Research Article A Statistical Approach to Optimizing...

Research ArticleA Statistical Approach to Optimizing Concrete Mixture Design

Shamsad Ahmad and Saeid A Alghamdi

Civil and Environmental Engineering Department King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia

Correspondence should be addressed to Shamsad Ahmad shamsadkfupmedusa

Received 19 November 2013 Accepted 24 December 2013 Published 9 February 2014

Academic Editors H Cui I B Topcu and H Wang

Copyright copy 2014 S Ahmad and S A Alghamdi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A step-by-step statistical approach is proposed to obtain optimum proportioning of concrete mixtures using the data obtainedthrough a statistically planned experimental program The utility of the proposed approach for optimizing the design of concretemixture is illustrated considering a typical case in which trial mixtures were considered according to a full factorial experimentdesign involving three factors and their three levels (33) A total of 27 concrete mixtures with three replicates (81 specimens) wereconsidered by varying the levels of key factors affecting compressive strength of concrete namely watercementitious materialsratio (038 043 and 048) cementitious materials content (350 375 and 400 kgm3) and finetotal aggregate ratio (035 040and 045) The experimental data were utilized to carry out analysis of variance (ANOVA) and to develop a polynomial regressionmodel for compressive strength in terms of the three design factors considered in this study The developed statistical model wasused to show how optimization of concrete mixtures can be carried out with different possible options

1 Introduction

Optimization of the concrete mixture design is a process ofsearch for a mixture for which the sum of the costs of theingredients is lowest yet satisfying the required performanceof concrete such as workability strength and durability Thebasic ingredients of concrete can be classified into two groupscement paste and aggregates Although the quality of cementpaste is governed mainly by the watercement ratio thequantity of cement paste required to achieve a targeted qualityof concrete depends on the characteristics of aggregatesThese characteristics mainly include surface area and voidsin aggregates While surface area is governed by the shapeand maximum size of aggregates the void content is affectedmainly by the particle size distribution of aggregates Therequirement of the paste can be reduced by reducing thevoid content of aggregates through proper packing of theaggregates [1ndash5] and also by increasing the aggregatecementratio [6] Goltermann et al [1] have suggested a packingmodel for the aggregate selection and combination to obtainaggregate mixes having the lowest void contents with maxi-mum packing degree (the ratio between bulk density and theaggregate grain density) Thus the packing degree according

to them is a characteristic of the specific aggregate type ormix and it indicates the void volume and the amount ofcement paste necessary in the concrete This indicates thata concrete mixture design can be optimized by adjusting thelevels of the keymixture factors such as water to cementitiousmaterials ratio coarse aggregate to total aggregate ratio andcementitious material content or aggregate to cementitiousmaterials ratio as reported by various researchers [7ndash12]

Attempts have been made in the past to optimize theconcrete mixture design using either the fully experimentalmethods or fully analytical methods or semiexperimen-tal (half-analytical) methods or statistical methods Fullyexperimental methods involve an extensive series of testssometimes conducted on a trial-and-error basis and theoptimization results are often applicable only to a narrowrange of local materials [13 14] In order to reduce thenumber of trial mixtures required to obtain an optimalmixture efforts have been made towards developing analyti-cal methods rationalizing the initial mixture proportioninginto a more logical and systematic process [15] Analyticalmethods help in searching for an optimum concrete mixturebased on detailed knowledge of specific weights of mixturecomponents and on certain basic formulas which result from

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 561539 7 pageshttpdxdoiorg1011552014561539

2 The Scientific World Journal

previous experience without conducting expensive and time-consuming experimental works [15 16] Semiexperimental(half-analytical) methods are based on combining the exper-imental database or experimentally developed predictionmodels and various analytical tools such as artificial neuralnetwork genetic algorithm and mathematical programming[17ndash19] Statistical methods also termed as statistical experi-ment design methods or statistical factorial design methodsor design of experiments methods or empirical methodsare also used frequently in obtaining the optimum concretemixture design [9 10 20ndash24] Statistical methods are animprovement over fully experimental methods in whichinstead of selecting one starting mix proportion and thenadjusting by trial and error for achieving the optimumsolution a set of trial batches covering a chosen range ofproportions for eachmixture component is defined accordingto established statistical procedures Trial batches are thencarried out test specimens are fabricated and tested andexperimental results are analyzed using standard statisticalmethods These methods include fitting empirical models tothe data for each performance criterion In these modelseach response (resultant concrete property) such as strengthslump or cost is expressed as an algebraic function of factors(individual component proportions) such as wc cementcontent chemical admixture dosage and percent pozzolanareplacement After a response can be characterized by anequation (model) several analyses are possible For examplea user could determine which mixture proportions wouldyield one or more desired properties A user also couldoptimize any property subject to constraints on other prop-erties Simultaneous optimization to meet several constraintsis also possible For example one could determine the lowestcost mixture with strength greater than a specified value aircontent within a given range and slumpwithin a given range

Fully analytical methods are less expensive and less timeconsuming but they have the disadvantage of being lessprecise because of the variations in the materials character-istics of the aggregates and cements Fully experimental orsemiexperimental (ie half-analytical) methods are reliableand accurate however they involve comprehensive labora-tory works [16] Statistical methods also require a certainamount of experimental works but they have an additionaladvantage in a sense that the expected properties (responses)can be characterized by an uncertainty (variability) This hasimportant implications for specifications and for productionof the cost-effective concrete mixture [10]

In the present work an effort has been made to exhibitthe application of a statistical approach proposed to obtainoptimum proportioning of concrete mixtures using the dataobtained through an experiment design considering water-cementitious materials ratio cementitious materials contentand fine aggregate to total aggregate ratio as design factorsThe experimental data were analyzed statistically and math-ematical polynomials regression was developed for concretestrength as a function of mixture variables The utility ofthe developed compressive strength model in optimizing themixture designs was illustrated considering different possibleoptions

2 Proposed Approach

Theproposed approach to optimizing the proportions of con-crete mixtures is based on the planned experimental works(within the domain of required characteristic performanceof concrete) and statistical analysis of the data generatedwhich would reduce the number of trial batches needed Theproposed approach consists of the following steps

21 Specification of the Characteristic Performance of ConcreteThe information pertaining to required workability strengthand exposure conditions (for durability requirements) shouldbe first collected The workability requirements depend onthe mode of transportation handling and placing andalso on type of construction [25] The strength is specifiedbased on the structural requirements for concrete protectedfrom exposure to freezing and thawing and application ofdeicing chemicals or aggressive substances However foraggressive exposure conditions the strength specified bythe structural designer should not be less than the min-imum design compressive strength recommended for thegiven exposure condition For example ACI 318 [26] hasspecified minimum design compressive strengths of 28 31and 35MPa respectively for concrete exposed to waterfreezing-thawing and chloridesThe durability requirementsof concrete mixtures are normally satisfied by ensuring thatthe cementitious materials content is not less than a specifiedminimum value and the watercementitious materials ratiois not more than the one specified for a given exposurecondition For example the cementitious materials contentshould not be less than 335 kgm3 and watercementitiousmaterials ratio should not be more than 040 (by mass) forsatisfying the durability requirements for concrete subjectedto severe exposure conditions such as severe freeze-thawdeicer and sulfate exposures [26]

22 Selection of the Levels of Key Mix Design Factors Selec-tion of the levels of the three key mixture design factorsnamely cementitious materials content watercementitiousmaterials ratio and finetotal aggregate ratio which mainlyaffect the quality of concrete will be made to ensure thatenough experimental data are generated for obtaining aregressionmodel for compressive strength which can be usedto optimize the mixture proportions meeting the specifiedcharacteristic performance of concrete

The minimum level of cementitious materials contentshould not be less than 335 kgm3 which is the minimumvalue to satisfy the durability requirements for aggressiveexposure conditions The maximum level of cementitiousmaterials content should be selected considering the riskof shrinkage The minimum level of watercementitiousmaterials ratio should be selected considering the strengthrequirements In case of choosing a very low level of thewatercementitious materials ratio the difficulty in trans-porting handling and placing concrete and extra cost ofsuperplasticizer to meet the workability requirements shouldbe consideredThemaximum level of the watercementitiousmaterials ratio should be within the maximum permissible

The Scientific World Journal 3

limit for the watercementitious materials ratio for the givenexposure condition The minimum and maximum levelsof the finetotal aggregate ratio should be selected withinthe optimum range for achieving maximum packing ofaggregates For example Soudki et al [9] have reportedoptimum finetotal aggregate ratio in the range of 040 and045

23 Experimental Work for Generating Data to Obtain Statis-tical Model for Optimization An experimental work shouldbe conducted involving designing preparing and testing var-ious trial mixtures according to the full factorial experimentdesign considering the various possible combinations of thelevels of the mixture variables within their selected rangesof variation The workability of each trial mixture should beequal to or more than the specified value In case if super-plasticizer is needed to achieve the intended workability thecost of superplasticizer should be added to the cost of cementAfter finalizing the dosage of superplasticizer based on therequired workability for each of the trial mixtures the cubicalor cylindrical specimens should be prepared cured for 28days and then tested for compressive strength for generatingdata to obtain statistical model for strength to be used foroptimization

24 Statistical Analysis of Experimental Data and Fitting ofthe Strength Model Analysis of variance (ANOVA) can beused for examining the significance of the factors consideredfor developing the strength model and subsequently fittingan empirical model for compressive strength in terms of thesignificant mixture factors using polynomial regression InANOVA the statistical terminologies used are as follows

Degree of Freedom (DF) Degree of freedom is the number ofvalues in the final calculation of a statistic that are free to varyDF = 119899 minus 1 where 119899 represents the number of groups

Error (Residual) It is the amount by which an observedvariate differs from the value predicted by the assumedstatistical model

Sum of Squares (SS) It is the squared distance between eachdata point (119883

119894) and the sample mean (119883) summed for all 119899

data points SS = sum119899119894=1(119883119894minus 119883)

2 where119883119894represents the 119894th

observation and119883 represents the sample mean

Mean Square (MS) It is the sum of squares divided by thedegrees of freedom

119865-Ratio It is ratio of MS of the concerned factor to the MS ofthe error A higher 119865-ratio indicates a significant effect of thefactor

119875-Value It is a measure of acceptance or rejection of astatistical significance of a factor based on a standard that nomore than 5 (005 level) of the difference is due to chanceor sampling error In other words if the 119875 value for a factor

Table 1 Test program

Factor 1 2 3 LevelCementitious materials content(119876119862) in kgm3 350 375 400 3

Watercementitious materialsratio (119877wcm) by mass 038 043 048 3

Finetotal aggregate ratio(119877FATA) by mass 035 040 045 3

is 005 or more it would not have effect on the dependentvariable

25 Optimization of Mixture Proportions Using the FittedStrength Model The statistical model for the compressivestrength derived utilizing the experimental model can beused to obtain the optimal mixture proportions satisfying thespecified characteristic performance of concrete as requiredconstraints The mixture satisfying all the constraints andhaving the lowest requirements of cement and superplasti-cizer would be considered as optimummixture

3 Experimental Program

31 Test Program For illustrating the utilization of the pro-posed approach to optimizing concrete mixture designan experimental program was considered A full factorialexperiment with 3 times 3 times 3 treatment combinations was usedresulting in a total of 27 concrete trial mixtures consideringthree typical levels of each of the three key factors affecting theperformance of concrete mixtures as shown in Table 1 Forthe levels of the watercementitious materials ratio selectedin the present work no superplasticizer was added Thecombinations of the levels of the three factors for all 27 trialmixtures are shown in Table 2

32 Materials and Mix Proportioning The cementitiousmaterials used in this study consisted of 92 Type I Portlandcement conforming to ASTM C 150 [27] and 8 silica fume(by mass) The crushed stone particles obtained from a localquery were used as coarse aggregate and local dune sand wasused as fine aggregate Potable water was used for mixing theconstituents of all the specimens The specific gravity waterabsorption and sieve analysis results for the used coarse andfine aggregates are presented in Table 3 The specific gravitiesof water and cementitious materials were taken as 1 and 315respectively

The proportioning of all 27 trial mixtures was carried outin terms of absolute volume using the specific gravities ofthe concrete ingredients and the values of watercementitiousmaterials ratio cementitious materials content and finetotalaggregate ratio for each of 27 mixtures as given in Table 2The water absorption values of fine and coarse aggregateswere used to determine the gross water content

33 Preparation and Testing of Specimens Considering threereplicates for each of the 27 mixtures a total number of


Table 2 Trial mixtures

Mixnumber

Watercementitiousmaterials ratio

(119877wcm)

Cementitiousmaterials content119876119862(kgm3)

Finetotalaggregate ratio

(119877FATA)1 038 350 035

2 038 350 040

3 038 350 045

4 038 375 035

5 038 375 040

6 038 375 045

7 038 400 035

8 038 400 040

9 038 400 04510 043 350 03511 043 350 04012 043 350 04513 043 375 03514 043 375 04015 043 375 04516 043 400 03517 043 400 04018 043 400 04519 048 350 03520 048 350 04021 048 350 04522 048 375 03523 048 375 04024 048 375 04525 048 400 03526 048 400 04027 048 400 045

81 cylindrical concrete specimens (size 75mm diameterand 150mm high) were cast for determining compressivestrength After casting the concrete specimenswere cured for28 days in a curing tank under laboratory conditions and thentested for compressive strength in accordance with ASTMC 39 [28] The average compressive strength of the threespecimens made from the same concrete mixture and testedat the same age was considered as characteristic compressivestrength of a mixture

4 Results and Discussion

Average 28-day compressive strength test results for all 27concrete mixtures along with the standard deviation of threereplicates of each mixture are presented in Table 4 The datagiven in Table 4 were utilized for statistical analysis to exam-ine the significance of themixture factors and subsequently toobtain a regression model for compressive strength in termsof the factors considered

Table 3 Specific gravity water absorption and sieve analysis ofaggregates

Sieve size Cumulative percentage retained(a) Coarse aggregate

(specific gravity = 255 water absorption = 11)19mm 5125mm 6095mm 95475mm 100

(b) Fine aggregate(specific gravity = 266 water absorption = 06)

118mm 0060mm 2442030mm 9049015mm 9659

Table 4 Compressive strength test results

Mixnumber

28-day averagecompressive strength 1198911015840

119888

(MPa)

Standard deviation of threereplicates of each mixture

(MPa)1 397 192 388 103 391 084 341 125 382 196 406 207 342 118 393 119 398 1610 279 1111 374 1812 385 1113 319 0814 371 1315 339 0216 265 1417 307 1718 365 1619 300 1520 321 1321 305 0822 207 1823 275 0824 299 0325 254 1126 310 0227 253 02

41 Statistical Analysis of Data and Fitting of the CompressiveStrength Model Analysis of variance (ANOVA) was carriedout to pinpoint the individual and interactive effects ofvariable factors on the dependent variable ANOVAof the testresults in the present studywas donewith the software named


Table 5 ANOVA for compressive strength test results

Factors Type Level Scale values119876119862

Fixed 3 0875 0938 1000119877wcm Fixed 3 0792 0896 1000119877FATA Fixed 3 0778 0889 1000Source DF SS MS 119865-ratio 119875 value Significance119876119862

2 39672 19380 1990 0199 No119877wcm 2 464501 232250 23260 0000 Yes119877FATA 2 135281 67640 6770 0019 Yes119876119862lowast 119877wcm 4 23686 5921 0590 0678 No119876119862lowast 119877FATA 4 4993 1248 0120 0969 No119877wcm lowast 119877FATA 4 22437 5609 0560 0697 NoError 8 79890 9986Total 26 770456

MINITAB [29] Based on theANOVA results the polynomialregression model for compressive strength was obtained

The results of ANOVA for compressive strength are pre-sented in Table 5 A factor was considered to have significanteffect on the compressive strength if 119875 value was found tobe less than 005 (95 confidence level) The 119875 value wasobtained from Fisherrsquos distribution table which depends onerror degree of freedom (DF) and the mean squares (MS)Table 5 shows that the 119877wcm and 119877FATA have significanteffects on compressive strength as their levels of significance119875 values are less than 005 Therefore these two significantvariables should be considered for obtaining the regressionmodel for compressive strength1198911015840

119888 Although the effect of119876

119862

on compressive strength is found to be insignificant becauseit varies within a narrow range of 350 to 400 kgm3 it isconsidered in the regression analysis as cement remains anindispensable material in concrete production

The polynomial regression model obtained for compres-sive strength using the data presented in Table 4 is presentedas follows

119891

1015840

119888= minus6124 minus 0056119876

119862minus 1987Exp (2083119877wcm)

+ 18345119877

0119

FATA (1198772= 080)

(1)

where 1198911015840119888is the 28-day compressive strength in MPa 119876

119862is

the cementitious materials content in kgm3 119877wcm is thewatercementitious materials ratio by mass 119877FATA is thefinetotal aggregate ratio by mass

The upper and lower bounds of each of the three variablesare given in Table 1

42 Optimization of Concrete Mixture Proportions UsingCompressive Strength Model The empirical model obtainedfor compressive strength can be used for optimization ofconcretemixture proportions using any suitable optimizationmethodtool The developed compressive strength modelwas utilized for optimization of concrete mixture designcorresponding to the following options (ie constraints)typically using theMicrosoft Excel Solver

(1) optimizing the levels of the 119877wcm and 119877FATA forachieving maximum possible compressive strengthat different values of cementitious materials con-tent within the selected range (ie 350 375 and400 kgm3)

(2) optimizing the levels of the 119877wcm and 119877FATA forachieving different target compressive strengths at dif-ferent values of cementitiousmaterials content withinthe selected range (ie 350 375 and 400 kgm3)

The optimization results presented in Table 6 indicatethat the maximum compressive strength corresponding toa cementitious materials content of 350 kgm3 is higherthan that at cementitious materials contents of 375 and400 kgm3 Further the maximum compressive strength atcementitious materials content of 375 kgm3 is higher thanthat corresponding to a cementitious materials content of400 kgm3This indicates that at the same optimum values of119877wcm and 119877FATA the compressive strength is more at lowercementitious materials content (ie at a higher aggregate tocement ratio) due to better aggregate packing as reported byNeville and Brooks [6] At all levels of the cementitious mate-rials content maximum compressive strengths correspondto minimum watercementitious materials ratio (038) andmaximumfinetotal aggregate ratio (045) within their rangesof variation considered in the present work

From Table 6 the concrete mixture having a maximumcompressive strength of 421MPa at aminimumcementitiousmaterials content of 350 kgm3 watercementitious materialsratio of 038 and finetotal aggregate ratio of 045 can betypically selected as the optimummixture However in caseswhere the compressive strength requirement is less thanthe maximum a set of watercementitious materials ratiocementitious materials content and finetotal aggregate ratioother than the optimum one can be selected for achieving theworkability and durability requirements

The data obtained from the optimization option (ii) aspresented in Table 6 were plotted to depict the variationsof compressive strength with watercementitious materialsratio and finetotal aggregate ratio at different cementitiousmaterials contents as shown in Figures 1 and 2 respectively It


Table 6 Optimization of concrete mixture design

Optimization option 119891

1015840

119888(MPa)

Optimum levels of the mixture variablesCost level

119876119862

(kgm3)119877wcm

(by mass)119877FATA

(by mass)(i) Optimizing the levels of 119877wcm and 119877FATA forachievingmaximum compressive strengths atdifferent levels of 119876

119862

421 350 038 045 Low407 375 038 045 Medium393 400 038 045 High

(ii) Optimizing the levels of 119877wcm and 119877FATAfor achieving different target compressivestrengths at different levels of 119876

119862

300 048 040350 350 044 042 Low400 040 044300 047 041350 375 043 042 Medium400 038 044300 046 041350 400 041 043 High400 038 045

can be seen from Figures 1 and 2 that at a given cementitiousmaterials content the compressive strength increases withthe decrease in thewatercementitiousmaterials ratio and theincrease in the finetotal aggregate ratio It can be observedfromFigure 1 that for the same value of compressive strengththe requirement for watercementitious materials ratio islower at higher values of the cementitious materials contentTherefore the plots presented in Figure 1 can be utilized toselect an adequate value of the watercementitious materialsratio and cementitious materials content for a given value ofthe target compressive strength satisfying the workability anddurability requirements For example in the case of a normalexposure a higher value of the watercementitious materialsratio and a lower value of cementitious materials content canbe selected which would give more workability at a lowercost whereas for harsh exposure conditions a lower valueof the watercementitious materials ratio and a higher valuecementitious materials content can be selected which wouldprovide better durability

5 Conclusions

A simplified step-by-step approach is proposed for optimiz-ing the concrete mixture design based on the analysis of thedata obtained through a statistically planned experimentalprogram The proposed approach consists of five steps asfollows (i) specification of the characteristic performance ofconcrete (ii) selection of the levels of key mix design factors(iii) experimental work considering trial mixtures using fullfactorial experiment design for generating data to obtainstatistical model for optimization (iv) statistical analysis ofexperimental data and fitting of the strength model and (v)optimization of mixture proportions using the fitted strengthmodel

The results of the experimental works conducted in thepresent study for demonstrating the utility of the proposedstatistical approach have indicated the significant effects

25

30

35

40

45

035 038 041 044 047 05Rwcm (by mass)

QC= 350kg per cubic meterQC= 375kg per cubic meterQC= 400kg per cubic meter

f998400 c

(MPa

)

Figure 1 Variation of compressive strength with 119877wcm at differentlevels of 119876

119862

25

30

35

40

45

035 038 041 044 047 05RFATA (by mass)

f998400 c

(MPa

)


Figure 2 Variation of compressive strength119877FATA at different levelsof 119876119862


of watercementitious materials ratio cementitious materi-als content and finetotal aggregate ratio on compressivestrength The optimum values of watercementitious mate-rials ratio and finetotal aggregate ratios have resulted in ahigher compressive strength at a lower cementitiousmaterialscontent resulting in significant cost saving in the concreteproduction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial supportreceived from King Abdulaziz City for Science amp Technol-ogy (KACST) Saudi Arabia for conducting this researchwork (Project no KACST AT-23-21) The support of theDepartment of Civil Engineering King Fahd University ofPetroleum and Minerals (KFUPM) Saudi Arabia is alsoacknowledged

References

[1] P Goltermann V Hohansen and L Palbol ldquoPacking of aggre-gates an alternative tool to determine the optimal aggregatemixrdquo ACI Materials Journal vol 94 no 5 pp 435ndash443 1997

[2] G Shakhmenko and J Birsh ldquoConcrete mix design and opti-mizationrdquo in Proceedings of the 2nd International Symposium inCivil Engineering pp 1ndash8 Budapest Hungary 1998

[3] M Glavind and C Munch-Petersen ldquoGreen concrete in Den-markrdquo Structural Concrete vol 1 no 1 pp 1ndash6 2000

[4] V K Senthil and S Manu ldquoParticle packing theories and theirapplication in concretemixture proportioning a reviewrdquo IndianConcrete Journal vol 77 no 9 pp 1324ndash1331 2003

[5] H He P Stroeven M Stroeven and L J Sluys ldquoOptimizationof particle packing by analytical and computer simulationapproachesrdquo Computers and Concrete vol 9 no 2 pp 119ndash1312012

[6] A M Neville and J J Brooks Concrete Technology LongmanSingapore 1996

[7] A F Abbasi M Ahmad and MWasim ldquoOptimization of con-crete mix proportioning using reduced factorial experimentaltechniquerdquo ACIMaterials Journal vol 84 no 1 pp 55ndash63 1987

[8] J M Shilstone Sr ldquoConcrete mixture optimizationrdquo ConcreteInternational vol 12 no 6 pp 33ndash39 1990

[9] K A Soudki E F El-Salakawy and N B Elkum ldquoFull factorialof optimization of concretemix design for hot climatesrdquo JournalofMaterials in Civil Engineering vol 13 no 6 pp 427ndash433 2001

[10] M J Simon ldquoConcrete mixture optimization using statisti-cal methodsrdquo Final Report FHWA-RD-03-060 InfrastructureResearch and Development Federal Highway AdministrationGeorgetown Pike McLean Va USA 2003

[11] P-K Chang ldquoAn approach to optimizing mix design forproperties of high-performance concreterdquoCement andConcreteResearch vol 34 no 4 pp 623ndash629 2004

[12] A Ozlem A K Ulas and S Bahar ldquoSelf-consolidating high-strength concrete optimization bymixture designmethodrdquoACIMaterials Journal vol 107 no 4 pp 357ndash364 2010

[13] P L J Domone and M N Soutsos ldquoAn approach to theproportioning of high-strength concrete mixesrdquo Concrete Inter-national vol 16 no 10 pp 26ndash31 1994

[14] S Ahmad ldquoOptimum concrete mixture design using locallyavailable ingredientsrdquo The Arabian Journal for Science andEngineering vol 32 no 1 pp 27ndash33 2007

[15] I-C Yeh ldquoComputer-aided design for optimum concrete mix-turesrdquo Cement amp Concrete Composites vol 29 no 3 pp 193ndash202 2007

[16] J Kasperkiewicz ldquoOptimization of concretemix using a spread-sheet packagerdquoACIMaterials Journal vol 91 no 6 pp 551ndash5591994

[17] B Y Lee J H Kim and J-K Kim ldquoOptimum concretemixture proportion based on a database considering regionalcharacteristicsrdquo Journal of Computing in Civil Engineering vol23 no 5 pp 258ndash265 2009

[18] I-C Yeh ldquoOptimization of concrete mix proportioning usinga flattened simplexmdashcentroid mixture design and neural net-worksrdquo Engineering with Computers vol 25 no 2 pp 179ndash1902009

[19] M A Jayaram M C Nataraja and C N Ravikumar ldquoElitistgenetic algorithm models optimization of high performanceconcrete mixesrdquoMaterials andManufacturing Processes vol 24no 2 pp 225ndash229 2009

[20] A Ghezal and K H Khayat ldquoOptimizing self-consolidatingconcrete with limestone filler by using statistical factorial designmethodsrdquo ACI Materials Journal vol 99 no 3 pp 264ndash2722002

[21] R Patel K M A Hossain M Shehata N Bouzoubaa andM Lachemi ldquoDevelopment of statistical models for mixturedesign of high-volume fly ash self-consolidating concreterdquo ACIMaterials Journal vol 101 no 4 pp 294ndash302 2004

[22] M Muthukumar and D Mohan ldquoOptimization of mechanicalproperties of polymer concrete and mix design recommenda-tion based ondesign of experimentsrdquo Journal of Applied PolymerScience vol 94 no 3 pp 1107ndash1116 2004

[23] L Xiaoyong and M Wendi ldquoOptimization for mix design ofhigh-performance concrete using orthogonal testrdquo Communi-cations in Computer and Information Science vol 232 no 2 pp364ndash372 2011

[24] M Sonebi and M T Bassuoni ldquoInvestigating the effect ofmixture design parameters on pervious concrete by statisticalmodelingrdquoConstruction andBuildingMaterials vol 38 pp 147ndash154 2013

[25] American Concrete Institute Standard practice for selectingproportions for normal heavyweight and mass concrete ACI21111991(reapproved 2009)

[26] American Concrete Institute ldquoBuilding code requirements forstructural concrete and commentaryrdquo ACI Standard 318-08ACI Committee 2008

[27] American Society for TestingMaterials ldquoStandard specificationfor Portland cementrdquo ASTM C150 vol 4 no 1 2012

[28] American Society for Testing Materials ldquoStandard test methodfor compressive strength of cylindrical concrete specimensrdquoASTM C39 vol 4 no 2 2012

[29] ldquoMINITAB Statistical Package Release 13 for Windows 95 and98rdquo Minitab 2000 httpwwwminitabcomen-us

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previous experience without conducting expensive and time-consuming experimental works [15 16] Semiexperimental(half-analytical) methods are based on combining the exper-imental database or experimentally developed predictionmodels and various analytical tools such as artificial neuralnetwork genetic algorithm and mathematical programming[17ndash19] Statistical methods also termed as statistical experi-ment design methods or statistical factorial design methodsor design of experiments methods or empirical methodsare also used frequently in obtaining the optimum concretemixture design [9 10 20ndash24] Statistical methods are animprovement over fully experimental methods in whichinstead of selecting one starting mix proportion and thenadjusting by trial and error for achieving the optimumsolution a set of trial batches covering a chosen range ofproportions for eachmixture component is defined accordingto established statistical procedures Trial batches are thencarried out test specimens are fabricated and tested andexperimental results are analyzed using standard statisticalmethods These methods include fitting empirical models tothe data for each performance criterion In these modelseach response (resultant concrete property) such as strengthslump or cost is expressed as an algebraic function of factors(individual component proportions) such as wc cementcontent chemical admixture dosage and percent pozzolanareplacement After a response can be characterized by anequation (model) several analyses are possible For examplea user could determine which mixture proportions wouldyield one or more desired properties A user also couldoptimize any property subject to constraints on other prop-erties Simultaneous optimization to meet several constraintsis also possible For example one could determine the lowestcost mixture with strength greater than a specified value aircontent within a given range and slumpwithin a given range

Fully analytical methods are less expensive and less timeconsuming but they have the disadvantage of being lessprecise because of the variations in the materials character-istics of the aggregates and cements Fully experimental orsemiexperimental (ie half-analytical) methods are reliableand accurate however they involve comprehensive labora-tory works [16] Statistical methods also require a certainamount of experimental works but they have an additionaladvantage in a sense that the expected properties (responses)can be characterized by an uncertainty (variability) This hasimportant implications for specifications and for productionof the cost-effective concrete mixture [10]

In the present work an effort has been made to exhibitthe application of a statistical approach proposed to obtainoptimum proportioning of concrete mixtures using the dataobtained through an experiment design considering water-cementitious materials ratio cementitious materials contentand fine aggregate to total aggregate ratio as design factorsThe experimental data were analyzed statistically and math-ematical polynomials regression was developed for concretestrength as a function of mixture variables The utility ofthe developed compressive strength model in optimizing themixture designs was illustrated considering different possibleoptions

2 Proposed Approach

Theproposed approach to optimizing the proportions of con-crete mixtures is based on the planned experimental works(within the domain of required characteristic performanceof concrete) and statistical analysis of the data generatedwhich would reduce the number of trial batches needed Theproposed approach consists of the following steps

21 Specification of the Characteristic Performance of ConcreteThe information pertaining to required workability strengthand exposure conditions (for durability requirements) shouldbe first collected The workability requirements depend onthe mode of transportation handling and placing andalso on type of construction [25] The strength is specifiedbased on the structural requirements for concrete protectedfrom exposure to freezing and thawing and application ofdeicing chemicals or aggressive substances However foraggressive exposure conditions the strength specified bythe structural designer should not be less than the min-imum design compressive strength recommended for thegiven exposure condition For example ACI 318 [26] hasspecified minimum design compressive strengths of 28 31and 35MPa respectively for concrete exposed to waterfreezing-thawing and chloridesThe durability requirementsof concrete mixtures are normally satisfied by ensuring thatthe cementitious materials content is not less than a specifiedminimum value and the watercementitious materials ratiois not more than the one specified for a given exposurecondition For example the cementitious materials contentshould not be less than 335 kgm3 and watercementitiousmaterials ratio should not be more than 040 (by mass) forsatisfying the durability requirements for concrete subjectedto severe exposure conditions such as severe freeze-thawdeicer and sulfate exposures [26]

22 Selection of the Levels of Key Mix Design Factors Selec-tion of the levels of the three key mixture design factorsnamely cementitious materials content watercementitiousmaterials ratio and finetotal aggregate ratio which mainlyaffect the quality of concrete will be made to ensure thatenough experimental data are generated for obtaining aregressionmodel for compressive strength which can be usedto optimize the mixture proportions meeting the specifiedcharacteristic performance of concrete

The minimum level of cementitious materials contentshould not be less than 335 kgm3 which is the minimumvalue to satisfy the durability requirements for aggressiveexposure conditions The maximum level of cementitiousmaterials content should be selected considering the riskof shrinkage The minimum level of watercementitiousmaterials ratio should be selected considering the strengthrequirements In case of choosing a very low level of thewatercementitious materials ratio the difficulty in trans-porting handling and placing concrete and extra cost ofsuperplasticizer to meet the workability requirements shouldbe consideredThemaximum level of the watercementitiousmaterials ratio should be within the maximum permissible




























Mixnumber


(119877wcm)



(119877FATA)1 038 350 035

2 038 350 040

3 038 350 045

4 038 375 035

5 038 375 040

6 038 375 045

7 038 400 035

8 038 400 040

9 038 400 04510 043 350 03511 043 350 04012 043 350 04513 043 375 03514 043 375 04015 043 375 04516 043 400 03517 043 400 04018 043 400 04519 048 350 03520 048 350 04021 048 350 04522 048 375 03523 048 375 04024 048 375 04525 048 400 03526 048 400 04027 048 400 045








118mm 0060mm 2442030mm 9049015mm 9659


Mixnumber


119888

(MPa)


(MPa)1 397 192 388 103 391 084 341 125 382 196 406 207 342 118 393 119 398 1610 279 1111 374 1812 385 1113 319 0814 371 1315 339 0216 265 1417 307 1718 365 1619 300 1520 321 1321 305 0822 207 1823 275 0824 299 0325 254 1126 310 0227 253 02










119862



119891

1015840

119888= minus6124 minus 0056119876

119862minus 1987Exp (2083119877wcm)

+ 18345119877

0119

FATA (1198772= 080)

(1)


119862is












1015840

119888(MPa)


119876119862

(kgm3)119877wcm

(by mass)119877FATA


119862

421 350 038 045 Low407 375 038 045 Medium393 400 038 045 High


119862

300 048 040350 350 044 042 Low400 040 044300 047 041350 375 043 042 Medium400 038 044300 046 041350 400 041 043 High400 038 045


5 Conclusions



25

30

35

40

45

035 038 041 044 047 05Rwcm (by mass)


f998400 c

(MPa

)


119862

25

30

35

40

45

035 038 041 044 047 05RFATA (by mass)

f998400 c

(MPa

)







Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































Mixnumber


(119877wcm)



(119877FATA)1 038 350 035

2 038 350 040

3 038 350 045

4 038 375 035

5 038 375 040

6 038 375 045

7 038 400 035

8 038 400 040

9 038 400 04510 043 350 03511 043 350 04012 043 350 04513 043 375 03514 043 375 04015 043 375 04516 043 400 03517 043 400 04018 043 400 04519 048 350 03520 048 350 04021 048 350 04522 048 375 03523 048 375 04024 048 375 04525 048 400 03526 048 400 04027 048 400 045








118mm 0060mm 2442030mm 9049015mm 9659


Mixnumber


119888

(MPa)


(MPa)1 397 192 388 103 391 084 341 125 382 196 406 207 342 118 393 119 398 1610 279 1111 374 1812 385 1113 319 0814 371 1315 339 0216 265 1417 307 1718 365 1619 300 1520 321 1321 305 0822 207 1823 275 0824 299 0325 254 1126 310 0227 253 02










119862



119891

1015840

119888= minus6124 minus 0056119876

119862minus 1987Exp (2083119877wcm)

+ 18345119877

0119

FATA (1198772= 080)

(1)


119862is












1015840

119888(MPa)


119876119862

(kgm3)119877wcm

(by mass)119877FATA


119862

421 350 038 045 Low407 375 038 045 Medium393 400 038 045 High


119862

300 048 040350 350 044 042 Low400 040 044300 047 041350 375 043 042 Medium400 038 044300 046 041350 400 041 043 High400 038 045


5 Conclusions



25

30

35

40

45

035 038 041 044 047 05Rwcm (by mass)


f998400 c

(MPa

)


119862

25

30

35

40

45

035 038 041 044 047 05RFATA (by mass)

f998400 c

(MPa

)







Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119862



119891

1015840

119888= minus6124 minus 0056119876

119862minus 1987Exp (2083119877wcm)

+ 18345119877

0119

FATA (1198772= 080)

(1)


119862is












1015840

119888(MPa)


119876119862

(kgm3)119877wcm

(by mass)119877FATA


119862

421 350 038 045 Low407 375 038 045 Medium393 400 038 045 High


119862

300 048 040350 350 044 042 Low400 040 044300 047 041350 375 043 042 Medium400 038 044300 046 041350 400 041 043 High400 038 045


5 Conclusions



25

30

35

40

45

035 038 041 044 047 05Rwcm (by mass)


f998400 c

(MPa

)


119862

25

30

35

40

45

035 038 041 044 047 05RFATA (by mass)

f998400 c

(MPa

)







Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1015840

119888(MPa)


119876119862

(kgm3)119877wcm

(by mass)119877FATA


119862

421 350 038 045 Low407 375 038 045 Medium393 400 038 045 High


119862

300 048 040350 350 044 042 Low400 040 044300 047 041350 375 043 042 Medium400 038 044300 046 041350 400 041 043 High400 038 045


5 Conclusions



25

30

35

40

45

035 038 041 044 047 05Rwcm (by mass)


f998400 c

(MPa

)


119862

25

30

35

40

45

035 038 041 044 047 05RFATA (by mass)

f998400 c

(MPa

)







Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A Statistical Approach to Optimizing...

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