Expert Systems with Applications 38 (2011) 5650–5661

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Expert Systems with Applications

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Neuro-fuzzy modeling of rotation capacity of wide flange beams

Abdulkadir Cevik ⇑Department of Civil Engineering, University of Gaziantep, Turkey

a r t i c l e i n f o a b s t r a c t

Keywords:Rotation capacityBeamsNeuro-fuzzyModeling

0957-4174/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.eswa.2010.10.070

⇑ Tel.: +90 (342) 3172409; fax: +90 (342) 3601107E-mail address: akcevik@gantep.edu.tr

This study is a pioneer work that investigates the feasibility of neuro-fuzzy (NF) approach for the mod-eling of rotation capacity of wide flange beams. The database for the NF modeling is based on experimen-tal studies from literature. The results of the NF model are compared with numerical results obtained by aspecialized computer programme and existing analytical and genetic programming based equations. Theresults indicate that the proposed NF model performs better. By using the proposed NF model, a widerange of parametric studies are also performed to evaluate the main effects of each variable on rotationcapacity.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The evaluation of rotation capacity of steel beams has been thesubject of numerous numerical and experimental studies as it is anextremely important phenomenon for plastic and seismic analysisand design of steel structures. Rotation capacity is used to expresswhether plastically designed sections possess the required ductil-ity or not. In the same way, the moment redistribution in a steelstructure also depends on the rotation capacity of the section(Dinno, 2002).

Although many studies have been conducted on this topic, thereis still lack of reliable analytical models to describe the concept ofrotation capacity of steel beams in all respects. Theoretical, empir-ical and approximate methods have been proposed so far for thedetermination of available rotation capacity of wide flange steelbeams in literature which have been reported by Gioncu and Petcu(1997a, 1997b). Besides, a relatively new approach is also growingrapidly as an alternative tool to handle the rotation capacity ofsteel beams: soft computing techniques. Guzelbey, Cevik, andGogus (2006) have proposed neural networks (NN) as an alterna-tive approach for the prediction of rotation capacity of wide flangebeams based on experimental results collected from literature. Theproposed NN model showed perfect agreement with experimentalresults (R = 0.997) where its accuracy was also quite high. More-over, Guzelbey et al. (2006) have also presented the proposed NNmodel in explicit form as a mathematical function. FurthermoreCevik proposed an alternative genetic programming (GP) basedformulation which performed quite well compared to existinganalytical equations (Cevik, 2007).

ll rights reserved.

.

This study investigates the feasibility of another soft computingtechnique namely as neuro-fuzzy (NF) approach for the modeling ofavailable rotation capacity of wide flange steel beams for the firsttime in this field. Similar to previous soft computing models, theproposed NF model is also based on experimental results collectedfrom literature. The results of the proposed NF model are comparedwith numerical results and existing analytical equations and foundto be more accurate. Main effects of each variable on rotation capac-ity are also obtained by using the proposed NF model.

2. Rotation capacity

2.1. Definition of rotation capacity

There are various definitions of rotation capacity in literature asa non-dimensional parameter.

According to Lay and Galambos (1965) rotation capacity is,R = hh/hp, in which hp is the elastic rotation at the initial attainmentof the plastic moment Mp and hh is the plastic rotation at the pointwhen moment drops below Mp.

A widely used definition for rotation capacity is proposed byASCE (Fig. 1):

R = h2/h1 where h1 refers to the theoretical rotation at which thefull plastic capacity is achieved and h2 is the rotation when themoment capacity drops below Mp on the unloading portion.

Kemp (1985) defined rotation capacity as R = hhm/hp in whichhhm is the plastic rotation up to the maximum moment on themoment rotation curve.

2.2. Rotation capacity in design codes

The behaviour of laterally restrained beams is commonlydivided into three or four classes of behaviour as illustrated in

Fig. 1. Definition of rotation capacity.

A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661 5651

Fig. 2. The Australian Standard AS 4100 and AISC LRFD have threeclasses (compact, non-compact, and slender). A compact or Class 1section is suitable for plastic design, and can sustain the plasticmoment (Mp) for a sufficiently large rotation capacity (R) to allowfor moment redistribution in a statically indeterminate system(Wilkinson & Hancock, 2002).

On the other hand, Eurocode 3 defines four classes of cross-sections to identify the extent to which the resistance and rotation

Fig. 2. Classical definition for rotation capacity based on normalised moment-rotation relationship.

Fig. 3. Input data me

capacity of cross sections is limited by its local buckling resistanceas follows:

– Class 1 cross-sections are those which can form a plastichinge with the rotation capacity required from plastic analy-sis without reduction of the resistance.

– Class 2 cross-sections are those which can develop their plas-tic moment resistance, but have limited rotation capacitybecause of local buckling.

– Class 3 cross-sections are those in which the stress in theextreme compression fibre of the steel member assumingan elastic distribution of stresses can reach the yield strength,but local buckling is liable to prevent development of theplastic moment resistance.

– Class 4 cross-sections are those in which local buckling willoccur before the attainment of yield stress in one or moreparts of the cross-section.

2.3. Analytical rotation capacity models

Rotation capacity was investigated analytically by manyresearchers of which the following studies are worth to mention.Lay presented the following equation for rotation capacity (Lay,1965):

R ¼ 1� k21

k21

!s� 1

F1

� �1

1ch� 1

!; ð1Þ

where k1 ¼ Lbry

ffiffiffiffieypp and F1 ¼ 1

2pþ1

2 1�2pð Þ 1� 1ffiffiffiffi

hstp

� �.

ch is the most probable effective stiffness of the bending yieldplane in the flange, hst is the young’s Modulus to strain hardeningmodulus ratio, s is the strain at onset of strain hardening to yieldstrain ratio and ey is yield strain.

On the other hand, Kemp (1996) proposed the most widely ana-lytical expression where he related beam rotation capacity to theeffective lateral slenderness ratio (ke). Following are the basicparameters used in the model:

� Yield stress factor for the flange or web cf = (Fyf/250)1/2 orcw = (Fyw/250)1/2(Fy in MPa).� Slenderness ratio in lateral torsional buckling (Li/ryc)cf where Li

is the length from the section of maximum moment to the adja-cent moment of inflection, and ryc is the radius of gyration of theportion of the elastic region in compression.� Flange slenderness factor in local buckling Kf = (b/t)cf/9 in the

range 0.9 < Kf < 1.5.

mbership values.

Fig. 4. The Sugeno fuzzy model (Jang et al., 1997).

5652 A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661

� Web slenderness factor in local buckling Kw = (hc/tw)cw/70 in therange 0.9 < Kw < 1.5.� Distortional restraint factor Kd of concrete slab in the negative

moment region of continuous composite beams (Kd ¼ 1 forplain steel beams and 0.71 for composite beams).

These parameters form the effective lateral slenderness ratio(ke) as follows:

Table 1Data pairs for Eq. (1).

a b yi

1 3 163 4 295 1 302 6 347 8 898 7 991 2 119 4 1012 5 297 8 891 1 69 9 1261 9 469 1 861 3 163 4 29

Fig. 5. Initial membe

ke ¼ Kf KdKwðLi=rycÞcf ð25 < ke < 140Þ: ð2Þ

And finally the beam rotation capacity is related to the effective lat-eral slenderness ratio (ke) in the following form:

R ¼ 3:01560ke

� �1:5

: ð3Þ

Li (2002) proposed an alternative expression for beam rotationcapacity with the following parameters given as follows:

k0f ¼b=tf

6 , 6 is the smallest beam flange slenderness ratio used inthe analysis.

k0w ¼hw=tw

24 , 24 is the smallest beam web slenderness ratio used inthe analysis.

k0w ¼Lb=ry

20 , 20 is the smallest overall slenderness ratio used in theanalysis

kt ¼ k0f k0wk0b: ð4Þ

Which is the beam overall slenderness. Li related the beam rotationcapacity to beam overall slenderness as follows:

R ¼ 12Cm

ktFy

347

� �2 ; ð5Þ

where Cm = 1 for uniform gradient beams and 2 for moment gradi-ent beams and Fy is the beam flange yield stress in MPa.

rship functions.

A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661 5653

On the other hand, there has also been attempt to use geneticprogramming for the formulation of rotation capacity. Cevik hasrecently proposed a genetic programming based formulation(Cevik, 2007) given as follows:

R ¼ �18:3þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Fyf þ Lþ d� Fyw þ 99:653

q� �� �62:14ðtwÞðbÞ=ðFyf ÞðLþ ðbÞðdÞÞ �

ffiffiffib3pð5626=FywÞ lnððFywÞðtf ÞÞ

h i: ð6Þ

In his pioneer work, Cevik (2007) has compared the GP based for-mulation with existing analytical equations of Kemp and Li andfound to be more accurate.

3. Fuzzy logic

Over the last decade, fuzzy logic invented by Zadeh (1965) hasbeen applied to a wide range of applications covering engineering,process control, image processing, pattern recognition and classifi-

Fig. 6. Final membe

Fig. 7. Fuzzy infer

cation, management, economics and decision making (Rutkowski,2004).

Fuzzy systems can be defined as rule-based systems that areconstructed from a collection of linguistic rules which can repre-sent any system with accuracy, i.e., they work as universal approx-imators. The rule-based system of fuzzy logic theory uses linguisticvariables as its antecedents and consequents where antecedentsexpress an inference or the inequality, which should be satisfiedand consequents are those, which we can infer, and is the outputif the antecedent inequality is satisfied. The fuzzy rule-based sys-tem is actually an IF–THEN rule-based system, given by, IF ante-cedent, THEN consequent (Sivanandam, Sumathi, & Deepa, 2007).

FL operations are based on fuzzy sets where the input data maybe defined as fuzzy sets or a single element with a membership va-lue of unity. The membership values (l1 and l2) are found from theintersections of the data sets with the fuzzy sets as shown in Fig. 3which illustrates the graphical method of finding membership val-ues in the case of a single input (Haris, 2006).

A fuzzy set contains elements which have varying degrees ofmembership in the set, unlike the classical or crisp sets where a

rship functions.

ence diagram.

Table 2Experimental database.

No. Ref. b (mm) d (mm) tf (mm) tw (mm) L (mm) fyf (MPa) fyw (MPa) Re

1 Lukey and Adams (1969)a 101.75 235.10 10.80 7.65 3480 283 308 11.802 88.00 235.10 10.80 7.65 2946 283 308 13.603b 51.3 191.3 5.28 4.45 1554 371 395 2.94 36.95 191.30 5.28 4.45 1036 371 395 10.405 43.05 191.30 5.28 4.45 1254 371 395 6.706 47.00 191.30 5.28 4.45 1396 371 395 3.407 48.40 191.30 5.26 4.45 1448 371 395 3.208 50.95 241.20 5.26 4.60 1372 371 350 4.209 36.85 241.20 5.26 4.60 960 371 350 13.70

10 42.95 241.20 5.26 4.60 1168 371 350 8.0011 46.75 241.20 5.26 4.60 1296 371 350 4.2012 44.45 241.20 5.26 4.60 1280 371 350 6.50

13 Kuhlmann (1989) andKuhlmann (1986)c

70.50 278.00 8.00 5.00 3404 236 217 8.0014 75.00 278.00 8.00 5.00 3704 236 217 7.0015b 80 277 8.5 5.5 4000 449 217 1.016 80.00 261.00 8.00 6.00 2540 287 260 12.7017 80.00 258.00 8.00 5.00 2636 287 252 8.6018 80.00 259.00 8.00 4.00 2716 287 252 4.6019 80.00 280.00 8.00 5.00 1796 287 252 13.5020 80.00 280.00 8.00 5.00 2196 287 252 11.5021 80.00 275.00 8.00 5.00 2598 287 252 7.8022 85.00 279.00 8.00 5.00 2802 236 217 5.5023 91.00 278.00 8.00 5.50 3002 236 217 8.9024 95.00 278.00 8.00 5.50 3400 236 217 7.6025 70.50 239.60 10.20 5.50 3000 333 709 5.1026 75.00 239.00 10.00 5.50 3200 333 709 3.8027 80.00 237.20 10.40 5.50 3508 333 709 3.6028 80.00 148.60 10.20 5.50 2304 333 709 10.5029 80.00 200.00 10.00 5.50 2204 333 709 9.5030 80.50 269.00 10.00 5.50 2100 333 709 6.6031 80.00 278.00 10.00 6.00 2000 333 341 12.0032 80.00 279.00 10.00 6.00 2402 333 349 8.7033 80.00 279.00 10.00 6.00 2804 333 349 7.2034 85.00 279.00 10.00 6.00 2406 333 349 10.0035 91.50 278.40 10.30 6.00 2500 333 349 6.7036 95.00 278.60 10.20 6.00 2700 333 349 5.20

37 Spangemacher (1991)d 109.30 186.50 16.30 9.80 3500 486 532 6.4038 109.30 184.90 16.20 9.40 3500 486 532 7.8039 109.50 186.10 16.30 9.60 3500 278 286 18.9040 109.20 185.20 16.10 9.40 3500 278 286 19.8041 110.00 188.00 10.50 7.50 3000 282 308 12.0042 110.50 189.00 11.00 7.40 4000 282 308 9.3043 112.80 188.30 11.00 7.50 3000 420 437 2.8044b 111 192.6 10.7 7.5 4000 420 437 1.545 139.70 241.20 17.80 10.90 3000 248 252 34.1046 139.50 246.60 17.70 10.80 4000 248 252 20.5047 141.70 246.40 17.40 11.40 3000 489 539 9.5048 142.00 249.70 17.40 11.50 4000 489 539 8.3049 140.00 240.80 12.60 8.00 3000 276 311 19.0050 140.00 243.40 12.80 7.50 4000 276 311 6.4051 140.50 250.40 12.60 9.00 3000 504 535 6.4052 140.50 249.60 12.70 9.30 4000 504 535 4.1053b 141.6 249.9 17.4 11.5 3000 489 535 0.954b 141.7 246.4 17.3 11.3 3000 489 535 2.055 141.50 246.90 17.30 11.35 3000 489 535 10.4056 117.80 185.40 10.30 7.25 4000 275 302 10.3057 117.80 186.10 11.10 7.65 4000 430 448 2.6058 150.30 320.00 15.00 10.00 3000 486 990 2.7059 150.30 320.00 15.00 10.00 3000 248 323 16.9060b 150.3 320 15 10 3000 817 813 0.9

61 Boreave et al. (1993)e 100.30 155.10 14.10 8.80 3000 303 342 16.8062 100.10 153.90 14.70 9.50 3000 375 421 12.1063 100.70 154.40 15.10 9.50 3000 445 462 10.0064 100.20 156.60 14.60 9.60 3000 261 291 24.3065 100.00 159.50 14.90 9.40 3000 409 426 9.20

66 Kemp (1985)f 75.00 217.80 8.09 6.65 3660 340 358 2.7067 72.50 217.40 10.57 6.82 3660 285 329 6.6068 53.00 273.90 7.05 5.85 3660 332 388 2.2069 74.50 217.90 8.56 6.78 1830 340 353 15.2070 74.50 217.10 1.44 6.78 1830 294 300 14.80

5654 A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661

Table 2 (continued)

No. Ref. b (mm) d (mm) tf (mm) tw (mm) L (mm) fyf (MPa) fyw (MPa) Re

71 70.00 209.50 10.77 6.76 1830 288 329 14.0072 77.00 120.30 9.83 7.44 3660 313 300 8.40

73 Suzuki et al. (1994)g 75.00 132.00 9.00 6.00 1200 291 340 33.4074 75.00 132.00 9.00 6.00 1200 527 340 19.2075 75.00 132.00 9.00 6.00 1200 291 509 22.3076 75.00 132.00 9.00 6.00 1200 526 509 9.4077 75.00 132.00 9.00 6.00 1800 291 340 27.2078 75.00 132.00 9.00 6.00 1800 291 509 18.5079 75.00 132.00 9.00 6.00 1800 526 509 7.7080 75.00 132.00 9.00 6.00 1800 687 340 9.4081 75.00 132.00 9.00 6.00 1800 291 686 15.70

a Tested on 12 specimens (rolled wide-flange beams).b Eliminated experiments from statistical analysis due experimental deficiencies.c 24 Specimens (built-up welded wide-flange sections) are tested.d Tested 34 rolled wide flange beams, where 24 of these are used as the remaining 10 specimens were not used on which shear failure is observed.e Five hot rolled section were tested.f 12 built-up welded I-beams was tested, but only seven beams due to missing data.g 9 built-up welded hybrid beams are tested.

Fig. 8. SB1.

Fig. 9. SB2.

A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661 5655

member either belongs to that set or does not (0 or 1). However afuzzy set allows a member to have a varying degree of membershipand this partial degree membership can be mapped into a functionor an universe of membership values (Ying Bai, Zhuang, & Wang,2006).

The implementation of fuzzy logic to real applications considersthe following steps (Ying Bai et al., 2006):

1. Fuzzification which requires conversion of classical data or crispdata into fuzzy data or membership functions (MFs).

2. Fuzzy Inference Process which connects membership functionswith the Fuzzy rules to derive the fuzzy output.

3. Defuzzification which computes each associated output.

3.1. Neuro-fuzzy systems

Fuzzy systems can also be connected with Neural Networks toform neuro-fuzzy systems which exhibit advantages of both ap-proaches. Neuro-fuzzy systems combine the natural languagedescription of fuzzy systems and the learning properties of neuralnetworks. Various neuro fuzzy systems have been developed thatare known in literature under short names. ANFIS developed byJang, Sun, and Mizutani (1997), (Adaptive Network-based FuzzyInference System) is one of these Neuro-fuzzy systems which allowthe fuzzy systems to learn the parameters using adaptive back-propagation learning algorithm (Rutkowski, 2004).

Mainly three types of fuzzy inference systems have been widelyemployed in various applications: Mamdani, Sugeno and Tsukam-oto fuzyy models. The differences between these three fuzzy infer-ence systems are due to the consequents of their fuzzy rules, andthus their aggregation and defuzzification procedures differaccordingly (Jang et al., 1997). In this study the Sugeno FIS is usedwhere each rule is defined as a linear combination of input vari-ables. The corresponding final output of the fuzzy model is simplythe weighted average of each rule’s output. A Sugeno FIS consistingof two input variables x and y, for example, a one output variable fwill lead to two fuzzy rules:

Rule 1: If x is A1, y is B1 then f1 = p1x + q1y + r1;Rule 2: If x is A2, y is B2 then f2 = p2x + q2y + r2,

where pi, qi, and ri are the consequent parameters of ith rule. Ai,Bi and Ci are the linguistic labels which are represented by fuzzysets shown in Fig. 4.

3.2. Solving a simple problem with ANFIS

To illustrate how ANFIS works for function approximation, letssuppose one is given a sampling of the numerical values from thesimple function below:

yi ¼ a2 þ 5b; ð7Þ

where a and b are independent variables chosen over randomlypoints in the real interval (AISC LRFD, 1997; Dinno, 2002) and. Inthis case, a sample of data in the form of 20 pairs (a,b,yi) are givenin Table 1. Where yi is the output of the function given in Eq. (7).The aim is to construct the ANFIS model fitting those values withinminimum error for Eq. (7) by using the simplest ANFIS model that isavailable where the number of rules are 2 for each variable, the in-put membership function triangular and the type of output mem-bership function is constant.

Initial and final membership values of rules for each input aregiven in Figs. 5 and 6, respectively. Suppose one will find the out-put for input values of 9 and 1. The inference diagram of the pro-posed ANFIS model is given in Fig. 7 for input values of 9 and 1with corresponding values of output membership which is chosenas constant. For the first input which is 9 the value of the member-ship function is observed to be 1 shown on left side of Fig. 7. For thesecond input which is 5 the value of the membership function isobserved to be 0.5 shown on left side of Fig. 7. As the minimumof the two membership values are taken, the first and second val-ues of output membership will be zero. The third and fourth outputmembership should be multiplied by 0.5. Thus the final output willbe: 86 � 0.5 + 126 � 0.5 = 106.

Table 3Minimum and maximum values of cross-section variables.

Variable Minimum value Maximum value

Half length of flange b (mm) 36.95 150.4Height of web d (mm) 120.3 320Thickness of flange tf (mm) 1.44 17.3Thickness of web tw (mm) 4 11.5Length of beam L (mm) 960 4000Yield strength of flange fyf (MPa) 236 817Yield strength of web fyw (MPa) 217 990

Table 4Features of the proposed ANFIS model.

Type SUGENO

Aggregation method MaximumDefuzzification method Weighted averageInput membership function type TriangularOutput membership function type Constant

Table 5Statistical parameters of the proposed ANFIS model.

Mean COV (%)

ANFIS testing set 0.77 30ANFIS training set 1.00 5.5

Table 6Values of output membership functions (128 constant output MF).

mf 1 135 mf 17 508 mf 33 87.4 mf 49 317 mmf 2 �192 mf 18 230 mf 34 350 mf 50 �151 mmf 3 �142 mf 19 214 mf 35 �451 mf 51 93.2 mmf 4 �177 mf 20 55.3 mf 36 �344 mf 52 �44.6 mmf 5 �196 mf 21 �650 mf 37 15.3 mf 53 81.7 mmf 6 �6.48 mf 22 21 mf 38 103 mf 54 17.1 mmf 7 �304 mf 23 91.6 mf 39 �982 mf 55 �13.6 mmf 8 �98.8 mf 24 102 mf 40 �461 mf 56 �24.5 mmf 9 44.7 mf 25 �120 mf 41 �108 mf 57 �251 mmf 10 223 mf 26 18.3 mf 42 �85.6 mf 58 �290 mmf 11 262 mf 27 39.7 mf 43 �22.4 mf 59 �39.9 mmf 12 60.1 mf 28 �3.74 mf 44 �54.9 mf 60 �41.3 mmf 13 �76.3 mf 29 281 mf 45 141 mf 61 24.5 mmf 14 164 mf 30 88.9 mf 46 8.42 mf 62 �127 mmf 15 151 mf 31 �96 mf 47 �89.9 mf 63 �135 mmf 16 78 mf 32 �4.43 mf 48 �52.5 mf 64 �73.6 m

Fig. 10. Geometry of cross-section variables.

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The exact result for a = 9 and b = 5 from Eq. (7) will be y = 92 +5(5) = 106.

4. Numerical application

The main aim of this article is the NF modeling of rotationcapacity of wide flange steel based on experimental results fromliterature. Therefore an extensive literature survey has been per-formed for experimental results shown in Table 2. The experimen-tal results in this field are dispersed. Standard beams are used inexperimental studies (SB1,SB2) shown in Figs. 8 and 9. SB1 is usedin the experimental studies given in Table 2. The geometry ofcross-section variables of tested beams is shown in Fig. 10. Theranges of variables i.e., the maximum and minimum values ofcross-section variables where the proposed NF Model will be validare given in Table 3. The experimental database presented in Table2 has been used for training (60 tests) and testing set (15 tests) ofthe proposed NF model. The NF model is constructed with trainingsets and the accuracy is verified by testing sets which the NF modelfaces for the fist time.

The simplest ANFIS model is selected to illustrate the effective-ness of the NF approach. The proposed ANFIS model uses Triangu-lar input membership functions with minimum number of ruleswhich is 2. The output membership function is chosen as the sim-plest one available which is a constant value. Moreover the numberof cycles the training epoch is taken as 1 which leads the initial andfinal membership values to be same. These conditions will lead tothe simplest available NF model. The proposed ANFIS model hasbeen constructed by MATLAB Fuzzy Logic Toolbox. Features and re-lated parameters of the ANFIS model are presented in Table 4. Sta-tistical parameters of the proposed ANFIS model for Test/Predictedresults are given in Table 5. Output membership function valuesare given in Table 6. The initial and final membership functionsfor inputs are presented in Fig. 11. The accuracies of Mean (Test/ANFIS) for training and testing sets are presented in Figs. 12 and13.

The prediction of the proposed NF model vs. actual experimen-tal values and their comparison with previously obtained numeri-cal results and existing analytical equations are given in Table A1.The numerical results are obtained by a specialized computer pro-gram (DUCTROT) (Petcu & Gioncu, 2003) developed by Petcu andGioncu which is based on local plastic mechanism. The validationof computer program is performed by Petcu and Gioncu comparingthe obtained values with experimental and numerical results. Theresult accuracy is confirmed by this comparison (Kuhlmann, 1986).The mean (Test/Predicted) values of NF model, DUCTROT, Kemp’sequation, GP formulation and Li’s equation are presented in Figs.

f 65 �99.5 mf 81 �177 mf 97 �320 mf 113 �128f 66 �191 mf 82 �101 mf 98 �11.9 mf 114 �293f 67 �80.1 mf 83 123 mf 99 �387 mf 115 39.7f 68 �74.1 mf 84 50.5 mf 100 �235 mf 116 �36.6f 69 646 mf 85 230 mf 101 �117 mf 117 42.5f 70 101 mf 86 �14 mf 102 566 mf 118 �327f 71 104 mf 87 303 mf 103 203 mf 119 1.1f 72 17 mf 88 218 mf 104 �14.1 mf 120 �45.1f 73 �260 mf 89 26.8 mf 105 650 mf 121 297f 74 �20.4 mf 90 �161 mf 106 104 mf 122 �418f 75 171 mf 91 46.9 mf 107 43.3 mf 123 31.5f 76 57.6 mf 92 42.3 mf 108 �34.2 mf 124 96.4f 77 �137 mf 93 9.65 mf 109 �69.4 mf 125 73.8f 78 �70.4 mf 94 �171 mf 110 �5.27 mf 126 �201f 79 22.3 mf 95 �53.8 mf 111 �134 mf 127 �55.6f 80 �5.06 mf 96 151 mf 112 �141 mf 128 228

Fig. 11. Initial and final membership values.

Fig. 12. (Test/ANFIS) of testing set.

A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661 5657

14–18, respectively. As seen, the NF model is more accurate(COV = 0.15) compared to all models given in Table A1.

5. Main effects of variables on rotation capacity

The Main Effect plot is an important graphical tool to visualizethe independent impact of each variable on rotation capacity. This

graphical tool enables a better and simple picture of the overallimportance of variable effects on the output which is the rotationcapacity for the case and will provide a general snapshot. In maineffects plot, the mean output (rotation capacity) is plotted at eachfactor level which is later connected by a straight line. The slope ofthe line for each variable expresses the degree of its effect on theoutput. To obtain the main effect plot a wide range of parametric

Fig. 13. (Test/ANFIS) of training set.

Fig. 14. Test/predicted (Re/Rfuz) values for NF model.

Fig. 15. Test/predicted (Re/Rduct) values for Ductrot program.

Fig. 16. Test/predicted (Re/Rkemp) values for Kemp’s equation.

Fig. 17. Test/predicted (Re/Rgp) values for GP formulation.

5658 A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661

study has been performed by using the well trained NF model.From main effect plot in Fig. 19, it can be concluded that all vari-ables used for NF modeling given in the experimental databasehave significant effects on rotation capacity. Variables that are ob-served to be directly proportional from Fig. 19 are half length of

flange (b) and Thickness of web (tw) where other variables (Heightof web (d), Thickness of flange (tf), Length of beam (L), Yieldstrength of flange (fyf) and web (fyw)) are found to be inversely pro-portional with rotation capacity. The evaluation of separate inter-action effects plot between any two variables is left as the scope

Fig. 18. Test/predicted (Re/Rli) values for Li’s equation.

A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661 5659

of another study. The main effects plot will also help furtherresearchers willing to perform experimental studies on rotationcapacity of wide flange beams in the phase of design ofexperiments.

Fig. 19. Main effects plot of var

Table A1Comparative analysis of proposed NF model with experimental, numerical results and ana

No. Ref. b(mm)

d(mm)

tf

(mm)tw

(mm)L(mm)

fyf

(MPa)fyw

(MPa)Re

1 Lukey–Adams

101.75 235.10 10.80 7.65 3480 283 308 11.802 88.00 235.10 10.80 7.65 2946 283 308 13.604 36.95 191.30 5.28 4.45 1036 371 395 10.405 43.05 191.30 5.28 4.45 1254 371 395 6.706 47.00 191.30 5.28 4.45 1396 371 395 3.407 48.40 191.30 5.26 4.45 1448 371 395 3.208 50.95 241.20 5.26 4.60 1372 371 350 4.209 36.85 241.20 5.26 4.60 960 371 350 13.70

10 42.95 241.20 5.26 4.60 1168 371 350 8.0011 46.75 241.20 5.26 4.60 1296 371 350 4.2012 44.45 241.20 5.26 4.60 1280 371 350 6.50

6. Conclusion

This paper presents a pioneer work for the modeling of avail-able rotation capacity of wide flange beams using Neuro-fuzzy ap-proach for the first time in literature. The proposed NF model is arule-based model based on experimental results collected from lit-erature. To show its effectiveness, the simplest possible NF modelis constructed. The results of the proposed NF model show verygood agreement with experimental results (COV = 0.15). Numericalresults of the same experimental database are obtained by a spe-cialized computer program (DUCTROT), Kemp’s and Li’s and Cevik’sgenetic programming based equations and the NF rule-based mod-el is found to perform better. It should be noted that existing ana-lytical models are not valid for some test results where ranges ofparameters exceed maximum specified limits. However the pro-posed NF model is valid for the ranges of the whole experimentaldatabase used for the modeling process. On the other hand, itshould be noted that the NF model is bound with the experimentaldatabase used for modeling. The outcomes of this study reveal thatNF may serve as a very effective and accurate tool for the modelingof rotation capacity of wide flange beams. However the effective-ness of NF approach for the modeling of structural engineeringproblems should be verified by further studies in this field.

iables vs. rotation capacity.

lytical equations.

Rfuz Re/Rfuz

Rduct Re/Rduct

Rgp R/Rgp

Rkemp R/Rkemp

Rli R/Rli

11.93 0.99 7.91 1.49 10.03 1.18 5.65 2.09 10.51 1.1213.32 1.02 10.74 1.27 12.6 1.08 8.77 1.55 14.09 0.9610.40 1.00 8.01 1.30 7.91 1.31 10.36 1 13.89 0.75

6.01 1.11 7.57 0.88 7.65 0.88 6.59 1.02 10.06 0.673.75 0.91 7.38 0.46 7.45 0.46 5.01 0.68 8.38 0.412.84 1.12 7.01 0.46 7.37 0.43 4.54 0.71 7.85 0.413.76 1.12 6.80 0.62 7.05 0.6 4.99 0.84 7.87 0.53

13.45 1.02 7.95 1.72 7.39 1.85 12.97 1.06 14.87 0.928.16 0.98 7.28 1.10 7.18 1.11 7.92 1.01 10.71 0.755.43 0.77 7.06 0.60 7.03 0.6 6.07 0.69 8.97 0.476.05 1.07 6.89 0.94 6.93 0.94 6.6 0.98 9.48 0.69

(continued on next page)

Table A1 (continued)

No. Ref. b(mm)

d(mm)

tf

(mm)tw

(mm)L(mm)

fyf

(MPa)fyw

(MPa)Re Rfuz Re/

RfuzRduct Re/

RductRgp R/

RgpRkemp R/

RkempRli R/

Rli

13 Kuhlmann 70.50 278.00 8.00 5.00 3404 236 217 8.00 8.21 0.97 7.92 1.01 7.65 1.05 5.27 1.52 10.1 0.7914 75.00 278.00 8.00 5.00 3704 236 217 7.00 6.73 1.04 6.93 1.01 6.52 1.07 4.29 1.63 8.8 0.816 80.00 261.00 8.00 6.00 2540 287 260 12.70 13.03 0.98 7.24 1.75 11.18 1.14 5.81 2.19 9.7 1.3117 80.00 258.00 8.00 5.00 2636 287 252 8.60 8.85 0.97 7.57 1.14 9.22 0.93 4.45 1.93 8 1.0818 80.00 259.00 8.00 4.00 2716 287 252 4.60 4.54 1.01 8.83 0.52 7.06 0.65 3.18 1.45 6.39 0.7219 80.00 280.00 8.00 5.00 1796 287 252 13.50 13.45 1.00 13.91 0.97 12.39 1.09 7.77 1.74 11.59 1.1620 80.00 280.00 8.00 5.00 2196 287 252 11.50 11.18 1.03 9.43 1.22 10.41 1.1 5.75 2 9.48 1.2121 80.00 275.00 8.00 5.00 2598 287 252 7.80 8.93 0.87 7.64 1.02 8.8 0.89 4.48 1.74 8.04 0.9722 85.00 279.00 8.00 5.00 2802 236 217 5.50 5.71 0.96 8.20 0.67 11.51 0.48 5.53 0.99 10.43 0.5323 91.00 278.00 8.00 5.50 3002 236 217 8.90 8.89 1.00 6.76 1.32 11.83 0.75 5.17 1.72 9.98 0.8924 95.00 278.00 8.00 5.50 3400 236 217 7.60 7.59 1.00 5.78 1.32 9.69 0.78 4.06 1.87 8.48 0.925 70.50 239.60 10.20 5.50 3000 333 709 5.10 5.12 1.00 4.69 1.09 3.62 1.41 2.8 1.82 8.49 0.626 75.00 239.00 10.00 5.50 3200 333 709 3.80 6.39 0.59 3.80 1.00 3.37 1.13 2.27 1.68 7.37 0.5227 80.00 237.20 10.40 5.50 3508 333 709 3.60 3.59 1.00 3.49 1.03 2.99 1.2 1.94 1.85 6.66 0.5428 80.00 148.60 10.20 5.50 2304 333 709 10.50 10.53 1.00 13.34 0.79 8.27 1.27 3.95 2.66 10.67 0.9829 80.00 200.00 10.00 5.50 2204 333 709 9.50 9.41 1.01 9.98 0.95 6.56 1.45 3.8 2.5 10.4 0.9130 80.50 269.00 10.00 5.50 2100 333 709 6.60 6.61 1.00 5.21 1.27 5.13 1.29 3.78 1.75 10.36 0.6431 80.00 278.00 10.00 6.00 2000 333 341 12.00 11.95 1.00 11.16 1.08 9.33 1.29 7.88 1.52 11.74 1.0232 80.00 279.00 10.00 6.00 2402 333 349 8.70 9.34 0.93 9.31 0.93 7.63 1.14 4.66 1.87 8.37 1.0433 80.00 279.00 10.00 6.00 2804 333 349 7.20 7.01 1.03 8.14 0.88 6.29 1.14 5.88 1.22 9.77 0.7434 85.00 279.00 10.00 6.00 2406 333 349 10.00 8.83 1.13 9.70 1.03 7.82 1.28 5.42 1.84 9.25 1.0835 91.50 278.40 10.30 6.00 2500 333 349 6.70 7.40 0.91 10.18 0.66 7.77 0.86 4.9 1.37 8.64 0.7836 95.00 278.60 10.20 6.00 2700 333 349 5.20 6.45 0.81 9.26 0.56 7.16 0.73 4.08 1.27 7.66 0.68

37 Spangema 109.30 186.50 16.30 9.80 3500 486 532 6.40 6.46 0.99 9.09 0.70 5.59 1.14 4.55 1.41 6.96 0.9238 109.30 184.90 16.20 9.40 3500 486 532 7.80 7.74 1.01 9.28 0.84 5.41 1.44 4.27 1.83 6.67 1.1739 109.50 186.10 16.30 9.60 3500 278 286 18.90 19.43 0.97 11.53 1.64 18.37 1.03 16.27 1.16 20.87 0.9140 109.20 185.20 16.10 9.40 3500 278 286 19.80 19.21 1.03 15.44 1.28 18.02 1.1 15.55 1.27 20.25 0.9841 110.00 188.00 10.50 7.50 3000 282 308 12.00 21.20 0.57 7.68 1.56 16.56 0.72 6.27 1.92 11.3 1.0642 110.50 189.00 11.00 7.40 4000 282 308 9.30 9.49 0.98 6.23 1.49 8.93 1.04 4.3 2.16 8.8 1.0643 112.80 188.30 11.00 7.50 3000 420 437 2.80 2.88 0.97 5.32 0.53 6.6 2.35 2.78 1.01 5.26 0.5345 139.70 241.20 17.80 10.90 3000 248 252 34.10 34.10 1.00 19.78 1.72 30.29 1.13 24.98 1.37 29.23 1.1746 139.50 246.60 17.70 10.80 4000 248 252 20.50 20.50 1.00 14.15 1.45 16.03 1.28 15.81 1.3 21.55 0.9547 141.70 246.40 17.40 11.40 3000 489 539 9.50 10.80 0.88 9.41 1.01 7.31 1.3 5 1.9 7.39 1.2948 142.00 249.70 17.40 11.50 4000 489 539 8.30 8.30 1.00 7.06 1.18 3.9 2.12 3.32 2.5 5.64 1.4749 140.00 240.80 12.60 8.00 3000 276 311 19.00 18.99 1.00 9.31 2.04 15.99 1.19 6.57 2.89 11.98 1.5950 140.00 243.40 12.80 7.50 4000 276 311 6.40 6.38 1.00 7.55 0.85 8.14 0.79 4.01 1.6 8.62 0.7451 140.50 250.40 12.60 9.00 3000 504 535 6.40 6.38 1.00 4.86 1.32 5.23 1.22 ** ** 3.95 1.6252 140.50 249.60 12.70 9.30 4000 504 535 4.10 9.90 0.41 3.69 1.11 2.79 1.47 ** ** 3.08 1.3355 141.50 246.90 17.30 11.35 3000 489 535 10.40 10.40 1.00 9.36 1.11 7.28 1.43 5.03 2.07 7.42 1.456 117.80 185.40 10.30 7.25 4000 275 302 10.30 10.18 1.01 5.25 1.96 9.58 1.08 3.63 2.84 7.98 1.2957 117.80 186.10 11.10 7.65 4000 430 448 2.60 2.57 1.01 3.74 0.69 4.07 0.64 1.69 1.54 3.73 0.758 150.30 320.00 15.00 10.00 3000 486 990 2.70 2.70 1.00 5.08 0.53 3.55 0.76 1.83 1.48 5.18 0.5259 150.30 320.00 15.00 10.00 3000 248 323 16.90 52.47 0.32 14.87 1.14 17.26 0.98 ** ** 1.83 9.23

61 Boerav 100.30 155.10 14.10 8.80 3000 303 342 16.80 16.98 0.99 13.61 1.23 19.51 0.86 12.81 1.31 17.86 0.9462 100.10 153.90 14.70 9.50 3000 375 421 12.10 11.95 1.01 11.74 1.03 13.89 0.87 9.55 1.27 13.16 0.9263 100.70 154.40 15.10 9.50 3000 445 462 10.00 10.05 1.00 9.90 1.01 10.31 0.97 7.16 1.4 9.59 1.0464 100.20 156.60 14.60 9.60 3000 261 291 24.30 24.26 1.00 16.52 1.47 28.78 0.84 21.6 1.12 27.09 0.965 100.00 159.50 14.90 9.40 3000 409 426 9.20 10.19 0.90 10.76 0.85 11.72 0.79 8.32 1.11 11.07 0.83

66 Kemp 75.00 217.80 8.09 6.65 3660 340 358 2.70 2.76 0.98 4.59 0.59 4.9 0.55 2.71 1 5.78 0.4767 72.50 217.40 10.57 6.82 3660 285 329 6.60 6.65 0.99 9.50 0.69 6.9 0.96 6.49 1.02 11.84 0.5668 53.00 273.90 7.05 5.85 3660 332 388 2.20 2.17 1.01 4.36 0.51 2.78 0.79 2.61 0.84 6.02 0.3769 74.50 217.90 8.56 6.78 1830 340 353 15.20 15.26 1.00 10.64 1.43 13.1 1.16 8.85 1.72 12.63 1.270 74.50 217.10 1.44 6.78 1830 294 300 14.80 30.60 0.48 11.84 1.25 13.75 1.08 ** ** 2.23 6.6371 70.00 209.50 10.77 6.76 1830 288 329 14.00 14.13 0.99 15.26 0.92 17.59 0.8 19.52 0.72 24.39 0.5772 77.00 120.30 9.83 7.44 3660 313 300 8.40 8.38 1.00 6.22 1.35 12.08 0.7 6.43 1.31 10.23 0.82

73 Suzuki 75.00 132.00 9.00 6.00 1200 291 340 33.40 33.55 1.00 23.05 1.45 31.14 1.07 22.61 1.48 27.08 1.2374 75.00 132.00 9.00 6.00 1200 527 340 19.20 19.22 1.00 20.35 0.94 14.45 1.33 9.28 2.07 8.26 2.3375 75.00 132.00 9.00 6.00 1200 291 509 22.30 25.69 0.87 24.75 0.90 23.25 0.96 16.71 1.33 27.08 0.8276 75.00 132.00 9.00 6.00 1200 526 509 9.40 9.40 1.00 13.16 0.71 10.82 0.87 6.88 1.37 8.29 1.1377 75.00 132.00 9.00 6.00 1800 291 340 27.20 25.88 1.05 13.60 2.00 23.67 1.15 12.31 2.21 18.05 1.5178 75.00 132.00 9.00 6.00 1800 291 509 18.50 20.45 0.90 17.21 1.08 17.69 1.05 9.09 2.03 18.05 1.0279 75.00 132.00 9.00 6.00 1800 526 509 7.70 9.50 0.81 6.93 1.11 8.19 0.94 7.27 1.06 18.05 0.4380 75.00 132.00 9.00 6.00 1800 687 340 9.40 9.38 1.00 5.73 1.64 7.34 1.28 3.74 2.51 5.52 1.781 75.00 132.00 9.00 6.00 1800 291 686 15.70 14.76 1.06 19.63 0.80 14.5 1.08 22.61 0.69 27.08 0.58

Mean 0.95 1.08 1.01 1.54 1.11StdDv 0.14 0.37 0.26 0.52 1.22COV 0.15 0.35 0.26 0.34 1.10

Bold Sets are used as Testing sets.** Exceeds max limitations for Kemp’s equation.

5660 A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661

A. Cevik / Expert Systems with Applications 38 (2011) 5650–5661 5661

Acknowledgement

This research was supported by Gaziantep University ProjectResearch Unit.

Appendix A

See Table A1.

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