FORMING A CORRELATION AMONG THE PROPERTIES OF FRESH SELF- COMPACTING CONCRETE USING FUZZY LEAST SQUARES MODEL

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1 Number 1, Year 016 Page FORMING A CORRELATION AMONG THE PROPERTIES OF FRESH SELF- COMPACTING CONCRETE USING FUZZY LEAST SQUARES MODEL Aleksadar Savić Uiversity of Belgrade, Faculty of Civil Egieerig, Assistat Professor Correspodig author: sasha@imk.grf.bg.ac.rs Maria Aškrabić Uiversity of Belgrade, Faculty of Civil Egieerig, Teachig Assistat Abstract: The research preseted i this paper was motivated by two problems regardig the properties of fresh self-compactig cocrete (SCC). The first problem is that mixture properties ca vary owig to differet factors (compositio, materials, ambiet, etc.) ad the secod problem is that there is usually a restricted group of experimetal results. Because of the importace of the properties of fresh SCC mixtures, differet tests for determiig its passig ad fillig ability, as well as segregatio resistace, have bee developed. I this paper, results from V-fuel ad slump flow tests coducted o seve differet cocrete mixtures are preseted. As a correlatio betwee the measuremets of V-fuel period t v ad the slump flow period t 500 is usually strog, i this paper, it was expaded usig approximate-distace fuzzy least squares method. The goal was to perceive the possible rage of these properties ad, whe applicable, to reduce the umber of tests ecessary durig the productio of SCC. Keywords: cocrete fresh state properties, self-compactig cocrete; fuzzy least square model FORMIRANJE VEZE IZMEĐU SVOJSTAVA SVJEŽEG SAMOZBIJAJUĆEG BETONA POMOĆU NEIZRAZITE METODE NAJMANJIH KVADRATA Sažetak: Istraživae prikazao u ovom radu povezao e s dva problema koi se avlau pri ispitivau svostava svežih samozbiaućih (SCC) betoa. Prvi problem e u vezi s čieicom da svostva mešavie mogu varirati zbog različitih čimbeika (sastav, materiali, uveti okolie itd.), dok e drugi problem mali bro eksperimetalih rezultata ispitivaa po mešavii. U sklopu ispitivaa svostava svežih SCC mešavia, osmišlei su različiti testovi sposobosti prolaska i ispuavaa oplate, kao i otporosti a segregaciu. U ovome radu prikazae su vriedosti rezultata ispitivaa metodama V-lievka i sliegaa razlievaem, izmeree a sedam različitih SCC mešavia. Uobičaeo dobra korelacia između izmereih veličia proširea e korišteem eizrazite metode amaih kvadrata. Svrha prikazaog postupka e spozavae mogućih graičih vriedosti spomeutih svostava SCC betoa i smaivae broa potrebih testova, kada e to moguće, pri proizvodi SCC betoa. Kluče rieči: svostva betoa u svežem stau; samozbiaući beto; eizrazita metoda amaiih kvadrata 66

2 Number 1, Year 016 Page INTRODUCTION Self-compactig cocrete (SCC) was itroduced i Japa i the mid-80s for the first time. SCC was a step forward i cocrete techology compared to ormally vibrated cocrete owig to its fresh state properties. This type of cocrete fills every part of the formwork ad surrouds re-bars solely by meas of its ow weight [1]. I order to produce SCC, coditios related to properties of fresh cocrete (fillig ad passig ability as well as segregatio resistace) have to be satisfied. Differet testig methods have bee itroduced i order to determie the properties of fresh SCC. I this paper, the results from two widely used testig methods, both showig the fillig ability of SCC, were preseted: slump flow ad V-fuel test. It should be oted that the properties of a fresh cocrete mixture ca vary because of differet factors (heterogeeity of compoet materials, temperature, time ad mixig method, the perso that is coductig the test, etc.). Nevertheless, durig cocrete productio, there is usually a restricted group of experimetal results cocerig these properties (1 results per mixture). Fuzzy set theory, proposed by L. Zadeh aroud 1965, represets the basis of the theory of possibility []. Fuzzy sets are those sets whose members have degrees of membership. While modelig a problem, oe is ofte forced to defie a variable i precise mathematical terms although that is ot its ature i reality. This happes i experimetal work too, whe beig forced to estimate the value of a variable, based o limited iput such as a restricted umber of samples. It is importat to differetiate the terms possibility ad probability. It is ofte the case that exteral circumstaces, i.e., those that make somethig possible, have a small probability [3]. For example, the problem i questio (the measuremets of properties of fresh SCC) ca be affected by may factors that do ot represet probabilistic methods, such as the experiece of the coductor of the experimet, his or her mood, cocetratio, axiety, iteractio with other people at the momet of measurig, ad time scales. The most commo approach to fuzzy data processig is either the adaptatio of existig algorithms by the extesio priciple, or the usage of fuzzy arithmetic o models already developed from crisp (determiistic) data. This study used the secod approach for creatig fuzzy least squares models that ca easily be applied to experimetal data aalysis [4]. I this study, the correlatio betwee measuremets of the V-fuel period t v ad the slump flow period t 500 was exteded usig the approximate distace fuzzy least squares method. METHODS.1 The method of least squares The least squares method is widely used i formig the best fittig correlatios betwee oe depedet ad oe idepedet variable. I liear regressio aalysis, after determiig the existece of a good liear correlatio (straight lie) betwee two observed parameters x ad y, the mea value of y is assumed to be depedet o the mea value of x: 0 y 1 x I other words, the depedet variable y ca be described as the sum of its mea value ad relative mistakes oted as E: y x, () 0 1 E o coditio that the mea value of E equals 0 (M(E) = 0). Coefficiets β 1 ad β are calculated from the sample ad oted as b 1 ad b 0, by miimizig the sum of the squares of distaces (d) betwee the measured ad the fittig values of y [5]: mi ( d ) mi ( y ( x)) (3) i i 0 1 Therefore, the fial explicit forms of the equatios for b 1 ad b 0 are give by (4) ad (5): (1) 67

3 Number 1, Year 016 Page b 1 xi yi ( xi )( i 1 i 1 i 1 xi ( xi ) i 1 i 1 b0 y b1 x y ) i (4) (5). Fuzzy set theory A fuzzy set А i a uiverse of discourse X is characterized by a membership fuctio μ A (x), which associates a real umber i the iterval [0,1] with each elemet x i X. The value of the fuctio μ A(x) is called the grade of membership of x i A []. If L ad R are eve, o icreasig real fuctios o [0, ), satisfyig L(0) = R(0) = 1, fuzzy umbers ca take the form: μ A (x) = { L (m x), μ m, a > 0, a R ( x m b ), μ m, b > 0. (6) This is a widely used represetatio of fuzzy umbers called a LR represetatio ad deoted by A = (m,a,b) LR, where m is a modal value of A, ad a ad b are called left ad right spread, respectively [6]. This presetatio is based o left (L) ad right (R) fuctios that defie the fuzzy umber. Aother otatio used i this paper is A = (α,β,δ), where β= m, α= m a ad δ= m + b. Triagular fuzzy umber A = ( α,β,δ)t, i the simplest fuzzy form, is defied by the followig L ad R fuctios: μ A (x) = L = x α β α 0, x α, α < x β R = δ x, β < x δ δ β { 0, x > δ. Workig with fuzzy umbers is possible by usig stadard mathematical operatios. If A(a,b,c) ad B(d,e,f) are two triagular fuzzy umbers, defied by the three poits explaied above, their sum, product, ad quotiet, respectively, would be equal to [7]: A + B = (a + d, b + e, c + f) (8) A B = (mi(a b, a f, c d, c f), (b d), max(a b, a f, c d, c f)) (9) A / B = (mi(a/b, a/f, c/d, c/f), (b/d), max(a/b, a/f, c/d, c/f)) (10).3 Fuzzy least square method Fuzzy regressio aalysis is used i the evaluatio of the fuctioal relatioship betwee the two variables i a fuzzy eviromet [8]. May models have bee developed i order to apply the method of least squares for the correlatio of two variables that ca be crisp or fuzzy, formig liear fuctio whose coefficiets ca also be crisp or fuzzy, depedig o the model. This paper describes the approximate-distace fuzzy least squares method, which was preseted by Dubois ad Prade [8]. If X = (,, ) is the fuzzy iput, A = (,, ), i = 0,, k the fuzzy x x x parameters ad Y = (,, ) is the fuzzy output the the followig correlatio has bee adopted: Y = A 0 + A 1 X k, = 1,, (11) All the umbers preseted are of triagular LR type. The difficulty i treatig a model of fuzzy iput-output data is that the product A i X i may ot be of LR type. This is why the suggested approximatio is used. Whe applied to A A X (,, ), the result is the LR fuzzy umber Y = (,, ), where: 0 1 = (1) a0 + x i ai ai ai (7) 68

4 Number 1, Year 016 Page = (13) a0 + a0 + x x = (14) It is easy to calculate the distace betwee two fuzzy umbers Y m = (α y, β y, δ y) (based o measured values) ad Y (fuzzy output) as follows: d ) ( ) ( ) (15) ( yi yi yi The the sum of squares of distaces U is calculated by [9]: N 1 U ( ) ( ) ( ) 1 y yi yi (16) 3 Ukow parameters A 0(α a0, β a0, δ a0) ad A 1(α, β, δ ) are calculated from six liear equatios formed by miimizig the value of U (16), usig coditio that the first derivative of a fuctio equals 0 at extreme poits: U N 0 for N,,,,, ) (17) ( a0 a0 a0 3 MATERIALS AND EXPERIMENTAL METHODS The experimetal program cosisted of measurig the fresh cocrete properties of seve differet SCC mixtures through V-fuel ad slump flow tests [10, 11] (Figure 1). The mixtures were divided ito two groups: two referet mixtures (E1 ad E), ad five mixtures (LP1, LP, LP3, LP4, ad LP5) cotaiig fly ash as a partial replacemet of mieral filler (limestoe powder). Slump flow test procedure cosists of the followig steps: Abrams coe is placed o the flat surface ad filled with fresh cocrete. After 30 s, the coe is carefully lifted, ad the timig is started at the same momet. The stopwatch is stopped whe the cocrete sample reaches a diameter of 500 mm (lower base diameter of Abrams coe is 00 mm). This period is oted as t 500. Whe the cocrete sample reaches the fial diameter ad stops movig, its diameter is measured. Figure 1 Dispositio of the slump flow test (left) ad V-fuel test (right) [1] The V-fuel test is coducted with the use of the equipmet show o the right i Figure 1. A V-fuel of kow dimesios ad a higed gate at the bottom is filled with fresh cocrete. The stopwatch is started oce the gate at the bottom has bee opeed ad cocrete starts pourig out. It is stopped whe it is possible to see vertically through the fuel ito the cotaier below for the first time, ad the measured period is oted as t v. Mixtures cotaiig fly ash were created by varyig the percetage ad the origi of the fly ash used, without chagig the cotet or ature of other compoet materials. This research was performed i the Laboratory for materials, Istitute for materials ad structures, Faculty of Civil Egieerig, Uiversity of Belgrade. The river aggregate, origiated from the Daube ad divided ito fractios I (0/4), II (4/8), ad III (8/16) - Figure, was used for the productio of SCC mixes. True ad bulk desities of aggregates ad other compoet materials are preseted i Table 1, while a series mix desig is preseted i Table (values are give i kg/m 3 ). 69

5 Sieve passig Y i (%) Number 1, Year 016 Page The amouts of cemet, sad, gravel, ad the total amout of filler were kept costat i all of the series. With the exceptio of the first referet mixture E1 i which this ratio was somewhat smaller, the water/cemet ratio was also kept costat Compoet materials I (0/4) II (4/8) III (8/16) Mixture (0/16) 0,15 0,5 0, Grai size diameter d i (mm) Figure Graulometry of used fractios ad aggregate mixture Table 1 Compoet materials [11] True desity Type /origi [kg/m 3 ] Bulk desity i loose state [kg/m 3 ] Cemet CEM I (PC 4.5 Lafarge Beoči) Limestoe powder Grait Peščar Lig River aggregate Daube Fly ash Kostolac Fly ash Kolubara Superplasticizer Gleium Sky 690, BASF Two referet mixtures E1 ad E did ot cotai fly ash, while i mixtures LP1 ad LP3 10 % of limestoe filler was replaced with fly ash. Mixtures LP ad LP4 had 0% of the limestoe filler replaced with fly ash, ad fially, mixture LP5 had equal amouts of the two filler compoets. It should be oted that the fly ash used i mixtures LP1 ad LP origiated from the Power plat Kostolac, while fly ash i mixtures LP3, LP4, ad LP5 origiated from the Power plat Kolubara. Table Series mix desig [11] Series E1 E LP1 LP LP3 LP4 LP5 Water W Cemet C Limestoe filler Fly ash Total amout of filler Sad (0/4 mm) Coarse aggregate (4/8 mm) Coarse aggregate (8/16 mm) Superplasticizer

6 Number 1, Year 016 Page RESULTS AND ANALYSIS 4.1 Experimetal results Results of the measuremets of t 500 ad t v are preseted i Table 3. Time measured i the V-fuel test (t v) raged withi the limits 9.7 s ad 7. s, while t 500 period raged from.6 s to 11.3 s. Accordig to the Europea stadard EN 06-9 [13], all the cocrete mixtures beloged to the VS/VF category, with t 500 loger tha s ad t v loger tha 9 s. The lowest values of both of the parameters were obtaied for referet series E with higher water/cemet (w/c) ratio tha series E1, while the highest values were measured for the series LP5 with the highest amout of fly ash. This ca be explaied by the opposite effects of icreasig water ad fly ash cotet. I the case of water cotet, higher values mea higher w/c ad thus lower t v ad t 500 values, provided adequate segregatio resistace is achieved. I the case of fly ash cotet, higher values lead to a drop i workability ad flowability that is reflected i the results obtaied from all mixtures cotaiig fly ash (LP1 to LP5). Sice fly ash was used i its origial state (without sievig or mechaical activatio), the stated results differ from those commoly foud i the literature [14,15], i which very fie fly ash was used. It was also oticed that the measured values have similar growth (series with measured lower t 500 values, also had lower t v values ad vice versa). Table 3 Results of slump flow test ad V-fuel test [11] Series E1 E LP1 LP LP3 LP4 LP5 t500 (s) tv (s) Least squares regressio method applied to the experimetal results A liear correlatio betwee the two measured parameters, t 500 ad t v, had bee established usig the least squares method as show i Figure 3. By comparig values of the correlatio coefficiet for differetly shaped fittig fuctios, it was established that liear regressio with R = was a satisfactory choice. Correlatio coefficiet is a quatity that cofers the quality of a least squares fittig to the origial data [5]. If t 500 is represeted by x ad t v as y, the correlatio coefficiet equals: R = i(x i x )(y i y ) i(x i x ) i(y i y) As show i Figure 3, liear regressio coefficiets b 1 ad b 0 were calculated to determie the correlatio betwee measured periods t 500 ad t v. They amouted to b 1 = ad b 0 = Fially, the adopted fuctio was: t v = 1.666t (19) (18) Figure 3 Correlatio betwee measured periods t 500 ad t v usig least squares method 71

7 Number 1, Year 016 Page This correlatio betwee t 500 ad t v leads to the coclusio that oe of these tests could be excluded ad its values estimated based o the other. A compariso of the boudaries for the VS/VF category with the obtaied correlatio leads to a similar coclusio (whe t 500 = s the t v = 1.666t = 10.8 s, where t v = 9 s is the lower boudary for the class). Oe must be careful i such a aalysis, sice good results obtaied i the V-fuel ad slump flow tests do ot guaratee good properties regardig stability ad passig ability. 4.3 Fuzzy least squares method applied to the experimetal results The aim of this paper was to establish the possible boudaries for the correlatio betwee two measured values, as show i Figure 4. These boudaries were established usig the approximate distace fuzzy least squares method. For the purpose of this calculatio, both t v ad t 500 were defied as triagular fuzzy umbers. Left ad right spreads were adopted based o the previous trials ad experieces with the fresh state behavior of preseted mixtures. The measured values preseted i Table 3 were take to be the most possible oes. Figure 4 Correlatio fuctio with possible upper ad lower boudary Time periods t 500 ad t v, preseted as fuzzy umbers with graphical represetatios are show i Table 4 for all of the series. Ukow parameters A 0 (α a0, β a0, δ a0) ad A 1 (α, β, δ ) are calculated by miimizig sum of squares of distaces betwee measured ad fitted Y values, oted as U (16,17). Applyig this methodology umerically, the fuzzy coefficiets of liear regressio are obtaied: A 0 = {6.938,7.457,8.196} ad A 1 = {1.593,1.666,1.699} (0) Fially, the results of applyig the fuzzy least square method are preseted graphically i Figure 5. Fuctio III is the same as the oe gaied from the least squares method, while fuctios I ad II were defied usig the lower ad upper boudaries of the calculated coefficiets (II: t v=1.593t ad III: t v=1.699t ). The fuctios preseted i Figure 5 simulate the boudaries of the possible outcomes of a higher umber of measuremets that might be performed o differet SCC mixtures with these compoet materials. This provides a good estimatio of t v ad t 500 rages i differet mixtures, ad at the same time decreases the umber of ecessary tests. The deviatios from the measured values did ot have a great ifluece o the adopted liear correlatio fuctio. 7

8 Number 1, Year 016 Page Table 4 Periods t 500 ad t v preseted as fuzzy umbers 73

9 Number 1, Year 016 Page Figure 5 Upper ad lower boudary of liear correlatios betwee the periods t 500 ad t v for the tested series 5 CONCLUSION As stated before, the properties of fresh cocrete, especially importat for SCC, are iflueced by differet factors. O the other had, the umber of the collected data poits is almost always isufficiet for a more detailed statistical aalysis. These facts lead us to the coclusio that, oce a good correlatio has bee established betwee the data, it ca be broadeed usig a probabilistic method i this case, fuzzy set theory. I this paper, seve experimetal results from V-fuel ad slump flow tests (t v ad t 500) obtaied from differet mixtures were aalyzed, ad the correlatio betwee measured values was formed. A umber of testig results were too small to provide complete cofidece i the regularity of the fittig fuctio. That is why the fuzzy least squares method was applied, with satisfyig results. The adopted fuzzy umbers had differet spreads that raged from 0.0 s to 1.08 s for t 500, ad from 0. s to 1.03 s for t v. Nevertheless, the calculated fuzzy coefficiets of the correlatio fuctio had left ad right spreads that amouted to oly ad for A 0, ad ad for A 1. It ca be cocluded that deviatios from the measured values did ot have a great ifluece o the adopted liear correlatio fuctio. I additio, the lower (fuctio I) ad upper limits (fuctio II) for determiig the possible outcomes for t v were established. This ca help i the evaluatio of ew cocrete mix desigs, e.g., with the use of differet amouts of fly ash, by usig oly oe cotrol testig method for cocrete fillig ability durig trial testig. ACKNOWLEDGMENTS The work reported i this paper is a part of the ivestigatio withi the research proect TR "Utilizatio of byproducts ad recycled waste materials i cocrete composites i the scope of sustaiable costructio developmet i Serbia: ivestigatio ad evirometal assessmet of possible applicatios", supported by the Miistry of Educatio, Sciece ad Techological Developmet, Republic of Serbia. This support is gratefully ackowledged. 74

10 Number 1, Year 016 Page Refereces [1] Europea Proect Group, 005: Europea Guidelies for Self-Compactig Cocrete: Specificatio, Productio ad Use [] Zadeh, L. A., 1965: Fuzzy Sets, Iformatio ad Cotrol, pp , [3] Praščević, Z. : Primea teorie mogućosti u plairau realizacie proekata, Izgrada 1/89, pp. 9-13, [4] Diamod, F. 1988: Fuzzy Least Squares, Iformatio Scieces, 46(3), pp , / (88) [5] Mališić, J.; Jevremović, V. 014: Statističke metode u meteorologii i ižeerstvu, Matematički fakultet Uiverziteta u Beogradu, Uiverzitetski cetar za primeeu statistiku, Uiverzitet u Novom Sadu, Beograd [6] Dubois, D.; Prade, H. 1980: Fuzzy Sets ad Systems: Theory ad Applicatios, Academic Press, New York [7] Larse H. L.: Fudametals of Fuzzy Sets ad Fuzzy Logic, Aalborg Uiversity Esberg, FL-14b pp. 1-6 [8] Yag, M. S.; Li, T. S. 00: Fuzzy least-squares liear regressio aalysis for fuzzy iput output data, Fuzzy Sets ad Systems, 16(3), pp , [9] Torfi, F.; Farahai, R. Z.; Mahdavi, I. 011: Fuzzy Least-Squares Liear Regressio Approach to Ascertai Stochastic Demad i the Vehicle Routig Problem, Applied Mathematics,, pp , /am [10] SRPS EN :01 Ispitivae svežeg betoa - Deo 8: Samougrađuući beto Ispitivae rasprostiraa slegaem, Istitut za stadardizaciu Srbie, [11] SRPS EN :01 Ispitivae svežeg betoa - Deo 9: Samougrađuući beto Ispitivae pomoću V- levka, Istitut za stadardizaciu Srbie, [1] Savić, A. 015: Istraživae svostava svežeg i očvrslog samozbiaućeg betoa sa mieralim dodacima a bazi idustriskih usprodukata, Doktorska disertacia, Uiverzitet u Beogradu, Građeviski fakultet, Beograd, metori: prof. dr Dragica Jevtić, prof. dr Tataa Volkov Husović [13] EN 06-9:010 Cocrete. Additioal rules for Self-Compactig Cocrete (SCC) [14] Brow, J. H. 198: The stregth ad workability of cocrete with PFA substitutio. Proceedigs, Iteratioal Symposium o the Use of PFA i Cocrete, Uiversity of Leeds, Eglad, pp [15] Wesche, K. 005: Fly Ash i Cocrete: Properties ad performace (Rilem Report 7), Report of Techical Committee 67-FAB Use of Fly Ash i Buildig, Taylor & Fracis e-library, 75