www.advancejournals.org Open Access Scientific Publisher Research Article FACTORIAL DESIGN AND OPTIMIZATION OF THE WEIGHT OF THE CUBE (KG) IN CONCRETE MIXTURE ABSTRACT Ejikeme Ifeanyi R 1, Ezeliora Chukwuemeka Daniel 2 1 Commissioner for Transport, Anambra State, Nigeria 2 Department of Mechanical Engineering, Nnamdi Azikiwe University Awka, Anambra State, Nigeria Correspondence should be addressed to Ezeliora Chukwuemeka Daniel Received August 10, 201; Accepted August 1, 201; Published September 0, 201; Copyright: 201 Ezeliora Chukwuemeka Daniel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Cite This Article: Ifeanyi R, E., Daniel, E.(201). Factorial Design and Optimization of the weight of the cube (kg) in Concrete Mixture. Advances in Engineering & Scientific Research, 1(1).1-1 This research work focused on the design, and optimization of the concrete mixture using factorial analysis. Data were collected for the concrete mixture proportion to observe the ratio of the mixture content. Factorial design analyses were used to design the concrete mixture. It was also used to optimize the concrete mixture of the variables. The results were discussed and were also recommended for concrete mixture. KEY WORDS: Optimization, Factorial Design, Concrete Mix, Response Surface Modelling and Analysis of Variance. INTRODUCTION INTRODUCTION TO CONCRETE MIXING The successful placement of concrete is dependent upon careful mixing, the proper equipment, and adequate transportation. This will define, analyze, and demonstrate the importance of each in the overall process of placing concrete. BATCHING AND MIXING CONCRETE Mixing concrete is simply defined as the "complete blending of the materials which are required for the production of a homogeneous concrete" [1]. This can vary from hand to machine mixing, with machine mixing being the most common. However, no successful mixture can be achieved without the proper batching of all materials. Batching is the "process of weighing or volumetrically measuring and introducing into the mixer the ingredients for a batch of concrete" [2]. Quality assurance, suitable arrangement of materials and equipment, and correct weighing of the materials are the essential steps that must be completed before any mixing takes place. The objective of the stud is to design, analyze and to optimize the weight of cube in concrete mixture content. CONCRETE MIXER A concrete mixer (also commonly called a cement mixer) is a device that homogeneously combines cement, aggregate such as sand or gravel, and water to form concrete. A typical concrete mixer uses a revolving drum to mix the components. For smaller volume works portable concrete mixers are often used so that the concrete can be made at the construction site, giving the workers ample time to use the concrete before it hardens. An alternative to a machine is mixing concrete by hand. This is usually done in a wheelbarrow; however, several companies have recently begun to sell modified tarps for this purpose [3]. 1
TYPES OF CONCRETE Advances in Engineering & Scientific Research There are many types of concrete, designed to suit a variety of purposes coupled with a range of compositions, finishes and performance characteristics [4]. MIX DESIGN between, the "slump" for easy mixing and placement and ultimate performance []. A mix is then designed using cement (Portland or other cementitious material), coarse and fine aggregates, water and chemical admixtures. The method of mixing will also be specified, as well as conditions that it may be used in. Modern concrete mix designs can be complex. The choice of a concrete mix depends on the need of the project both in terms of strength and appearance and in relation to local legislation and building codes []. The design begins by determining the requirements of the concrete. These requirements take into consideration the weather conditions that the concrete will be exposed to in service, and the required design strength. The compressive strength of a concrete is determined by taking standard melded, standard-cured cylinder samples. This allows a user of the concrete to be confident that the structure will perform properly. Various types of concrete have been developed for specialist application and have become known by these names. Concrete mixes can also be designed using software programs. Such software provides the user an opportunity to select their preferred method of mix design and enter the material data to arrive at proper mix designs []. Many factors need to be taken into account, from the cost of the various additives and aggregates, to the trade-offs The research method used is the analyses and optimization of the weight of cube in concrete mixture. Data, Analyses and Results Table 1: Level of factors and test for weight of cube kg 2 Level of factors and test Xnar Highest level (+) Xim Lowest level (-) Xer Central Level (0) average Interval of Change Δ X 1 = C Cement kg/m 3 20 24 X 2 = w water content kg/m 3 X 3 = Fa fine paragraph kg/m 3 90 414 2 X 4 = Ca coarse Aggregate kg/m 0 1380 93 11 Density of the Cube Kg/M 3 4 1 138 213 Test No X 1 X 2 X 3 X 4 Y3 1 20 414 93 88 2 20 90 93 109 3 20 90 93 10 4 20 90 93 1 414 93 90 1380 81 20 90 1380 99 8 20 90 1380 0 9 20 2 11 10 2 11 2 11 24 2 11 82 12 24 2 11 93 13 24 414 93 1 14 90 93 1 1 20 414 1380 110 1 24 2 11 19 1 20 414 93 10 18 20 90 93 101 19 24 2 11 9 20 24 2 11 90 21 24 90 93 89
22 24 414 11 102 23 24 2 1380 10 24 24 2 93 19 2 24 2 11 1 Factorial Fit: Y3 versus X1, X2, X3, X4 Estimated Effects And Coefficients For Y3 (Coded Units) Term Effect Coef SE Coef T P Constant.848 0.031 239.8 0.000 X1-0.0314-0.01 0.1009-0.1 0.89 X2-0.0308-0.014 0.0492-0.31 0.1 X3-0.1488-0.044 0.11-0.42 0.82 X4-0.138-0.093 0.04-1.48 0.19 X1*X2-0.404-0.203 0.189-1.28 0.229 X1*X3 10.9191.49 20.0932 0.2 0.91 X1*X4 0.24 0.1228 0.19 0.3 0.44 X2*X3 11.10.29 20.1412 0.28 0.88 X2*X4 0.0821 0.0410 0.1394 0.29 0.4 X3*X4-0.0088-0.0044 0.1849-0.02 0.981 X1*X2*X3 0.113 0.0 0.00 0.9 0.3 X1*X2*X4-0.14-0.083 0.14-0.2 0.13 X1*X3*X4 10.8392.419 19.9904 0.2 0.92 X2*X3*X4 10.94.48 20.243 0.2 0.92 S = 0.09089 PRESS = * R-Sq = 81.42% R-Sq(pred) = *% R-Sq(adj) =.41% 3
Analysis of Variance for Y3 (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 4 0.239442 0.00418 0.0121 1.34 0.322 X1 1 0.0002281 0.0002281 0.02 0.89 X2 1 0.01211 0.0009233 0.0009233 0.10 0.1 X3 1 0.002923 0.001820 0.001820 0.18 0.82 X4 1 0.0023 0.020384 0.020384 2.20 0.19 2-Way Interactions 0.1839 0.084 0.0114411 1.21 0.34 X1*X2 1 0.02 0.014841 0.014841 1.4 0.229 X1*X3 1 0.01821 0.00099 0.00099 0.0 0.91 X1*X4 1 0.00842 0.003124 0.003124 0.39 0.44 X2*X3 1 0.03292 0.0001 0.0001 0.08 0.88 X2*X4 1 0.01443 0.000819 0.000819 0.09 0.4 X3*X4 1 0.0040 0.000003 0.000003 0.00 0.981 3-Way Interactions 4 0.0103 0.01034 0.0038 0.40 0.80 X1*X2*X3 1 0.008813 0.00849 0.00849 0.90 0.3 X1*X2*X4 1 0.0028 0.00232 0.00232 0.2 0.13 X1*X3*X4 1 0.000001 0.000928 0.000928 0.0 0.92 X2*X3*X4 1 0.00093 0.00092 0.00092 0.0 0.92 Residual Error 10 0.09421 0.09420 0.009421 Lack of Fit 3 0.03994 0.039940 0.0132313 1.0 0.24 Pure Error 0.04 0.04 0.0092 4 Total 24 0.03
Obs StdOrder Y3 Fit SE Fit Residual St Resid 1 1.9000.8000 0.08 0.10000 1.4 2 2.000.000 0.0909-0.00000 * X 3 3.000.119 0.090 0.03481 0.44 4 4.000.119 0.090-0.019-0.82.2000.2000 0.0909-0.00000 * X.9000.9000 0.0909 0.00000 * X.000.000 0.08-0.10000-1.4 8 8.8000.000 0.08 0.10000 1.4 9 9.0000.0000 0.0909-0.00000 * X 10 10.0.0 0.0909-0.00000 * X 11 11.2000.0193 0.081-0.08193-1.0 12 12.4000.93 0.081-0.1193-1.0 13 13.4000.2212 0.0830 0.0188 0.3 14 14.4000.348 0.0928 0.0042 0.3 1 1.0000.0000 0.0909 0.00000 * X 1 1.000.8443 0.03203 0.0 0.2 1 1.000.8000 0.08-0.10000-1.4 18 18.000.119 0.090 0.03481 0.44 19 19.000.93 0.081-0.0193-0.22 20 20.000.0193 0.081-0.0193-0. 21 21.0000.910 0.09388 0.00894 0.3 22 22.0000.0000 0.0909-0.00000 * X 23 23.000.13 0.004 0.0333 0.48 24 24.000.2 0.0430-0.002-0.03 2 2.2000.8443 0.03203 0.13 1.48 X denotes an observation whose X value gives it large leverage.
Estimated Coefficients for Y3 using data in uncoded units Term Constant -1098.2 X1 2.02 X2 9.249 X3 2.0299 X4 1.123 X1*X2 0.0009183 X1*X3-0.00381 Coef X1*X4 0.00211848 X2*X3-0.19310 X2*X4-0.100393 X3*X4-0.0021200 X1*X2*X3 8.989E-0 X1*X2*X4 X1*X3*X4-8.949E-0 3.982E-0 X2*X3*X4 0.0001828 Least Squares Means for Y3 Mean SE Mean X1 20.00 0.099.9 0.1139 X2.00 0.081.9 0.089 X3 414.9 0.1924 90.10 0.144 93.4 0.041 1380.1 0.09 X1*X2 20.412 0.241.88 0.243 20.89 0.142.30 0.099 X1*X3 20 414 13.134 20.209 414 2.184 19.832 20 90 2.0 20.291 90 12.94 20.0124 X1*X4 20 93.93 0.188 93.1 0.19 20 1380.408 0.200 1380.23 0.2 X2*X3 414 13.22 20.3012 414 2.091 19.91 90 1.93 20.2933 90 13.048 20.0140 X2*X4 93.10 0.114 93.98 0.191 1380.490 0.192 1380.41 0.114 X3*X4 414 93.24 0.0830 90 93.84 0.031
414 1380.94 0.3943 90 1380.43 0.3293 X1*X2*X3 20 414 18.441 40.209 414 8.014 0.4982 20 414.82 0.3141 414-3.4 40.113 0 90-3.1 40.80 90.3 0.084 20 90.0 0.09 90 18.34 40.030 X1*X2*X4 20 93.33 0.0443 93.88 0.3441 20 93.82 0.3141 93.343 0.111 20 1380.092 0.49 1380.888 0.938 20 1380.2 0.09 1380.3 0.220 X1*X3*X4 20 414 93.902 0.319 414 93.4 0.3400 20 90 93.83 0.00 90 93.48 0.113 20 414 1380 18.3 40.209 414 1380-3.18 39.193 20 90 1380-3.0 40.832 90 1380 18.423 40.02 X2*X3*X4 414 93.84 0.3418 414 93.02 0.321 90 93. 0.04 90 93.93 0.0929 414 1380 18.09 40.8132 414 1380-3.421 40.122 90 1380-3.30 40.80 90 1380 18.03 40.029
Predicted Response for New Design Points Using Model for Y3 Point Fit SE Fit 9% CI 9% PI 1.8000 0.08 (.903, 8.0029) (.80, 8.11494) 2.000 0.0909 (.33,.933) (.4440, 8.093) X 3.119 0.090 (.4903,.39) (.3,.8482) 4.119 0.090 (.4903,.39) (.3,.8482).2000 0.0909 (.033,.433) (.9440,.93) X.9000 0.0909 (.33,.8033) (.2840,.8993) X.000 0.08 (.903,.9029) (.480, 8.01494) 8.000 0.08 (.903,.9029) (.480, 8.01494) 9.0000 0.0909 (.383,.8133) (.2940,.9093) X 10.0 0.0909 (.133,.33) (.0440,.93) X 11.0193 0.081 (.423,.311) (.349,.8410) 12.93 0.081 (.433,.91) (.314,.81910) 13.2212 0.0830 (.30,.9081) (.439, 8.0044) 14.348 0.0928 (.3209,.000) (.23082,.84014) 1.0000 0.0909 (.483,.9133) (.3940, 8.0093) X 1.8443 0.03203 (.130,.9) (.34,.81222) 1.8000 0.08 (.903, 8.0029) (.80, 8.11494) 18.119 0.090 (.4903,.39) (.3,.8482) 19.93 0.081 (.433,.91) (.314,.81910) 20.0193 0.081 (.423,.311) (.349,.8410) 21.910 0.09388 (.38189,.80023) (.2901,.8919) 22.0000 0.0909 (.383,.8133) (.2940,.9093) X 23.13 0.004 (.32,.01) (.234,.92) 24.2 0.0430 (.12,.348) (.41384,.8912) 2.8443 0.03203 (.130,.9) (.34,.81222) X denotes a point that is an outlier in the predictors. Values of Predictors for New Observations 8 New Obs X1 X2 X3 X4 1 20 414 93 2 20 90 93 3 20 90 93 4 20 90 93 414 93 90 1380 20 90 1380 8 20 90 1380 9 20 2 11 10 2 11 11 24 2 11 12 24 2 11 13 24 414 93 14 90 93 1 20 414 1380
Term Percent Advances in Engineering & Scientific Research 1 24 2 11 1 20 414 93 18 20 90 93 19 24 2 11 20 24 2 11 21 24 90 93 22 24 414 11 23 24 2 1380 24 24 2 93 2 24 2 11 Figure 1: Effects Plot for Y3 99 9 90 80 0 0 0 40 30 20 10 Normal Plot of the Standardized Effects (response is Y3, Alpha = 0.0) Effect Type Not Significant Significant Factor Name A X1 B X2 C X3 D X4 1-3 -2-1 0 1 Standardized Effect 2 3 Figure 2: Effects Pareto for Y3 D AB ABC AD ABD C B BD BC AC ACD BCD A CD 0.0 Pareto Chart of the Standardized Effects (response is Y3, Alpha = 0.0) 2.228 0. 1.0 1. Standardized Effect 2.0 2. Factor Name A X1 B X2 C X3 D X4 9
Mean Frequency Residual Percent Residual Advances in Engineering & Scientific Research Figure 3: Residual Plots For Y3 99 90 Normal Probability Plot Residual Plots for Y3 0.1 Versus Fits 0 0.0 10 1-0.1 0.0 Residual 0.1-0.1.20.3.0. Fitted Value.80 12 9 Histogram 0.1 Versus Order 0.0 3 0-0.12-0.08-0.04 0.00 0.04 Residual 0.08 0.12-0.1 2 4 8 10 12 14 1 18 Observation Order 20 22 24 Figure 4: Main Effects Plot for Y3... X1 Main Effects Plot for Y3 Data Means X2.4. 20 X3 X4...4 414 90 93 1380 10
Figure : Interaction Plot for Y3 Interaction Plot for Y3 Data Means 93 1380 X1.8..4 X1 20.8..4 X2 X2 X3.8..4 X3 414 90.8..4 X4 X4 93 1380 20 414 90 Figure : Cube Plot (data means) for Y3 Cube Plot (data means) for Y3.000.000.2000.0000 X2.1.4000 90.9000.8000 20 X1 X3 414 93 X4 1380 11
Figure : Contour Plots of Y3 Contour Plots of Y3.0..0..0 0 00 0 00 40 220 X2*X1 20 X3*X2 0 00 0 00 40 1 1200 1100 1000 220 X3*X1 20 X4*X2 1 1200 1100 1000 1 1200 1100 1000 220 X4*X1 20 X4*X3 Y3 < 2 2 4 4 8 8 10 10 12 > 12 Hold Values X1 23. X2 X3 2 X4 11 480 0 40 Figure 8: Surface Plots of Y3 Surface Plots of Y3.8 Y3..4 200 20 X1 X2 12 Y3 9 20 3 00 200 480 X3 20 X1.8 Y3. 1400.4 1200 X4 200 20 1000 X1 Hold Values X1 23. X2 X3 2 X4 11 12 Y3 9 3 X2 20 00 480 X3. Y3. 1400. 1200 X4 1000 X2. Y3.. 1400.4 1200 X4 480 00 1000 X3 20 12
Table 2: Factorial Design table for weight of cube kg S/N FITS3 RESI3 COEF3 EFFE3 1.8 0.1.849-0.03138 2. -2.E-14-0.019-0.03081 3.1192 0.034808-0.014-0.1488 4.1192-0.019-0.044-0.138.2-1.2E-14-0.0928-0.403.9.11E-1-0.2038 10.9190. -0.1.492 0.2488 8. 0.1 0.12294 11.104 9. -8.9E-1.289 0.08208 10.3-1.2E-14 0.041034-0.00881 11.01931-0.08193-0.0044 0.11339 12.931-0.1193 0.09-0.143 13.2211 0.01883-0.0831 10.83919 14.3481 0.00419.41994 10.94 1. 1.9E-14.482 1.84431 0.09 1.8-0.1 18.1192 0.034808 19.931-0.0193 20.01931-0.0193 21.9109 0.008941 22. -1.2E-14 23.13 0.0333 24.21-0.002 2.84431 0.139 RESPONSE OPTIMIZATION Parameters Goal Lower Target Upper Weight Import Y3 Target.14 8 1 1 X1 = 20 X2 =.02212 X3 = 88. X4 = 93.02 Y3 =.140, desirability = 0.999989 Composite Desirability = 0.999989 X1 = X2 =.00244 X3 = 90.000 13
X4 = 1380 Advances in Engineering & Scientific Research X1 = 299.8 X2 =.88 14 Y3 =.139, desirability = 0.99999 Composite Desirability = 0.99999 X1 = 23. X2 = X3 = 2 X4 = 108.99 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 X1 = 299.812 X2 = X3 = 89.994 X4 = 139.8 Y3 =.44, desirability = 0.88348 Composite Desirability = 0.88348 X1 = 20.412 X2 = X3 = 414.008 X4 = 139.4 Y3 =.1820, desirability = 0.99301 Composite Desirability = 0.99301 X3 = 89.992 X4 = 9.30 Y3 =.131, desirability = 0.999849 Composite Desirability = 0.999849 X1 = 20 X2 =.98818 X3 = 90.000 X4 = 1380 Y3 =.14, desirability = 0.99984 Composite Desirability = 0.99984 X1 = 20.014 X2 =.0034 X3 = 90 X4 = 93.014 Y3 =.149, desirability = 0.99 Composite Desirability = 0.99 X1 = X2 =.0 X3 = 414.003 X4 = 93
Y3 =.140, desirability = 0.999991 Advances in Engineering & Scientific Research X3 = 14.8 Composite Desirability = 0.999991 X1 = 292.8 X2 =.833 X3 = 41.141 X4 = 93 Y3 =.144, desirability = 0.99989 Composite Desirability = 0.99989 X1 = 20.00 X2 =.0003 X3 = 88.24 X4 = 93 Y3 =.140, desirability = 0.999999 Composite Desirability = 0.999999 X1 = 209.33 X2 =.83 X3 = 414.230 X4 = 93 Y3 =.8812, desirability = 0.29881 Composite Desirability = 0.29881 X1 = X2 =.0193 X4 = 93 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 X1 = 299.90 X2 =.10018 X3 = 414.02 X4 = 103.40 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 X1 = 20.041 X2 =.93 X3 = 89.93 X4 = 1033.11 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 X1 = 20.041 X2 =.0184 X3 = 90 X4 = 93 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 1
X1 = 299.98 X2 =.03 X3 = 89.91 X4 = 1034.9 Advances in Engineering & Scientific Research Global Solution X1 = X2 =.0193 X3 = 14.8 X4 = 93 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 Y3 =.14, desirability = 1.000000 Composite Desirability = 1.000000 Figure 9: Optimization Plot Optimal D High Cur 1.0000 Low X1 X2 X3 X4.0.0 90.0 1380.0 [.0] [.019] [14.84] [93.0] 20.0.0 414.0 93.0 Composite Desirability 1.0000 Y3 Targ:.14 y =.14 d = 1.0000 1 DISCUSSION AND CONCLUSION From the result, a factorial design model was developed to show the new observed independent variables. It was observed that the coefficient of determination (R-sq) of the model developed was 81%. This shows a good correlation of both the dependent and independent variables. Further analyses were made to sow the effect of each of the independent variable to the model and the interaction of the independent variable. However, contour plots were also used to show the impact of the independent variables while the surface plot shows the area of the variables (both dependent and independent variables). Furthermore, an optimization response technique was also applied to observe the global response or the optimum response of both the dependent and independent variables.in conclusion, the application of the factorial design technique shows us a means of designing and optimizing the weight of the cube in concrete mixture. This will help and also serve as a guide line for standard concrete mixture. The research work is also recommended for wider use and applicability for concrete mixture in establishments. REFERENCES [1] Mindess, Sidney and J. Francis Young. Concrete. New Jersey: Prentice-Hall, Inc., 1981.
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