.0 0.7 1.818 1.6 1.1 0.9 1.719 1.7 1.0 0.8 0.320 1.9 0.9 0.7 NA21 5.7 0.6 0.2 0.722 2.2 0.8 1.3 1.523 2.5 1.0 1.0 1.924 2.5 1.0 0.8 1.125 1.9 1.3 1.1 0.426 2.6 1.0 0.8 NA27 4.1 0.9 1.1 1.728 3.6 0.3 1.0 1.429 3.2 1.3 1.1 1.830 3.6 1.1 1.1 1.831 1.2 1.2 1.1 NA32 1.3 1.5 0.9 NA33 1.5 1.6 1.3 NA34 1.6 1.6 1.1 0.235 1.4 1.8 1.0 NA36 1.1 1.1 1.7 1.037 0.8 1.8 0.4 NA38 0.9 1.6 0.5 NA39 12.9 0.7 1.0 0.840 10.7 0.8 0.9 0.6

Zhou and Young [35] 41 7.3 1.0 1.0 1.442 7.3 1.1 0.5 2.843 2.4 0.6 1.4 NA44 10.1 1.0 1.0 1.745 6.1 0.7 1.1 2.646 11.2 0.9 1.3 2.147 3.3 0.7 0.7 1.148 19.6 0.9 0.8 1.349 2.7 0.9 1.0 2.250 1.7 0.6 0.9 NA51 1.8 1.2 1.0 NA52 6.8 0.9 1.0 0.753 3.1 1.1 0.7 1.754 2.2 0.7 1.1 0.655 4.5 1.0 1.1 0.3

Gardner et al. [6] 56 12.5 0.9 0.9 1.557 7.2 1.3 1.1 3.058 3.5 1.7 1.1 3.7

Landolfo et al. [44] 59 15.9 1.0 0.7 0.760 4.0 1.0 1.0 2.261 12.7 1.1 1.0 1.562 1.7 2.0 0.9 1.463 21.9 1.0 1.0 0.664 2.7 1.2 1.0 1.5

n NA:Not available. The model yields invalid result.

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152150

Finally, in order to assess the performances of the predictionmodels, the following statistical parameters were calculated andpresented in Table 7.

Mean absolute percent error;

MAPE¼ 1n

∑n

i ¼ 1

mi�pimi

��������� 100 ð27Þ

Mean square error;

MSE¼∑ni ¼ 1ðmi�piÞ2

nð28Þ

Root mean square error

RMSE¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑n

i ¼ 1ðmi�piÞ2n

sð29Þ

where m and p are values of measured (mi) and predicted (pi )values, respectively.

As it can be recognized from Table 7, the lowest errors wereobserved for the proposed models. In particular, the lowest MAPEwas calculated for the NN model while the highest one was by farobserved for D'Aniello et al. model [7].

8. Conclusions

Novel and efficient approaches for the explicit formulation ofrotation capacity of steel beams made of cold-formed rectangularand square hollow sections are presented in this study. Theproposed formulations are based on the most popular soft-computing techniques, namely gene expression programming(GEP) and artificial neural networks (NNs). To this aim, availableexperimental data presented in the existing literature were usedto derive those formulations. In order to evaluate their efficiencyand advantages, the performance of the proposed models wascompared to that provided by the D'Aniello et al. [7], which was

assumed as benchmark because that analytical model is based onthe same parameters that those implemented in the present study.Based on the analysis of the results, the following conclusions canbe drawn:

� It is proved that soft-computing techniques may be beneficialways to derive comprehensive mathematical formulations offlexural rotation capacity of hollow section steel beams havingdifferent geometrical and mechanical properties.

� The correlation and accuracy of the proposed models are foundto be good enough to be utilized for prediction purposes. Thecorrelation coefficients for training database are 0.972 and0.961 for the GEP and NN models, respectively. Moreover, fortesting databases correlation coefficients of 0.946 for theformer and 0.954 for the latter were achieved. Even thoughthe database for testing data set were not used for training,high level of prediction was obtained for both training andtesting data sets associated with low mean absolute percentageof error and high coefficients of correlation. This can beconsidered as the generalization capability of the developedmodel.

� The comparison with analytical formulation proposed byD'Aniello et al. [7] for the rotation capacity emphasized thatthe GEP and NN models provide the best prediction of theexperimental data. Indeed, it was observed that D'Aniello et al.model could either underestimate or overestimate rotationcapacity with relatively higher errors. While the proposedformulations demonstrated relatively better accuracy, particu-larly for higher levels of R values (R43).

� Statistical analysis based on MAPE and MSE values revealedthat both of the proposed GEP and NN formulations havecomparatively lower errors than D'Aniello et al. model. Obser-ving the overall tendency of the estimation performance, itcan be concluded that NN model may be considered as morepreferable one.

References

[1] Wilkinson T. The plastic behaviour of cold formed rectangular hollow sections.Australia: Department of Civil Engineering, University of Sydney; 1999 (Ph.D.thesis).

[2] CEN (European Communities for Standardization). EN 1993-1-1Eurocode 3:design of steel structures. Part 1: general rules and rules for buildings; 2005.

[3] CEN (European Communities for Standardization). EN 1998-1Eurocode 8:design of structures for earthquake resistance. Part 1: general rules, seismicactions and rules for buildings; 2005.

[4] Gioncu V, Mazzolani FM. Ductility of seismic resistant steel structures. London,Great Britain: Spon Press; 2002.

[5] Mazzolani FM, Piluso V. Theory and design of seismic resistant steel frames.London: E & FN Spon an imprint of Chapman & Hall; 1996.

[6] Gardner L, Saari N, Wang F. Comparative experimental study of hot-rolled andcold-formed rectangular hollow sections. Thin-Walled Structures 2010;48(2010):495–507.

[7] D'Aniello M, Landolfo R, Piluso V, Rizzano G. Ultimate behavior of steel beamsunder non-uniform bending. Journal of Constructional Steel Research 2012;78:144–58.

[8] Gioncu V, Petcu D. Available rotation capacity of wide-flange beams andbeam-columns Part 1. theoretical approaches. Journal of Constructional SteelResearch 1997;43(1-3):161–217.

[9] Gioncu V, Petcu D. Available rotation capacity of wide-flange beams andbeam-columns Part 2. Experimental and numerical tests. Journal of Construc-tional Steel Research 1997;43(1-3):219–44.

[10] Gioncu V, Mosoarca M, Anastasiadis A. Prediction of available rotation capacityand ductility of wide-flange beams: Part 1: DUCTROT-M computer program.Journal of Constructional Steel Research 2012;69:8–19.

[11] Anastasiadis A, Mosoarca M, Gioncu V. Prediction of available rotation capacityand ductility of wide-flange beams: Part 2: Applications. Journal of Construc-tional Steel Research 2012;68:176–91.

[12] OPCM 3274. First elements in the matter of general criteria for seismicclassification of the national territory and of technical codes for structuresin seismic zones. Official Gazette of the Italian Republic, Rome, and FurtherModifications 2003.

Fig. 9. Prediction performances of the proposed soft-computing based models andsemi-empirical model by D'Aniello et al. [7].

Table 7Statistical analysis of outputs obtained from prediction models.

Statistical parameters Prediction models

GEP model NN model D'Aniello et al.model [7]

Mean absolute percent error (MAPE) 35.4 20.2 68.4Mean square error (MSE) 1.55 2.00 29.41Root mean square error (RSME) 1.25 1.41 5.42

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152 151

[13] Lopes SM. do Carmo RNF. Deformable strut and tie model for the calculation ofthe plastic rotation capacity. Computers and Structures 2006;84:2174–83.

[14] Beg D, Zupancic E, Vayas I. On the rotation capacity of moment connections.Journal of Constructional Steel Research 2004;60:601–20.

[15] Mazzolani FM, Piluso V. Prediction of the rotation capacity of aluminium alloybeams. Thin-Walled Structures 1997;27:103–16.

[16] Wilkinson T, Hancock G. Predicting the rotation capacity of cold-formed RHSbeams using finite element analysis. Journal of Constructional Steel Research2002;58:1455–71.

[17] Lagaros ND, Papadrakakis M. Applied soft-computing for optimum design ofstructures. Structural and Multidisciplinary Optimization 2012;45:787–99.

[18] Saka MP. Optimum design of pitched roof steel frames with haunched raftersby genetic algorithm. Computers and Structures 2003;81:1967–78.

[19] Gholizadeh S, Pirmoz A, Attarnejad R. Assessment of load carrying capacity ofcastellated steel beams by neural networks. Journal of Constructional SteelResearch 2011;67:770–9.

[20] Fonseca ET, Vellasco PCGS, de Andrade SAL, Vellasco MMBR. Neural networkevaluation of steel beam patch load capacity. Advances in Engineering Soft-ware 2003;34:763–72.

[21] Gandomi AH, Alavi AH, Kazemi SS, Alinia MM. Behavior appraisal of steelsemi-rigid joints using linear genetic programming. Journal of ConstructionalSteel Research 2009;65:1738–50.

[22] Güneyisi EM, DAniello M, Landolfo R, Mermerdaş K. A novel formulation of theflexural overstrength factor for steel beams. Journal of Constructional SteelResearch 2013;90:60–71.

[23] Rondal J, Ph Boeraeve Sedlacek G, Stranghöner N, Langenberg P. RotationCapacity of Hollow Beam Sections. Final Report, CIDECT Research Project No.2P, Liege University:1995.

[24] Sedlacek G, Feldmann M. Background document 5.09 for chapter 5 of Euro-code 3 part 1.1 — the b/t ratios controlling the applicability of analysis modelsin Eurocode 3 Part 1.1. Aachen; 1995.

[25] Sedlacek, G; Dahl, W; Stranghöner, N, Kalinowski, B. Investigation of therotation behaviour of hollow section beams. EUR 17994 EN. Final report ofproject no 7210-SA/119; 1998.

[26] Kemp A. Interaction of plastic local and lateral buckling. Journal of StructuralEngineering ASCE 1985;111(10):2181–96.

[27] Yura JA, Galambos TV, Ravindra MK. The bending resistance of steel beams.Journal of Structural Division ASCE 1978;104(ST9):1355–70 (Proc. Paper no.14015, September).

[28] AISC 360-05. Specification for structural steel buildings. Illinois, USA: Amer-ican Institute of Steel Construction; 2005.

[29] Bild S, Roik K, Sedlacek G, Stutzki Ch, Spangemacher R. Background documentfor chapter 5 of Eurocode 3 — the b/t-ratios controlling the applicability ofanalysis models in Eurocode 3. Aachen: Draft; 1989.

[30] DM. Nuove Norme Tecniche per le Costruzioni; 2008 [in Italian].[31] Korol RM, Hudoba J. Plastic behavior of hollow structural sections. Journal of

Structural Division ASCE 1972;98(5):1007–23.[32] Hasan SW, Hancock GJ. Plastic bending tests of cold-formed rectangular

hollow sections. Steel Construction: Journal of the Australian Institute of SteelConstruction 1989;23(4):2–19.

[33] Zhao XL, Hancock GJ. Tests to determine plate slenderness limits for cold-formed rectangular hollow sections of grade C450. Steel Construction: Journalof the Australian Institute of Steel Construction 1991;25(4):2–16.

[34] Stranghoner N, Sedlacek G, Boeraeve Ph. Rotation requirement and rotationcapacity of rectangular, square and circular hollow section beams. In:Proceedings of the 6th International Symposium on Tubular Structures, 1994.

[35] Zhou F, Young B. Tests of cold-formed stainless steel tubular flexural members.Thin-Walled Structures 2005;43:1325–37.

[36] Zadeh LA. Soft-computing and fuzzy logic. IEEE Software 1994;11(6):48–56.[37] Koza JR. Genetic programming: on the programming of computers by means of

natural selection. Massachusetts Institute of Technology, USA, 1992, [MIT Press].[38] Goldberg D. Genetic Algorithms in search, optimization and machine learning.

MA: Addison-Welsley; 1989.[39] Ferreira C. Gene expression programming: a new adaptive algorithm for

solving problems. Complex Systems 2001;13(2):87–129.[40] Schalkoff RJ. Artificial neural networks. Columbus: McGraw-Hill; 1997.[41] Haykin S. Neural networks: a comprehensive foundation. New Jersey: Mac-

millan College Publications Cooperation; 2000.[42] Hebb DO. The organization of behavior. New York: John Wiley and Sons Inc; 1949.[43] Sušac MZ, Šarlija N, Benšić M. Selecting neural network architecture for

investment profitability predictions. Journal of Information And Organiza-tional Sciences, 2005;29(2):83–95.

[44] Landolfo R, D'Aniello M, Brescia M, Tortorelli S. Rotation capacity andclassification criteria of steel beams. The development of innovativeapproaches for the design of steel–concrete structural systems — the line5 of the ReLUIS-DPC 2005–2008 Project. Doppiavoce, Napoli; 2011. p. 37–88.

M. D'Aniello et al. / Thin-Walled Structures 77 (2014) 141–152152