068 042 Kala Omishore-Fuzzy-Full

8/12/2019 068 042 Kala Omishore-Fuzzy-Full

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FUZZY AND STOCHASTIC THEORIES AND THEIR APPLICATIONS IN

STRUCTURAL ENGINEERING

Z. Kala & A. Omishore, Brno University of Technology, Faculty of Civil Engineering, CZ

ABSTRACT

The analysis of the influence of stochastic and fuzzy uncertainties of imperfections on the stochastic andfuzzy uncertainties of the load-carrying capacity of a steel plane frame is presented. On a simple exampleof a steel plane frame, the analyses of the possibilities of system imperfections implementation asrandom variables are performed. The influence of uncertainty of random characteristics of the correlationcoefficient between the random initial inclinations of the frame columns on the load-carrying capacity ofthe frame is studied. In the next stage, the system imperfections are considered as fuzzy numbers. Thefuzzy load-carrying capacity was determined as a fuzzy number, utilizing the general extension principle.The comparison of fuzzy and stochastic load-carrying capacity with the design standard load-carryingcapacity acc. to EUROCODE 3 is performed in the final stage.

1. INTRODUCTION

Each engineering structure is to a certain degreeatypical. The correct estimation of randomvariables is possible only for exactly determinedmass events provided that we have at our disposalan adequate number of experimental results. Thisis quite accurately fulfilled for geometric andmaterial characteristic of mass-productionhot-rolled steel profiles [4, 6].These apparently credible data are not complete inthe event that they are to be used as input randomvariables in stochastic models of complex, hard todescribe and hard to measure systems. Detailedstatistical information would require, for e.g.knowledge of correlation matrices, parameters ofrandom fields [2]. Some variables cannot bemeasured at all (e.g. system imperfections), or themeasurement is burdened with high statisticalerror (e.g. residual stress). The quality of models is,in these cases, given above all by the manner inwhich the applied methodology compensates with

the formalization and effective utilization ofuncertainty, which is characteristic in thedescription of such systems.Let us consider the frame illustrated in Fig. 1 theload-carrying capacity of which is significantlyinfluenced by initial imperfections. Theimperfections will be considered as both randomvariables and fuzzy numbers in the presentedarticle. The random uncertainty is analysedutilizing the stochastic simulation methods LatinHypercube Sampling (LHS). The fuzzy

uncertainty will be analysed utilizing the so-calledextension principle. The output variables are

random and fuzzy load-carrying capacity. Theapplication limits and restrictions of mathematicalstatistics in the formalization of uncertainty arediscussed. The fuzzy and stochastic analyses arecompared with the load-carrying capacityevaluated according to the EUROCODE 3:stability solution with buckling length and thegeometric non-linear solution with initialimperfections.

Figure 1. Geometry of steel plane frame

Figure 2. System imperfection acc. to [9]

3rdInternational ASRANet Colloquium

10 12thJuly 2006, Glasgow, UK.


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2. SYSTEM IMPERFECTIONS

2.1 RANDOM SYSTEM IMPERFECTION

In view of the fact that we have inadequate

number of statistically exploitable data for systemimperfections e1 and e2 at our disposal, we shallassume that the mean value me1= me2= 0(perfectly vertical column). We shall also requirethat 95% of the realizations of random variable eaare found within the tolerance interval + h/500,(Fig. 2) and that of random variable eb are foundwithin the tolerance interval of + h/100 (Fig. 3).These tolerance limits are determined by thetolerance standard [9].

Figure 3. Tolerance system limits acc. to [9]

The evaluation of this problem is ambiguousbecause probability density functions of variablese1 and e2 and their cross-correlation k12 are notknown. This problem can be heuristically analysedas a parametric study (sometimes called the what-if-study) for the correlation k121; +1 by theMonte Carlo (MC) method.One million random realizations of variablese1and e2 of the Gaussian distribution of meanvalue 0, standard deviation 1 and correlation k12were generated by the MC method in the first step.Obtained data file was divided into two. The firstfile contains the random realization of the paire1and e2 with the same sign (shape-wise acc. toFig. 2) and the second file contains the remainingrealizations with opposite sign (shape-wise acc. toFig. 3). Subsequently, the standard deviation for

both files was designated (multiples of realizationse1, e2) in such a manner that 95 % of therealizations of valid observations of eawere withinthe tolerance interval of + h/500 and 95 % of

realizations of ebwere within the tolerance limitsof + h/100. The iterative analysis of the problemrepetitive analysis utilizing the MC and theinterval-halving methods - was necessary. Afterobtaining the standard deviations of the first and

the second file, the union of both files wasperformed and variables e1 and e2 werestatistically evaluated. The histograms obtainedfor 1 million runs were approximated bycontinuous probability density functions, seeFig. 4 and 5.

Figure 4. Tolerance system limits

Figure 5. Tolerance system limits

Multiples of the realizations of variables of theseparated files (from the MC solution) independence on the correlation k12 for columnheight of 1 m are illustrated in Fig. 6.In practice the stochastic modelling ofimperfections is performed in this the followingmanner. By means of the MC method, therealizations of e1and e2with Gaussian distribution


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of mean value 0, standard deviation 1 and withselected correlation k12 is simulated. Therealization of variables e1> 0 and e2> 0, or e1< 0and e2< 0 is multiplied by the value obtained fromthe solid line in Fig. 6. The realization of variables

e1> 0 and e2> 0, or e1< 0 and e2< 0 is multipliedby the value obtained from the broken line inFig. 6. The obtained values correspond to acolumn with height of 1 m, for other height valuesit is necessary to multiply the obtained values witha number that is equal to the column height inmeters. Imperfections implemented in such amanner satisfy the requirements of the tolerancestandard [9], i.e. 95 % of realizations of eaand ebare found within the tolerance limits acc. to Fig. 2and 3. The behaviour of the standard deviation and

skewness of variables e1and e2 in dependence onthe correlation k12is illustrated in Fig. 7.

Figure 6. Parameters of system imperfections

Figure 7. Standard deviation &skewness ofe1, e2

The solution in Fig. 6 and 7 is heuristic. There isinadequate information in [9] for the generalsolution. This means that the probability densityfunctions (histograms) of the probabilisticvariables e1 and e2 and their cross-correlation k12

are unknown. Input random variables cannot bereliably deduced purely from the informationprovided by the tolerance standards [9].The random realization of system imperfectionscannot be obtained in practice from measurementson a higher number of structures. We generallyhave only imprecise information, e.g. fromtolerance standards or from a small amount ofmeasurements, the evaluation of which is

burdened with high statistical error.

2.2 FUZZY SYSTEM IMPERFECTION

The problem is relatively easy-to-solve providedthat we consider the system imperfections as fuzzynumbers, the supports of which are determined bythe tolerance limits. For symmetrically loadedframes with symmetric boundary conditions it isnecessary to consider three basic fuzzy sets ofsystem imperfections, see Fig. 2, 3, and 8. Theload-carrying capacity of a frame withimperfections acc. to Fig. 3 is higher than that of aframe without imperfections. On the contrary, theload-carrying capacity of a frame withimperfections acc. to Fig. 8 is lower than that of aframe without imperfections. It is thereforenecessary to divide imperfections in differentdirections into the case of inclination of columntop ends (difference in rotation) towards eachother (see Fig. 3) and away from each other(see Fig. 8).

Figure 8. Tolerance system limits acc. to [9]


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Since the frame is symmetric it suffices toconsider three sets of system imperfections. Onthe contrary it would also be necessary to dividethe set A into two sets.The membership function is determined for each

set according to the tolerance limits; see Fig. 9, 10,11. Due to the fact that we do not have additionalrelevant and exploitable information at ourdisposal, the membership functions wereconsidered as triangular.

Figure 9. Membership function acc. to [9]



3. NONLINEAR COMPUTATIONAL

MODEL

Member geometries may be modelled by means ofa beam element with initial curvature in the form

of a parabola of the 3rd

degree [1]. The structurewas meshed into 11 beam elements (Fig. 12).

Figure 12. Meshing of beam elements

The steel frame was solved utilizing the nonlinearEuler incremental method in combination with the

Newton-Raphson method. Geometrical and

material nonlinearities were considered.The first criterion for the load-carrying capacity isgiven by the loading at which plasticization of theflange is initiated. The second criterion for theload-carrying capacity is represented by a loadingcorresponding to a decrease of the determinant tozero (stability condition).These criteria present only one variant of thedefinition of the ultimate limit state; a moregeneral approach is presented upon reaching theallowable strain value [3].

The ultimate one-parametric loading is defined asthe lowest value of load-carrying capacity. This

phenomenon occurs at high yield point values withsmall geometrical member imperfections. In eachrun of the simulation method, the load-carryingcapacity was determined with an accuracy of0.1 %. The load-carrying capacity was evaluatedfor the basic element material only [7].The first author (Kala) of this article is thecompiler of the utilized software programme. Thesoftware programme was compiled in the

programming language Pascal. The programmewas compiled within the framework of scientificresearches and is not intended for commercial use.


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4. LOAD-CARRYING CAPACITY

4.1 LOAD-CARRYING CAPACITY ASRANDOM VARIABLES

Ten random variables were considered in thestochastic analysis, see Tab. 1. According tosensitivity analysis results [1] the dominantvariables are those of the flange thickness t2,Youngs ModulusE, yield strengthfy(steel S235)and of the imperfections e1, e2. Statisticcharacteristics were considered according to theexperimental results of material and geometriccharacteristics of structural steel from a dominantCzech manufacturer [4].

Table 1: Input random variables

No. Member Symbol Mean value Std. deviation1. t2

* 5.6601 mm 0.26106 mm2. E * 210 GPa 12.6 GPa3. fy

*297.3 MPa 16.8 MPa4.

Left

Column

e1 Chap. 2.1 for { }1;0;112 k5. t2

*9.7314 mm 0.44884 mm6.

BeamE * 210 GPa 12.6 GPa

7. t2* 5.6601 mm 0.26106 mm

8. E * 210 GPa 12.6 GPa9. fy

*297.3 MPa 16.8 MPa

10.

Right

Column

e2 Chap. 2.1 for { }1;0;112 k* Gauss distribution

Load-carrying capacity for k12 {-1; 0; 1} isillustrated in Fig. 3. 300 runs of the LHS methodwere performed. The design of the utilizedsoftware programme is depicted in Fig. 14.

Figure 13: Hermite density plots

Three files of random load-carrying capacity wereapproximated by the Hermite four parametric(mean, std. dev., skewness and kurtosis)

distribution utilizing the Statrel programme.Results in Fig. 13 represent the three mostimportant representatives from the set of all resultsof set k12-1; 1.This simple example demonstrates the general

drawbacks of present-day probabilistic methods.In the event that we do not have at our disposalsufficient valid observations of input randomvariables, the input data are vaguely (fuzzy)uncertain. The problem of insufficient validobservations is encountered in a number ofengineering applications. Additional observationsare due to economic or technical reasonsunavailable. If we do not have at our disposal anadequate number of observations at reproducibleconditions, the variability of natural processes

cannot be eliminated by satisfying therequirements of reproducibility by formulating anexcess of general conditions. We would obtainresults the dispersion of which could be so highthat it could completely devalue obtained results.If we do not have accurate information at ourdisposal, the model presents the source of vague(fuzzy) uncertainty, which could significantlydominate over the stochastic uncertainty incomplex systems.The correlation k12is a typical fuzzy characteristic.A possibility of the evaluation of this problem is

presented through the utilization of so-calledfuzzy-random variables. A more detailedmathematical description of fuzzy randomfunctions is contained e.g. in [5].

Figure 14: Design of my own program (in Czech)

Input realizations by LHS methods were evaluated

utilizing the programme (Fig. 14) compiled by thefirst author (Kala) of this article. The programmewas compiled in the Delphi interface.


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4.2 LOAD-CARRYING CAPACITY ASFUZZY VARIABLES

The fuzzy analysis was evaluated according to thegeneral extension principle utilizing the so-called

response function. Basic fuzzy arithmetic(addition, subtraction, multiplication, division) canthen be performed utilizing this function. Let bean arithmetic operation (e.g. addition, division)and Z1,Z2 R be fuzzy numbers. The extension

principle then allows the extension of operation to the operation with fuzzy numbers in thefollowing manner:

(1)

The result of operation is a fuzzy number

Z1 Z2, consisting of elements z =xy with amembership function that is given by theminimum of membership values of operators xinto fuzzy numberZ1andyinto fuzzy numberZ2.The extension principle in the form of -cuts wasused for the analysis.In the first step only the system imperfections withmembership functions (Fig. 9, 10, 11) were

considered as input fuzzy numbers. Othervariables were considered as singletons of theircharacteristic values (fy = 235 MPa, E = 210 GPa,etc.). The fuzzy load-carrying capacities obtainedare shown in Figs. 15, 16, 17.

Figure 15: Load-carrying capacity fuzzy set A

Figure 16: Load-carrying capacity fuzzy set B

Figure 17: Load-carrying capacity fuzzy set C

The resulting load-carrying capacity obtainedthrough the union of the three fuzzy sets isdepicted in Fig. 18. The solution is easy, clear andcovers all the possible combinations of initialsystem imperfection implementation.

Figure 18: Load-carrying capacity - Sets CBA


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4.3 VAGUENESS OF REQUIREMENTS OFEUROCODE 3

EUROCODE 3 [8] lists, for the analysis of theload-carrying capacity of systems with

compressed columns, two basic conventionalmethods: the stability solution with bucklinglength and the geometric non-linear solution.Furthermore cases when the combination of theaforementioned methods and also simplified

procedures according to the 1storder theory can beutilized are listed.The estimation of internal forces and memberdimensions by the geometric non-linear solution islimited by the normative estimation of initialgeometric imperfections for the solved structure.

Although EUROCODE 3 [8] lists the initialgeometric imperfections for basic frame types, theimperfections for atypical structures are not listed.This limits the practical applicability of thegeometric non-linear solution, which iscomputationally more accurate but suffers from aninadequacy of input data.On the contrary, the stability solution is moregeneral and provides solution for all types ofstructures frequently occurring in buildings. The

problem of the influence of axial forces on thestress state of bars in structural systems issimplified to the estimation of one variable of the

buckling length. This approach is sufficientlytransparent; however in the case of higher numberof load cases, performing the stability solution foreach combination (sometimes of the order of athousand combinations) is due to the time factor

practically unmanageable. This then leads tosimplifications in which the buckling length isimplemented by one value for all the combinations.

The buckling length is a typical vague (fuzzy)characteristic that cannot be measured andstatistically evaluated. It therefore is not a randomuncertainty but uncertainty due to the vaguerequirements of standards and human activity.The basic combinations of member and systemimperfections acc. to [8] are illustrated in Fig. 19(imperfections of cross-beam were neglected).The vagueness on the requirements of the standard[8] rests on the fact that it does not strictlydesignate what combination of initial

imperfections to consider. The uncertainty of thestability solution of more complex systems isgiven by the uncertainty of the buckling length.

Figure 19: System and members imperfections

Due to economic reasons, the designer is oftenforced to design structures of very low weight.

The weight of structures designed according tovalid standards can be minimized utilizingknowledge on the vagueness of standardregulations and computational models, whereasthe final effects could be very substantial evenfrom the point of view of loss of human life, seerecent collapse of roofs under the weight of snowat the following locations: Exhibition ground inWarsaw, emporium in Moscow, hall in Humpolec(in Czech Republic) and the winter stadium inGermany.

4.4 COMPARISON OF FUZZY,STOCHASTIC AND EUROCODE 3ANALYSES

The comparison of the load-carrying capacitiesgiven as fuzzy numbers, random variables anddeterministic according to the standard is difficult.Each method is based on a different theory and

processes qualitatively different information andhence, it has different predicative capabilities.

Let us compare the results of stochastic analysis inFig. 13 with the results of fuzzy analysis. Thefuzzy numbers of input imperfections are definedformatively identical with the Gaussiandistribution with parameters from Tab. 1 for this

purpose. The core of each fuzzy set is identical toits mean value and the degree of membership of 1is allocated to it. The fuzzy number of load-carrying capacity can be evaluated according tothe extension principle utilizing the geometricnon-linear solution described in Chapter 3. Thecomparison of the stochastic, fuzzy anddeterministic solution EC3 [8] is shown in Fig. 20.


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Figure 20: Comparison of solutionsThe design standard values of the load-carryingcapacity were evaluated acc. to [8] provided thatthe yield strength was reached in the most stressedsection of the frame. All the combinations fromFig. 19 were considered in the evaluation of theminimum and maximum of the geometric non-linear solution.

5. CONCLUSIONS

The differences between individual solutions areapparent from Fig. 20. Differences between designload-carrying capacities evaluated from thegeometric non-linear and stability solution ofEUROCODE 3 [8] are very alarming. The basic

probabilistic background of Eurocode is inEN 1990. If the design probability of failure is7.2E-5 (d= 3.8) then the design load-carryingcapacity evaluated according to EN 1990 should

be approximately equal to 0.1 percentile, whichevidently is not fulfilled. The quantification ofthese differences would require a more detailedstudy inclusive of the random load effects.Three density functions advert to the differencesdue to unfamiliarity of the correlation coefficientk12. We can assume that other variants ofcorrelation implementation among the inputvariables in Tab. 1 would lead to even greaterdifferences among the probability densityfunctions, i.e. to the broadening of the grey regionin Fig. 20. The credibility of the stochastic

solution is dependent on the validity of inputrandom variables and their correlation.Supplementary observations are often due totechnical or economic reasons unavailable.

ACKNOWLEDGEMENTS

This research was supported by grantKJB201720602 AVR, research center project1M68407700001 and GACzR 103/05/0417.

REFERENCES

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068 042 Kala Omishore-Fuzzy-Full