This paper deals with the dynamic analysis of a 2D concrete frame with uncertainties which are an integral part of any real structure. The uncertainties can be modeled by a stochastic or a fuzzy approach. The fuzzy approach is used and the influence of uncertain input data (modulus of elasticity and density) on output data is studied. Fuzzy numbers are represented by ?-cuts. In order to reduce the volume of computation in the fuzzy approach, the response surface function concept is applied. In this way the natural frequencies and mode shapes described by fuzzy numbers are obtained. The results of fuzzy dynamic analysis can be used, e.g., in seismic design of structures based on the response spectrum.

Kruis, J.

Štemberk, P.

English

This paper deals with the dynamic analysis of a 2D concrete frame with uncertainties which are an integral part of any real structure. The uncertainties can be modeled by a stochastic or a fuzzy approach. The fuzzy approach is used and the influence of uncertain input data (modulus of elasticity and density) on output data is studied. Fuzzy numbers are represented by ?-cuts. In order to reduce the volume of computation in the fuzzy approach, the response surface function concept is applied. In this way the natural frequencies and mode shapes described by fuzzy numbers are obtained. The results of fuzzy dynamic analysis can be used, e.g., in seismic design of structures based on the response spectrum

P. Štemberk

J. Kruis

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Acta Polytechnica

Fuzzy Dynamic Analysis of a 2D Frame

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionConcrete, as a convenient building material, inherentlyinvolves uncertainty about its composition, which is difficultto eliminate completely. However, this uncertainty can be as-sessed by statistical, fuzzy, or other suitable tools. In the case ofconcrete structures, such as frames made of reinforced con-crete, it is costly to obtain a sufficient experimental data setwhich would yield the desired statistical characteristics of thematerial parameters. Instead, the knowledge gained by prac-ticing engineers can be included as fuzzy numbers in thematerial modeling.For design purposes, traditionally, we may wish to conducta statistical analysis, using the statistical characteristics of sev-eral measured events. In the case of an earthquake, however,the measured data for each site of interest is not particularlydense, leaving the statistical characteristics with little rele-vance. On the other hand, the expected seismic load at a sitecan alternatively be expressed by fuzzy sets [1], which take intoaccount the scarcity of seismic stations and information aboutlocal sub-soil composition.In this paper, an approach to dynamic analysis based onfuzzy set theory is presented as a germane alternative to classi-cal stochastic dynamic analysis. The material parameters ofreinforced concrete are considered to be fuzzy quantities witha given distribution, i.e., fuzzy numbers with a desired shapeof the membership function [2]. The dynamic analysis is,then, performed with the help of fuzzy arithmetic on either-cuts or computation-efficient (L, R) numbers [3]. The resultof such an analysis is in the form of fuzzy numbers, which isless expensive than the stochastic approach in terms of com-putation time, but still provides an idea of the distribution ofthe sought quantity. In order to further improve the computa-tional efficiency, inspired by [4], the concept of a surfaceresponse function is utilized, [5, 6]. This approach is demon-strated in an illustrative example of a 2D frame, where theeffect of uncertain material parameters transpires in the cor-responding distributions of the natural modal shapes andnatural frequencies of the analyzed two-dimensional frame. Amethodology for a possible application to seismic design isalso explained.It is believed that this approach enables practicing engi-neers and other people with knowledge to contribute activelyto analyses of seismic-sensitive structures.2 Dynamic finite element analysisThe finite element method applied to dynamical prob-lems of structures results in the formM C Kdddd22r rr f( ) ( )( ) ( )ttttt t   , (1)where M denotes the mass matrix, C stands for the dampingmatrix, K denotes the stiffness matrix, f(t) expresses the loadvector and r(t) is the vector of the nodal displacements whichare computed, t stands for time. Eq. (1) represents a semidi-screte problem where the spatial coordinates are discretizedwhile the time is still assumed to be continuous [7].The analysis of the natural frequencies (eigenvalues) andnatural mode shapes (eigenvectors) of an undamped struc-ture is based on the simplified relation Eq. (1), which has theform( )K M 0 02 v . (2)The nonzero vector v is the eigenvector containing thenatural mode shapes and 0 stands for the natural frequency.Eq. (2) represents a generalized problem of eigenvalues. Themost common method for solving of such problems is sub-space iteration [8].3 Fuzzification of dynamic finiteelement analysisThe uncertainty, that is present in the input parameterscan be tackled with the help of fuzzy set theory [1]. In thistheory, uncertain quantities are defined in terms of fuzzysets. Unlike in the classical set theory, here membership of anelement in a fuzzy set also assumes values between 0 and 1,where 0 means “does not belong” and 1 means “definitelybelongs” to a fuzzy set. Usually, fuzzy sets represent a vagueverbal evaluation. In cases when a fuzzy set represents anumeral, it is called a fuzzy number.3.1 Fuzzy numbersThe notion of a fuzzy number arises from experienceof everyday life where many phenomena which can bequantified are not characterized in terms of absolutely precisenumbers.©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 117Acta Polytechnica Vol. 44  No. 5–6/2004Fuzzy Dynamic Analysis of a 2D FrameP. Štemberk, J. KruisThis paper deals with the dynamic analysis of a 2D concrete frame with uncertainties which are an integral part of any real structure. Theuncertainties can be modeled by a stochastic or a fuzzy approach. The fuzzy approach is used and the influence of uncertain input data(modulus of elasticity and density) on output data is studied. Fuzzy numbers are represented by -cuts. In order to reduce the volume ofcomputation in the fuzzy approach, the response surface function concept is applied. In this way the natural frequencies and mode shapesdescribed by fuzzy numbers are obtained. The results of fuzzy dynamic analysis can be used, e.g., in seismic design of structures based on theresponse spectrum.Keywords: fuzzy numbers, natural frequency, mode shape, response surface function.Fuzzy numbers are fuzzy sets which are defined on the setof real numbers. Their membership function assigns the de-gree of 1 to the central, also called nominal, modal or mean,value and lower degrees to other numbers which reflect theirproximity to the central value according to the used member-ship function. The membership function should thus de-crease from 1 to 0 on both sides of the central value. Suchfuzzy sets are called fuzzy numbers. An example of a fuzzynumber is shown in Fig. 1, where  represents the member-ship function and a1 and a2 stand for two real numbers onthe real axis. The intervals defined for a specific value of themembership function, e.g.   0.7, represent the so-called-cuts. A fuzzy number can be equally expressed by either anominal value and amembership function on each side of thenominal value or by a set of -cuts.3.2 Fuzzy arithmeticA fuzzy arithmetic operation depends on the definition ofa fuzzy number. In the cases when fuzzy numbers are definedby a set of -cuts, the problem of fuzzy arithmetic is reducedto the well-known arithmetic operations on intervals, whichare applied to each -cut. Implicitly, this means a sequence ofbinary combinations on each -cut in order to obtain themin-imum and the maximum value for each -cut. The finiteelement method converts a problem into a system of linearequations, in this case a system of fuzzy linear equations,which comprises an extensive number of arithmetic opera-tions. This fact makes the formulation in the above termsmerely unsolvable due to the number of all necessary binaryoperations.To eliminate the drawback of the -cut formulation, newtechniques for solving fuzzy linear equation systems havebeen developed, e.g. [9]. However, these techniques are noteasily applicable to robust problems, such as fuzzy dynamicfinite element analysis. Therefore, another technique for re-ducing the large number of binary combinations has beenexploited.This technique was originally developed for otherproblems, such as statistical analysis.3.3 Surface response functionFuzzy analyses, as well as stochastic analyses, suffer fromnon-occurrence of analytical solutions in the case of non-de-terministic input data. This situation can be remedied by thefollowing. Let~ ~x X denote the vector of input data from thespace of input data ~X and~ ~y Y denote the vector of outputdata from the space of output data ~Y . Both, stochastic andfuzzy analyses require knowledge of the response, which canbe written in the form~ (~)y x F , (3)where F denotes the response of a system (structure) to theinput data collected in vector ~x . This represents a mappingfrom the space ~X to the space ~Y . The non-occurrence of ananalytical solution requires the application of a suitable nu-merical method which discretizes the problem and solves itnumerically. The space ~X is discretized by an n-dimensionalspace, X, and similarly the space ~Y by an m-dimensionalspace, Y. A stochastic analysis based on simulation methodsgenerates thousands or millions of samples of input data (thevectors x) and then deterministic computation follows. Fuzzyanalysis based on -cuts requires computation of all combina-tions of input data, which also leads to thousands or millionsof samples. Both approaches yield the response of a systembased on a huge amount of output data (thousands or mil-lions of vectors y) obtained from many executions of stan-dard (deterministic or crisp) solutions.In order to reduce the necessary number of computationruns, the concept of a response surface function has been usedmany times. The basic idea of the response function is to ap-proximate operator F by a suitable function which should beas simple as possible. The function for the k-th output param-eter can be written in the formf x a b x c x xk k iki ijki jjninin( ) ( ) ( ) ( )( )   111, (4)where the superscript identifies an output parameter and ndenotes the dimension of the space of the input data, X. Theunknown coefficients are obtained from the least squaremethod in the following way. Let the set of input parameterscontain s samples. Each sample is located in the vector x [i],where the superscript identifies a sample. The standard com-putation gives the output data, which are collected in the vec-tors y [i]. The coefficients of the response function minimizethe following expression F a b c f yk k i k ijk k i kiis( ) ( ) ( ) ( ) ( ) [ ] [ ]( , , ) ( )  x21. (5)In many cases, it is not necessary to use quadratic terms.Considering only the linear terms simplifies furthercomputations.4 Numerical exampleAs an example, the natural frequency analysis of a two-di-mensional frame with four floors made of reinforced concreteis considered. The overall height of the frame is 16 metersand the width is 5+ 5 meters. The dimensions of the beamsand columns are identical (0.5×0.5 m). It is assumed that thebuilding was erected in four consecutive lifts. Each lift consistsof placing concrete in three columns and in the beam whichconnects the upper ends of the columns.118 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 5–6/2004...Fig. 1: A normal fuzzy number and its -cutsTherefore, it is further assumed that there are only fourtypes of concrete whose composition can possibly differ. Theinfluencing material parameters are the modulus of elasticity,E and the density, . E and  are fuzzy input parameters withnominal values of 30 GPa and 2500 kg/m3, respectively, whichcan change by 10 % and with a linear membership function(triangular fuzzy numbers).©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 119Acta Polytechnica Vol. 44  No. 5–6/2004Fig. 2: Mode shapeFig. 3: Enlarged section of mode shape 1Fig. 4: Mode shape 2Fig. 5: Mode shape 3Fig. 6: Mode shape 4Fig. 7: Mode shape 5For illustrative purposes, we need 125 response surfacefunctions to describe the first five natural vibrationmodes, i.e.a response surface function to express each natural frequencyand the horizontal and vertical displacements in each joint(three joints on each of the four floors) for each natural modeshape. In order to obtain sufficient input and output data forcalculation of the coefficients of the response surface func-tions it was decided to take three values (minimum, modalvalue, maximum) for each material parameter, E and , i.e.,(  6561) independent runs of the dynamic finite elementanalysis. The specific form of Eq. (4) in this example wasf x b E b E b E b Eb bk k k k kk( ) ( ) ( ) ( ) ( )( )( )      1 1 2 2 3 3 4 45 1 6 2 7 3 8 4 9( ) ( ) ( ) ( ).k k k kb b b    (6)The first five mode shapes are shown in Figs. 2 to 7, wherethe dotted lines represent all possible envelopes of response,in other words, the minimum and maximum values, whichcorrespond to the values obtained for -cuts (  0). Thefinite element model of this frame discretized each frame sec-tion (beam and column) by five beam elements, however, onlythe joint displacements are shown. This is why no significantdifference is distinguishable in Figs. 6 and 7. In Fig. 3, asection of the frame is enlarged and the vertical displace-ments are 1000 times increased compared to the horizontaldisplacements so that we can see the distribution of possibledisplacements of the frame. The distribution of the first fivenatural frequencies is shown in Fig. 8. It was observed that theresponse function gives very good results for the dominantdisplacements (at point A, which is the top left joint) in thelower natural mode shapes, which are important for seismicdesign. For vertical displacements, which do not play animportant role in seismic design (at point B, which is theintermediate joint of the first floor), the response functioncould not fit the proper shape of the membership function,which is evidenced in Figs. 9 and 10.5 Possible applicationsIn the design of earthquake resistant structures, it is essen-tial not to neglect any uncertainty as it may lead to an errone-ous conclusion due to the dynamic simulation which mayamplify such uncertainty beyond all limits. For these reasons,it seems reasonable to express uncertain numerical data interms of fuzzy numbers and to use them as such in analyses tocover all possible solutions.In the previous section, an approach to natural vibrationanalysis was shownwhich provides input data for further anal-yses considering, e.g. earthquake induced vibrations. Spec-tral-analysis based methods require only the maximum valuesobtained for each natural mode in order to evaluate theexcited vibration. Therefore, it is desirable to verify whetherthese values can be satisfactorily expressed by surface re-sponse functions which were obtained only by binary combi-nations of material parameters with three values (minimum,modal value, maximum). The resulting surface responsefunction was also obtained for five values, corresponding tothe -cut values with a equal to 0, 0.5 and 1. However, thismeant 52×4 ( 390625) independent runs of the dynamicfinite element analysis. The improvement was negligible, andcompared with the computational effort it proved trulyunnecessary.120 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 5–6/2004Fig. 8: Distribution of natural frequenciesFig. 9: Distribution of horizontal displacementFig. 10: Distribution of vertical displacement6 ConclusionsThe fuzzy approach has been applied to dynamic analysisof a 2D concrete frame with uncertainties. The natural fre-quencies and the natural mode shapes have been computedand described by fuzzy numbers. The response surface func-tion has been applied in order to reduce the number of re-quired computations. The results have been compared with afull analysis based on an evaluation of all combinations (in theorder of thousands) and very good accordance has been ob-tained. The first lower natural mode shapes are naturally de-scribedmore precisely than the higher ones. The errors of theobtained results for lower natural mode shapes are less than5%. There are some quantities belonging to higher modeswhere the response surface function gives results unaccept-able from the point of the fuzzy set theory. These difficultiesshould be studied in future.7 AcknowledgmentThis work was supported by project No. 103/04/1320 ofthe Grant Agency of the Czech Republic, which is gratefullyacknowledged.References[1] Zadeh L. A.: “Fuzzy sets.” Information Control, Vol. 8(1965) No. 3, p. 338–352.[2] Valliappan S., Pham T. D.: “Construction the member-ship function of a fuzzy set with objective and subjectiveinformation.” Microcomputers in Civil Engineering, Vol. 8(1993), p. 75–82.[3] Kaufman A., Gupta M. M.: “Introduction to FuzzyArithmetic: Theory and Application.” New York, VanNostrand Reinhold Company, Inc.1985.[4] Akpan U. O., Koko T. S., Orisamolu I. R., Gallant B. K.:“Practical fuzzy finite element analysis of structures.”Finite Elements in Analysis and Design, Vol. 38 (2001),p. 93–111.[5] Bucher C. G., Chen Y. M., Schueler G. I.: “Time VariantReliability Analysis Utilizing Response Surface Ap-proach.” In: Reliability and Optimization of StructuralSystems , 88. (Editor P. Thoft-Christensen), 1988.[6] Rajashekhar M. R., Ellingwood B. R.: “A new look at theresponse surface approach for reliability analysis.” Struc-tural Safety, Vol. 12 (1993), p. 205–220.[7] Bathe K. J.: “Finite element procedures.” Prentice-Hall,Inc. 1996.[8] Bittnar Z., Šejnoha J.: “Numerical Methods in StructuralMechanics.” ASCE Press 1996.[9] Buckley J. J., Qu Y.: “Solving systems of linear fuzzyequations.” Fuzzy Sets and Systems, Vol. 39 (1991),p. 33–43.Ing. Petr Štemberk, Ph.D.phone: +420 224 354 634Fax: +420 233 335 797e-mail: stemberk@fsv.cvut.czDepartment of Concrete StructuresIng. Jaroslav Kruis, Ph.D.phone: +420 224 354 369Fax: +420 224 310 775e-mail: jk@cml.fsv.cvut.czDepartment of Structural MechanicsCzech Technical University in PragueFaculty of Civil EngineeringThákurova 7166 29 Praha 6, Czech Republic©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 121Acta Polytechnica Vol. 44  No. 5–6/2004

Fuzzy Dynamic Analysis of a 2D Frame

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