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].

Types of concrete

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

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 [5].

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.

Many factors need to be taken into account, from the cost of the various additives and aggregates, to the trade-offs between, the “slump” for easy mixing and placement and ultimate performance [6].

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.

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 [7].

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

Level of factors and test X1 = C Cement kg/m3 X2= w water content kg/m3 X3 = Fa fine paragraph kg/m3 X4 = Ca coarse Aggregate kg/m0 Density of the Cube Kg/M3
Xnar Highest level (+) 300 7 690 1380
Xim Lowest level (-) 207 5 414 953
Xer Central Level (0) 254 6 552 1167
average
Interval of Change Δ 46 1 138 213
Test No X1 X2 X3 X4 Y3
1 207 5 414 953 88
2 207 7 690 953 109
3 207 5 690 953 160
4 207 5 690 953 156
5 300 7 414 953 65
6 300 5 690 1380 81
7 207 7 690 1380 99
8 207 7 690 1380 50
9 207 6 552 1167 67
10 300 7 552 1167 62
11 254 5 552 1167 82
12 254 7 552 1167 93
13 254 6 414 953 166
14 300 5 690 953 157
15 207 7 414 1380 110
16 254 6 552 1167 179
17 207 5 414 953 105
18 207 5 690 953 101
19 254 7 552 1167 95
20 254 5 552 1167 90
21 254 7 690 953 89
22 254 6 414 1167 102
23 254 6 552 1380 105
24 254 6 552 953 195
25 254 6 552 1167 165

Factorial Fit: Y3 versus X1, X2, X3, X4

Estimated Effects And Coefficients For Y3 (Coded Units)

Term Effect Coef SE Coef T P
Constant 7.5848 0.0317 239.58 0.000
X1 -0.0314 -0.0157 0.1009 -0.16 0.879
X2 -0.0308 -0.0154 0.0492 -0.31 0.761
X3 -0.1488 -0.0744 0.1761 -0.42 0.682
X4 -0.1386 -0.0693 0.0467 -1.48 0.169
X1*X2 -0.4074 -0.2037 0.1589 -1.28 0.229
X1*X3 10.9191 5.4595 20.0932 0.27 0.791
X1*X4 0.2456 0.1228 0.1957 0.63 0.544
X2*X3 11.1057 5.5529 20.1412 0.28 0.788
X2*X4 0.0821 0.0410 0.1394 0.29 0.774
X3*X4 -0.0088 -0.0044 0.1849 -0.02 0.981
X1*X2*X3 0.1153 0.0577 0.0607 0.95 0.365
X1*X2*X4 -0.1746 -0.0873 0.1674 -0.52 0.613
X1*X3*X4 10.8392 5.4196 19.9904 0.27 0.792
X2*X3*X4 10.9754 5.4877 20.2463 0.27 0.792

S = 0.0970879 PRESS = *

R-Sq = 81.42% R-Sq(pred) = *% R-Sq(adj) = 55.41%

Analysis of Variance for Y3 (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 4 0.239442 0.0504618 0.0126155 1.34 0.322
X1 1 0.0002281 0.0002281 0.02 0.879
X2 1 0.012115 0.0009233 0.0009233 0.10 0.761
X3 1 0.002923 0.0016820 0.0016820 0.18 0.682
X4 1 0.002375 0.0207384 0.0207384 2.20 0.169
2-Way Interactions 6 0.158639 0.0686466 0.0114411 1.21 0.374
X1*X2 1 0.077266 0.0154841 0.0154841 1.64 0.229
X1*X3 1 0.018271 0.0006959 0.0006959 0.07 0.791
X1*X4 1 0.008452 0.0037124 0.0037124 0.39 0.544
X2*X3 1 0.032792 0.0007165 0.0007165 0.08 0.788
X2*X4 1 0.014453 0.0008169 0.0008169 0.09 0.774
X3*X4 1 0.007405 0.0000053 0.0000053 0.00 0.981
3-Way Interactions 4 0.015035 0.0150346 0.0037587 0.40 0.805
X1*X2*X3 1 0.008813 0.0084955 0.0084955 0.90 0.365
X1*X2*X4 1 0.005528 0.0025632 0.0025632 0.27 0.613
X1*X3*X4 1 0.000001 0.0006928 0.0006928 0.07 0.792
X2*X3*X4 1 0.000693 0.0006925 0.0006925 0.07 0.792
Residual Error 10 0.094261 0.0942607 0.0094261
Lack of Fit 3 0.039694 0.0396940 0.0132313 1.70 0.254
Pure Error 7 0.054567 0.0545667 0.0077952
Total 24 0.507376

Obs StdOrder Y3 Fit SE Fit Residual St Resid
1 1 7.95 7.85 0.06865 0.1 1.46
2 2 7.75 7.75 0.09709 0 * X
3 3 7.65 7.61519 0.0559 0.03481 0.44
4 4 7.55 7.61519 0.0559 -0.06519 -0.82 ;
5 5 7.25 7.25 0.09709 0 * X
6 6 7.59 7.59 0.09709 0 * X
7 7 7.65 7.75 0.06865 -0.1 -1.46
8 8 7.85 7.75 0.06865 0.1 1.46
9 9 7.6 7.6 0.09709 0 * X
10 10 7.35 7.35 0.09709 0 * X
11 11 7.52 7.60193 0.05816 -0.08193 -1.05
12 12 7.45 7.56693 0.05816 -0.11693 -1.5
13 13 7.74 7.72212 0.0835 0.01788 0.36
14 14 7.54 7.53548 0.09628 0.00452 0.36
15 15 7.7 7.7 0.09709 0 * X
16 16 7.65 7.58443 0.03203 0.06557 0.72
17 17 7.75 7.85 0.06865 -0.1 -1.46
18 18 7.65 7.61519 0.0559 0.03481 0.44
19 19 7.55 7.56693 0.05816 -0.01693 -0.22
20 20 7.55 7.60193 0.05816 -0.05193 -0.67
21 21 7.6 7.59106 0.09388 0.00894 0.36
22 22 7.6 7.6 0.09709 0 * X
23 23 7.55 7.51663 0.06704 0.03337 0.48
24 24 7.65 7.65255 0.0453 -0.00255 -0.03
25 25 7.72 7.58443 0.03203 0.13557 1.48

X denotes an observation whose X value gives it large leverage.

Estimated Coefficients for Y3 using data in uncoded units

Term Coef
Constant -1098.25
X1 2.05727
X2 97.249
X3 2.02959
X4 1.15253
X1*X2 0.0009183
X1*X3 -0.0038176
X1*X4 0.00211848
X2*X3 -0.179310
X2*X4 -0.100393
X3*X4 -0.00212050
X1*X2*X3 8.98695E-06
X1*X2*X4 -8.79495E-06
X1*X3*X4 3.95582E-06
X2*X3*X4 0.00018625

Least Squares Means for Y3

Mean SE Mean
X1 207 7.6 0.0969
300 7.569 0.1139
X2 5 7.6 0.0581
7 7.569 0.0589
X3 414 7.659 0.1924
690 7.51 0.1644
953 7.654 0.0451
1380 7.515 0.0659
X1*X2 207 5 7.412 0.2471
300 5 7.788 0.2473
207 7 7.789 0.1542
300 7 7.35 0.0969
X1*X3 207 414 13.134 20.2069
300 414 2.184 19.8632
207 690 2.067 20.2917
300 690 12.954 20.0124
X1*X4 207 953 7.793 0.1588
300 953 7.516 0.1695
207 1380 7.408 0.25
300 1380 7.623 0.3002
X2*X3 5 414 13.227 20.3012
7 414 2.091 19.9571
5 690 1.973 20.2933
7 690 13.048 20.014
X2*X4 5 953 7.71 0.1714
7 953 7.598 0.1691
5 1380 7.49 0.1692
7 1380 7.541 0.1164
X3*X4 414 953 7.724 0.083
690 953 7.584 0.0531
414 1380 7.594 0.3943
690 1380 7.437 0.3293
X1*X2*X3 207 5 414 18.441 40.2096
300 5 414 8.014 0.4982
207 7 414 7.827 0.3141
300 7 414 -3.646 40.1173
07 5 690 -3.617 40.5807
300 5 690 7.563 0.0684
207 7 690 7.75 0.0595
300 7 690 18.346 40.0307
X1*X2*X4 207 5 953 7.733 0.0443
300 5 953 7.688 0.3441
207 7 953 7.852 0.3141
300 7 953 7.343 0.1151
207 5 1380 7.092 0.4965
300 5 1380 7.888 0.6938
207 7 1380 7.725 0.0595
300 7 1380 7.357 0.225
X1*X3*X4 207 414 953 7.902 0.3197
300 414 953 7.546 0.34
207 690 953 7.683 0.056
300 690 953 7.485 0.1136
207 414 1380 18.366 40.2069
300 414 1380 -3.178 39.5193
207 690 1380 -3.55 40.5832
300 690 1380 18.423 40.0267
X2*X3*X4 5 414 953 7.846 0.3418
7 414 953 7.602 0.3215
5 690 953 7.575 0.0554
7 690 953 7.593 0.0929
5 414 1380 18.609 40.8132
7 414 1380 -3.421 40.1227
5 690 1380 -3.63 40.586
7 690 1380 18.503 40.0295

Predicted Response for New Design Points Using Model for Y3

Point Fit SE Fit 95% CI 95% PI
1 7.85 0.06865 (7.69703, 8.00297) (7.58506, 8.11494)
2 7.75 0.09709 (7.53367, 7.96633) (7.44407, 8.05593)
3 7.61519 0.0559 (7.49063, 7.73976) (7.36557, 7.86482)
4 7.61519 0.0559 (7.49063, 7.73976) (7.36557, 7.86482)
5 7.25 0.09709 (7.03367, 7.46633) (6.94407, 7.55593)
6 7.59 0.09709 (7.37367, 7.80633) (7.28407, 7.89593)
7 7.75 0.06865 (7.59703, 7.90297) (7.48506, 8.01494)
8 7.75 0.06865 (7.59703, 7.90297) (7.48506, 8.01494)
9 7.6 0.09709 (7.38367, 7.81633) (7.29407, 7.90593)
10 7.35 0.09709 (7.13367, 7.56633) (7.04407, 7.65593)
11 7.60193 0.05816 (7.47235, 7.73151) (7.34977, 7.85410)
12 7.57E+00 0.05816 (7.43735, 7.69651) (7.31477, 7.81910)
13 7.72E+00 0.0835 (7.53607, 7.90817) (7.43679, 8.00744)
14 7.54E+00 0.09628 (7.32096, 7.75000) (7.23082, 7.84014)
15 7.7 0.09709 (7.48367, 7.91633) (7.39407, 8.00593)
16 7.58443 0.03203 (7.51307, 7.65579) (7.35664, 7.81222)
17 7.85 0.06865 (7.69703, 8.00297) (7.58506, 8.11494)
18 7.61519 0.0559 (7.49063, 7.73976) (7.36557, 7.86482)
19 7.56693 0.05816 (7.43735, 7.69651) (7.31477, 7.81910)
20 7.60193 0.05816 (7.47235, 7.73151) (7.34977, 7.85410)
21 7.59106 0.09388 (7.38189, 7.80023) (7.29015, 7.89197)
22 7.6 0.09709 (7.38367, 7.81633) (7.29407, 7.90593)
23 7.51663 0.06704 (7.36725, 7.66601) (7.25374, 7.77952)
24 7.65255 0.0453 (7.55162, 7.75348) (7.41384, 7.89126)
25 7.58443 0.03203 (7.51307, 7.65579) (7.35664, 7.81222)

X denotes a point that is an outlier in the predictors.

Values of Predictors for New Observations

New Obs X1 X2 X3 X4
1 207 5 414 953
2 207 7 690 953
3 207 5 690 953
4 207 5 690 953
5 300 7 414 953
6 300 5 690 1380
7 207 7 690 1380
8 207 7 690 1380
9 207 6 552 1167
10 300 7 552 1167
11 254 5 552 1167
12 254 7 552 1167
13 254 6 414 953
14 300 5 690 953
15 207 7 414 1380
16 254 6 552 1167
17 207 5 414 953
18 207 5 690 953
19 254 7 552 1167
20 254 5 552 1167
21 254 7 690 953
22 254 6 414 1167
23 254 6 552 1380
24 254 6 552 953
25 254 6 552 1167

Figure 1: Effects Plot for Y3

Figure 2: Effects Pareto for Y3

Figure 3: Residual Plots For Y3

Figure 4: Main Effects Plot for Y3

Figure 5: Interaction Plot for Y3

Figure 6: Cube Plot (data means) for Y3

Figure 7: Contour Plots of Y3

Figure 8: Surface Plots of Y3

Table 2: Factorial Design table for weight of cube kg

S/N FITS3 RESI3 COEF3 EFFE3
1 7.85 0.1 7.584759 -0.03138
2 7.75 -2.5E-14 -0.01569 -0.03081
3 7.615192 0.034808 -0.0154 -0.1488
4 7.615192 -0.06519 -0.0744 -0.13856
5 7.25 -1.2E-14 -0.06928 -0.40736
6 7.59 7.11E-15 -0.20368 10.91905
7 7.75 -0.1 5.459527 0.245588
8 7.75 0.1 0.122794 11.10574
9 7.6 -8.9E-15 5.552869 0.082068
10 7.35 -1.2E-14 0.041034 -0.00881
11 7.601931 -0.08193 -0.0044 0.115339
12 7.566931 -0.11693 0.057669 -0.17463
13 7.722117 0.017883 -0.08731 10.83919
14 7.535481 0.004519 5.419594 10.97545
15 7.7 1.69E-14 5.487725
16 7.584431 0.065569
17 7.85 -0.1
18 7.615192 0.034808
19 7.566931 -0.01693
20 7.601931 -0.05193
21 7.591059 0.008941
22 7.6 -1.2E-14
23 7.51663 0.03337
24 7.652551 -0.00255
25 7.584431 0.135569

Response optimization

Parameters

Goal Lower Target Upper Weight Import

Y3 Target 7 7.6164 8 1 1

Local Solution

X1 = 207

X2 = 5.02212

X3 = 688.766

X4 = 953.026

Predicted Responses

Y3 = 7.61640 , desirability = 0.999989

Composite Desirability = 0.999989

Local Solution

X1 = 300

X2 = 5.00244

X3 = 690.000

X4 = 1380

Predicted Responses

Y3 = 7.61639 , desirability = 0.999979

Composite Desirability = 0.999979

Local Solution

X1 = 253.5

X2 = 6

X3 = 552

X4 = 1068.99

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Local Solution

X1 = 299.812

X2 = 5

X3 = 689.994

X4 = 1379.58

Predicted Responses

Y3 = 7.54457 , desirability = 0.883468

Composite Desirability = 0.883468

Local Solution

X1 = 207.412

X2 = 7

X3 = 414.008

X4 = 1359.54

Predicted Responses

Y3 = 7.61820, desirability = 0.995301

Composite Desirability = 0.995301

Local Solution

X1 = 299.877

X2 = 6.88675

X3 = 689.992

X4 = 956.630

Predicted Responses

Y3 = 7.61631 , desirability = 0.999849

Composite Desirability = 0.999849

Local Solution

X1 = 207

X2 = 6.98818

X3 = 690.000

X4 = 1380

Predicted Responses

Y3 = 7.61646 , desirability = 0.999854

Composite Desirability = 0.999854

Local Solution

X1 = 207.014

X2 = 5.00734

X3 = 690

X4 = 953.014

Predicted Responses

Y3 = 7.61497, desirability = 0.997675

Composite Desirability = 0.997675

Local Solution

X1 = 300

X2 = 5.76055

X3 = 414.003

X4 = 953

Predicted Responses

Y3 = 7.61640 , desirability = 0.999991

Composite Desirability = 0.999991

Local Solution

X1 = 292.875

X2 = 5.83366

X3 = 417.141

X4 = 953

Predicted Responses

Y3 = 7.61644 , desirability = 0.999896

Composite Desirability = 0.999896

Local Solution

X1 = 207.006

X2 = 5.00063

X3 = 688.624

X4 = 953

Predicted Responses

Y3 = 7.61640 , desirability = 0.999999

Composite Desirability = 0.999999

Local Solution

X1 = 209.353

X2 = 5.83676

X3 = 414.230

X4 = 953

Predicted Responses

Y3 = 7.88612 , desirability = 0.296881

Composite Desirability = 0.296881

Local Solution

X1 = 300

X2 = 5.01973

X3 = 614.875

X4 = 953

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Local Solution

X1 = 299.960

X2 = 5.10018

X3 = 414.027

X4 = 1063.40

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Local Solution

X1 = 207.041

X2 = 6.93777

X3 = 689.973

X4 = 1033.11

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Local Solution

X1 = 207.041

X2 = 5.01846

X3 = 690

X4 = 953

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Local Solution

X1 = 299.958

X2 = 5.03567

X3 = 689.971

X4 = 1034.59

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Global Solution

X1 = 300

X2 = 5.01973

X3 = 614.875

X4 = 953

Predicted Responses

Y3 = 7.6164 , desirability = 1.000000

Composite Desirability = 1.000000

Figure 9: Optimization Plot

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.