Engineering Structures 28 (2006) 153–161www.elsevier.com/locate/engstruct

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Prediction of wind loadson a large flat roof using fuzzy neural networks

J.Y. Fua,b, Q.S. Lia,∗, Z.N. Xiec

aDepartment of Building and Construction, City University of Hong Kong, Hong Kongb Department of Civil Engineering, Jinan University, Guangzhou 510632, Chinac Department of Civil Engineering, Shantou University, Shantou 515063, China

Received 7 February 2005; received in revised form 27 July 2005; accepted 9 August 2005Available online 15 September 2005

Abstract

Fuzzy neural networks (FNN) have capability to develop complex, nonlinearfunctional relationships between input–output patterns based olimited and sometimes inconsistent data, and are therefore suitable to predict wind loads on buildings on the basis of data obtained from motests in wind tunnels. In this study, simultaneous pressure measurements are made on a large flat roof model in a boundary layer wAn FNN approach is developed for prediction of mean pressure distributions on the roof model, and parts of the wind tunnel test resultsas the training sets for the FNN to recognize the pressure distribution patterns. The procedure is further extended to predict the powand cross-power spectra of fluctuating wind pressures for some typical tap locations in the roof corners and leading edge areas undwind directions. It is found that the developed FNN approach can generalize functional relationships of wind loads varying with inciddirections and spatial locations on the roof, and can successfully predict the wind loads on the roof which are not fully covered bytunnel measurements. It is demonstrated from this study that the adoption of the FNN approach can lead to a significant reduction of thmeasurement programs (e.g., incident wind direction configurations and number of required pressure taps) in wind tunnel tests.c© 2005 Elsevier Ltd. All rights reserved.

Keywords: Fuzzy neural network; Wind loads; Flat roof; Wind tunnel test

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1. Introduction

In recent years, more and more long-span roof structuhave been built with increasing span and structural refinement.Roofs of such structures usually have the characteristics of lmass, high flexibility, slight damping and low natural frequencAs the span increases, the natural frequencies generdecrease, and the susceptibility of a roof structure with lospan to resonant excitation by turbulent wind action increaConsequently, these structures have become progressivelywind sensitive, and wind loads generally control the designsuch structures.

In designing such large roof structures, it is usuanecessary to conduct wind tunnel tests to determine wind loon rigid models by taking pressure measurements. In such wtunnel experiments, it is necessary to install as many prestaps as possible on the model surfaces in order to cap

∗ Corresponding author. Tel.: +852 2784 4677; fax: +852 2788 7612.E-mail address:bcqsli@cityu.edu.hk (Q.S. Li).

0141-0296/$ - see front matterc© 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2005.08.006

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the detailed characteristics of wind loads on the structuThese experiments could be time consuming and expenbut mayonly cover a limited number of basic configuratiosince wind pressure distributions on a long-span roof strucdepend on several factors such as incident wind directionwind turbulence characteristics, upstream terrain roughnroof shape configurations, etc. Once experimental resultsobtained, one strategy tries to look for an equation whichfits the results inorder to incorporate an empirical equationthe analysis and design of structures [1]. Such a procedure cabe very cumbersome, especially for a large and complexstructure.

Hence, there is a need to explore an effective wfor predicting wind loads on large roof structures fromcomparatively reduced pressure measurement programmwind tunnel tests. The ability of the artificial neural netwo(ANN) approach to train a given data set, and on that bato predict missing data, makes it an attractive methodknowledge acquisition for problems which are difficultsolve by conventional theoretical and numerical metho

154 J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161

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In fact, ANN has been successfully applied to solve a numof wind engineering problems. Turkkan and Srivastava2]used the neural network approach to predict the wind ldistribution for air-supported structures. Khanduri et al.3]applied the backpropagation neural networks to investigatewind interference problem among tall buildings. Sandri anMehta [4] applied a neural network for predicting wind-inducedamage to buildings. Chen et al. [5] adopted an artificialneural network approach for prediction of mean and romean-square pressure coefficients on gable roofs of lowbuildings. On the other hand, a fuzzy system based onpioneering work of Zadeh [6] in fuzzy set theory has been aactive research area with wide applications in civil engineeringsuch as earthquake intensity evaluation [7], a fuzzy modelfor load combinations [8] and structural reliability assessmen[9,10], etc.

In practice, there exists a veryclose relationship betweeneural networks and fuzzy systems, since they both work wdegrees of imprecision in a space that is not defined by sdeterministic boundaries. Hence, fuzzy and neural technolocan be fused into a unified methodology known as fuzzy nenetworks.

A literature search indicates that generally the authof the previous investigationson the application of neuralnetworks and fuzzy systems to wind engineering have maconcentrated their studies to the predictions of presdistributions on various building models. The usage ofneural network method to predict power spectra or cropower spectra of fluctuating wind pressures on buildingsstructures has received little, if any, attention in the literaturthe past.

The objective of the present work is to develop a FuzNeural Network (FNN) approach which is accurate and robfor the prediction of wind loads on a large flat roof. In ordto obtain wind-induced pressure data for training and testinthe FNN, simultaneous pressure measurements are madelarge flat roof model in boundary layer wind tunnel tests. Tstudy is concerned with not only the prediction of the mepressure distributions on the roof model, but also the pospectra and cross-power spectra of fluctuating wind pressusing the FNN approach.

2. Wind tunnel experiment

2.1. Experimental arrangements

Wind tunnel experiments were carried out in the boundlayer wind tunnel at Shantou University with a working secti3 m wide× 2 m high and 20 m long. Spires and roughneelements were used to generate a simulated atmospboundary layer of suburban terrain specified in the China LCode [11] as exposure B. This terrain type specifies a mewind speed profile with a power law exponent ofα = 0.16 and aturbulence intensity of 18% atthe model height. The measuremean wind speeds and turbulence intensities at various heover the test section are illustrated inFig. 1. The spectrum oflongitudinal wind velocity at the model height (z = 140 mm

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Fig. 1. Mean wind speed and turbulence intensity profiles.

Fig. 2. Spectra of longitudinal wind velocity at the model height.

in the wind tunnel andz = 28 m in full scale) is shown inFig. 2. The Davenport, Kaimal, Karman and Harris type speare also expressed as non-dimensional forms in the figurcomparative purposes.

A flat roof model with square shape shown inFig. 3 wasmade to represent a typical large flat roof structure wcantilevered parts. The roof was erected at a height of 140above the wind tunnel floor. The model was made of Plexigand its dimensions are shown inFig. 3. Fig. 4shows aphoto ofthe model mounted in the wind tunnel. The geometric lenscale was selected as 1:200, so the prototype dimensionsroof are thus 120 m× 120 m at a height of 28 m.

There were 144 and 108 pressure taps made on the upplower roof surfaces, respectively, for pressure measuremIn order to obtain the pressure differences between thesides of the cantilevered roof parts, 108 pairs of the prestaps were arranged at the same locations on both uppelower cantilevered roof surfaces. The other 36 pressureon the upper surface were located on the enclosed se

J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161 155

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Fig. 3. Model dimensions.

Fig. 4. The model in wind tunnel.

of the roof. Simultaneous pressure measurements werefrom the pressure taps. The layout of the pressure tapthe upper roof surface is shown inFig. 5. In the wind tunneltests, wind direction was defined as an angleβ from the eastalong anti-clockwise direction andβ varied from 0◦ to 360◦with increment of 22.5◦. Data sampling frequency was 330 Hwith sampling length of 32 768, and the measurements wcarried out when the mean wind speed at the model height10.5 m/s.

2.2. Definitions of parameters

The pressure coefficient of the pressure tapi on the roofsurface is defined as follows:

Cpi (t) = pi (t) − p∞p0 − p∞

(1)

wherepi (t) is the measured surface pressure at the tapi , p0 andp∞ are the total pressure and the static pressure at refeheight, respectively.

The convention for positive roof pressure andcorresponding pressure coefficient, on either the upper or lo

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Fig. 5. Layout of the pressure taps on the upper roof surface. (All dimensare in millimeters,β and A denote the wind attack angle and the area marby dotted lines, respectively.)

roof surface, is always from the air side onto the roof surfaThe net wind-induced pressure on the roof due to the combaction of pressures on the upper and lower roof surfacesdefined as positive in the downward direction.

�Cpi (t) = Cupi (t) − Cl

pi (t) = pui (t) − pl

i (t)

p0 − p∞(2)

where �Cpi (t) is the pressure coefficient difference of thtap i between the upper and lower roof surfaces.pu

i (t) andpl

i (t) are respectively the measured upper and lower surpressures at the tapi . In this study, the pressure coefficienCpi (t) defined in Eq.(1) and the pressure difference coefficie�Cpi (t) defined in Eq.(2) are both expressed asCpi (t). But,the pressure difference coefficients are used for the cantilevroof parts and the pressure coefficients are adopted for the usurface of the enclosed section.

Time-history signalsof wind-induced force on a roof surfacwereobtained from an integration of the simultaneous recof wind pressures over the corresponding surface, andmean pressure coefficientC̄p was determined from the pressurmeasurements.

Considering pressure signal attenuation due to the ustubing systems for pressure measurements, correction tosignal output of fluctuating wind pressure is necessary [12,13].Thus, the power spectrum of fluctuating wind pressure shobe determined by

SCp( f ) = SCpx( f )

|H ( f )|2 (3)

whereSCpx( f ) is the signal output of a pressure transducer, aH ( f ) is the frequency response function of the tubing syste

3. Fuzzy neural networks

There have been various applications of fuzzy neunetworks (FNN) in civil engineering [14,15]. Usually, models

156 J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161

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Fig. 6. Topology of the four-layer FNN.

based on neural networks for forecasting problems arecomplex than systems based on fuzzy logic. However,simplicity is achieved at the cost of an explicitly definrelationship between the individual inputs and overall moparameters.

This section presents a brief introduction to a fuzzy nenetwork approach [16] that combines the capability of neuralnetworks with fuzzy logic reasoning attributes. In any cathe FNN can be viewed as elaborate, nonlinear systems wgiven an external input, will estimate the corresponding outeven if the inputs are noisy or incomplete.

3.1. Fuzzy neural network model

The topology of the FNN used in the present workshown inFig. 6, which is comprised of four different layersan input layer, a membership layer, an inference layer, adefuzzification layer (an output layer). The network consof n input variables withn neurons in the input layer, andmnumber of rules withm neurons in the inference layer; ththe number of neurons in the membership layer isn × m.Where m can be generated by using the K-mean clustetechnique [16], of course, the values ofm can also be adjustewith practicalrequirements.

In the membership layer, each node performs a membefunction, and the activation function ofui consists of a seof membership functions, i.e.ui = (ui1, ui2, . . . , uim). TheGaussian function is adopted as the membership function.the j th node ofui :

ui j = exp

(− (xi − mij )

2

σ 2i j

), 1 ≤ i ≤ n, 1 ≤ j ≤ m (4)

whereui j is the value of the fuzzy membership function of ti th input variablexi corresponding to thej th rule;mij andσi j

are, respectively, the mean and the standard deviation oGaussian function in the j th term of thei th input variablexi ,andmi = (mi1, mi2, . . . , mim), σi = (σi1, σi2, . . . , σim).

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The activation function in the inference layer uses muplicative inference. For thei th rule node, the output is given b

πi = u1i × u2i × · · · × uni =n∏

j =1

u j i , (1 ≤ i ≤ m). (5)

The defuzzification layerhas the connecting weights(ωi ) tothe output from the inference layer, and these weights sigthe strength of each rule in the output of the model. The ouy is given as

y = ω1π1 + ω2π2 + · · · + ωmπm. (6)

More detailed descriptions of the FNN can be found in [16].

3.2. Learning algorithm of FNN

The learning algorithm [16] of the FNN used in this study,which is based on the conventional BP algorithm [17,18], wasadopted to train the network, and aims to minimize the msquare error (MSE), which is defined as

Ep = 1

2(y − Y)2 (7)

wherey is the model output,Y is the desired output.According to the gradient descent method [18], the weights

in the output layer are updated by the following equation:

ωi (n + 1) − ωi (n) = −η∂ Ep

∂ωi(8)

whereη is the learning-rate parameter of the weights and is apositive constant to be determined.

The update laws ofmij andσi j can also be obtained by thgradient descent method, i.e.

mij (n + 1) − mij (n) = −η∂ Ep

∂mij(9)

σi j (n + 1) − σi j (n) = −η∂ Ep

∂σi j. (10)

Here, assuming the learning-rate parameters of the mean athe standarddeviation of the Gaussian function are alsoη.

Then, substituting Eqs.(8) and(9), or (10), respectively, intoEq.(7), the following equations are obtained:

ωi (n + 1) − ωi (n) = −η(y − Y)πi (11)

mij (n + 1) − mij (n)

= −η(y − Y)ω j

n∏l=1,l �=i

ul j 2xi − mij

σ 2i j

e− (xi −mi j )2

σ2i j (12)

σi j (n + 1) − σi j (n)

= −η(y − Y)ω j

n∏l=1,l �=i

ul j 2(xi − mij )

2

σ 3i j

e− (xi −mi j )2

σ2i j . (13)

The overall process of this learning algorithm is shoin Fig. 7. Software (made by C and MATLAB programminglanguages) on the basis of this algorithm was developed forstudy.

J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161 157

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Fig. 7. Flow chart of the learning algorithm for the FNN.

4. Prediction of wind loads on the roof

The process of training the FNN using the above-mentiodata obtained from the wind tunnel tests is presented belowprediction of the wind loads on the roof.

4.1. Prediction of mean pressure coefficients

In this case, in order to determine whether the FNNsufficiently generalized to be of practical use, the mean presscoefficients for zoneA in the roof shown inFig. 5 for anincident wind direction of 292.5◦ were chosen to train the FNNand test its performance. The contours of the measured mpressure coefficients on the roof for this wind directionshown inFig. 8. ThezoneA was chosen because it is locatein the area under high negative pressure actions associatedcorner vortices. It is noted that the wind pressure distributiin this area are difficult to predict by conventional theoreticanumerical methods.

The zoneA contained 60 pressure taps. 80% of the t(those in the zone A without marked with•, as shown inFig. 5) were used in training of the FNN,and the data fromthe remaining pressure taps were used for evaluation ofprediction accuracy of the developed FNN.

During the training process, the input and output dwere thepositions and the mean pressure coefficients of thpressure taps in training, respectively. By training the FNN wthe input–output example patterns, the multivariate nonlinfunctional relationships were captured and generalized. Tgiven the values of the input variables(x, y) from the testingpressure taps (those in the zoneA markedwith •, as shown inFig. 5), the corresponding output,Cp, can be obtained. It musbe mentioned that the testing data for the FNN prediction wnot used in the training process.

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Fig. 8. Contours ofCp for a wind direction of 292.5◦ .

Fig. 9. Comparison ofCp between the experimental data and the FNprediction for a wind direction of 292.5◦ .

Fig. 9shows the comparison ofCp for the wind direction of292.5◦ between the experimental data and the FNN predicfor the testing pressure taps in the zoneA. It can be seen fromthis figure that the FNN predictions are in good agreement wthe experimental data. Actually, the average of the mean sqerror (MSE) of all the 12 testing pressure taps is less thanfor the prediction of Cp. This indicates that the prediction othe mean pressure coefficients for these taps in the area uhigh negative pressure actions associated with corner vorticesis acceptable.

4.2. Prediction of mean pressure distributions

Wind pressure distributions on large cantilevered rostructures depend on several factors such as incidwind direction and turbulence characteristics, roof shapconfigurations, building plan dimensions, and upstream teretc. Under similar flow conditions, the flow pattern arounbuilding strongly depends on wind direction and roof shaconfigurations. Accordingly, the distributions of wind loads

158 J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161

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roofs change significantly with these parameters. In the prestudy, wind direction,β, and thepositions of pressure tap(x, y) were considered as input variables in the FNN, andoutput wasCp.

Considering the symmetry of the flat roof, attention wpaid to the experimental data obtained for the wind directivaryingfrom 270◦ to 360◦. These available experimental datawere allocated into two subsets: training data and tesdata. The training data which consist of the experimeinput–output data pairs from 270◦ to 360◦ (except 315◦), wereused to train the FNN functions. The experimental data forwind direction (315◦) that werenot used in the training processwere chosen as the testing data and were used for evaluthe prediction performance. In other words, the developed FNmodel will be used to interpolate or expand the current datainto this new wind direction (315◦).

Fig. 10 shows comparison of the distribution ofCp onthe roof surface between the experimental data and the Fprediction for wind direction of 315◦.

It can be seen fromFig. 10(a) and (b) that the FNNpredictions are in good agreement with the experimentalover the whole roof surface, including the leading edge andcorner areas. Pressure distributions on such areas are difto predict by conventional theoretical or numerical methobecause of our imperfect knowledge about the structureseparating/reattaching flows and conical vortices and theisensitivity to incident flow characteristics [19]. The worstsuctions often occur in these regions due to the strong flseparation at the leading edges [20–22]. The effects of theconical vortices are clearly visible in the contour distributionCp predicted by the FNN, as shown inFig. 10(b). It is observedthat the sharp pressure gradients in the leading edge and rocorner areas were captured by the FNN.

Fig. 10(a) and (b) also reveal an important fact, i.for β = 315◦, the contour distribution ofCp should besymmetrical due to the symmetry of the flat roof, but thecontour distribution ofCp from the wind tunnel tests is not fullysymmetrical, as shown inFig. 10(a), which implies that theexperimental measurements may contain some noisy or ersignals. However, it is observed fromFig. 10(b) that the FNNpredictions tend to yield a smooth contour plot and the condistribution of Cp predicted by the FNN is symmetrical, whicindicates that the trained FNN did not memorize every dpoint, but recognized and generalized the informativemeaningful patterns [5,18]. Therefore, the performance of thtrained FNN is quite satisfactory for the prediction.

4.3. Prediction of power spectra of fluctuating wind pressufor some typical taps

Since the pressure distributions on the corner and leadedge areas are not easily predicted by conventional theoreor numerical methods, it is difficult to predict power specof fluctuating pressures in these regions by such methTherefore, it is desirable to seek an effective approach forpurpose. In this study, the FNN is applied to predict the pospectral density of fluctuating wind pressures for some typ

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Fig. 10. Contours ofCp for a wind direction of 315◦: (a) experimental data,(b) FNN prediction.

tap locations in the corner and leading edge areas to exathe predictive performance of the FNN. To the authors’ beknowledge, this is probably the first attempt to use the nenetwork method to predict power spectra of fluctuating wpressures on buildings.

For a corner pressure tap (marked with�10 in Fig. 5),wind direction,β, andfrequency,f , were considered as inpuvariables in the FNN, and theoutput was the power spectrdensity of fluctuating wind pressures which was expresse

a non-dimensional form,f ·SCp( f )

σ2 . The experimental data forthe wind direction varying from 180◦ to 360◦ were consideredherein. These available experimental data were also allocinto two subsets: training data and testing data. The traindata which consist of the experimental input–output data pfrom 180◦ to 360◦ (except 225◦, 270◦ and 315◦), were usedto train the FNN functions. The experimental data of the thrtypical wind directions (225◦, 270◦ and 315◦) that were not

J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161 159

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Fig. 11. Comparison of the power spectral density for tap�10 under threetypical wind directions: (a) 225◦; (b) 270◦; (c) 315◦.

used inthe training process were chosen as the new testand were used for evaluation of the performance of the traFNN.

Fig. 11(a)–(c) show comparison of the power spectdensity of fluctuating wind pressure for the tap marked wit�10 in Fig. 5 between the experimental data and the FN

tad

l

Fig. 12. Comparison of the power spectra density for tap�8 under a winddirection of 270◦.

prediction under three typical wind directions (225◦, 270◦ and315◦).

It can be seen fromFig. 11(a)–(c) that all the FNNpredictions are in excellent agreement with the experimentadata. It is found that the FNN has learned smaller-scfluctuations of signals and reproduced them in the simulatand the FNN predictions tend to yield a smooth contplot. This is because the FNN has the ability to captureunderlying nonlinear dynamics [5,23]. With enough neuronsin theory, the FNN can approximate any nonlinear continufunction to any desired degree of accuracy. This is one ofmost significant features of the FNN.

In addition,Fig. 11(c) demonstrates an important fact, i.for the experimental data, it is found that at higher frequerange (about 60–100 Hz), there is a small growth in specamplitudes because these experimental measurements macontain some noisy or erratic signals; but the FNN predictishow that at these higher frequencies, the amplitudepressure spectra still decrease as the frequency increases,is consistent with the observation made in Ref. [24], though thedeclines are not very obvious. This indicates that the traFNN did not memorize these noisy or erratic signals, andable to provide a satisfactory prediction.

For the leading edge areas, the power spectral densitfluctuating wind pressures for some typical taps (marked w�7–�11 inFig. 5) under a wind direction of 270◦ were chosento train the FNN and test itsperformance. The positions othese taps(x, y), and thefrequency, f , were considered asinput variables in the FNN, and the output is the power spedensity.

In this case, the power spectral density of fluctuating windpressures for the previously “unseen” taps will be predictedusing the experimental data from some adjacent taps. Fopreviously “unseen” tap�8, the four adjacent taps are�7,�9,�10 and�11. The results are shown inFig. 12. It can also beseen that the FNN predictions are in good agreement withexperimental data and the FNN predictions also tend to yiesmooth curve plot, which indicates that the FNN is sufficiengeneralized to be of practical use.

160 J.Y.Fu et al. / Engineering Structures 28 (2006) 153–161

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4.4. Prediction of cross-power spectra of fluctuating wipressures between different taps

The cross-power spectral density of fluctuating wipressures between different taps plays important rin determining wind-induced dynamic responses of lacantilevered roofs. It is thus desirable to provide supredictions. An attempt is made herein to predict the cropower spectral density under one wind direction between“seen” tap and the “missing” tap by using the cross-powspectral density data between some “seen” taps.

For a winddirection of 270◦, the “seen” taps are�7,�8,�9and�11, and the “missing” tap is�10, as depicted inFig. 5.The positions of these taps(x, y), and thefrequency,f , wereconsidered as input variables in the FNN, and the outputthe cross-power spectral density between these taps, which waalso expressed as the non-dimensional form.

The cross-power spectral density of fluctuating wipressures between the “missing” tap�10 and the “seen”taps �8 and �11 were obtained by the FNN, as shownFig. 13(a) and (b), respectively. It is clear that the two prediccross-power spectral densities are both close to the ameasurements. All the results indicate that the FNN has a gprediction performance, and thus it can be concluded thaFNN approach is robust when it is applied to predict the wloads on a large cantilevered roof.

5. Conclusions

The application of a fuzzy neural network approachthe prediction of mean pressure distributions, power speof fluctuating wind pressures for some typical taps, and crpower spectra of fluctuating wind pressures between diffetaps on a large cantilevered flat roof was observed tosuccessful. To the authors’ best knowledge, this was probabthe first attempt to use the neural network method to prepower spectra of fluctuating wind pressures on buildings.

The present results indicate that the developed FNNcapable of generalizing nonlinear functional relationshamong a number of variables such as wind-induced presapproaching wind direction, positions of pressure taps andfrequency so that it is able to predict wind loads on a lacantilevered flat roof with good accuracy for any combinationof these variables. It was observed that this approachthe potential to reduce the pressure measurement progr(approaching wind direction configurations and numberrequired pressure taps on building models) in wind tuntests. In fact, theFNN approach can alsobe used for solvingother similar wind engineering problems which involve mavariables and the relationships among these variablesunknown and complex.

Acknowledgement

The work described in this paper was supported bgrant from the Research Grant Council of Hong Kong SpeAdministrative Region, China (Project No: CityU 1266/03E)

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Fig. 13. Comparison of the cross-power spectra for a wind direction of 2◦between tap�10 and (a) tap�8; (b) tap�11.

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