A stepbystep statistical approach is proposed to obtain optimum proportioning of concrete mixtures using the data obtained through a statistically planned experimental program. The utility of the proposed approach for optimizing the design of concrete mixture is illustrated considering a typical case in which trial mixtures were considered according to a full factorial experiment design involving three factors and their three levels (3^{3}). A total of 27 concrete mixtures with three replicates (81 specimens) were considered by varying the levels of key factors affecting compressive strength of concrete, namely, water/cementitious materials ratio (0.38, 0.43, and 0.48), cementitious materials content (350, 375, and 400 kg/m^{3}), and fine/total aggregate ratio (0.35, 0.40, and 0.45). The experimental data were utilized to carry out analysis of variance (ANOVA) and to develop a polynomial regression model for compressive strength in terms of the three design factors considered in this study. The developed statistical model was used to show how optimization of concrete mixtures can be carried out with different possible options.
Optimization of the concrete mixture design is a process of search for a mixture for which the sum of the costs of the ingredients is lowest, yet satisfying the required performance of concrete, such as workability strength and durability. The basic ingredients of concrete can be classified into two groups: cement paste and aggregates. Although the quality of cement paste is governed mainly by the water/cement ratio, the quantity of cement paste required to achieve a targeted quality of concrete depends on the characteristics of aggregates. These characteristics mainly include surface area and voids in aggregates. While surface area is governed by the shape and maximum size of aggregates, the void content is affected mainly by the particle size distribution of aggregates. The requirement of the paste can be reduced by reducing the void content of aggregates through proper packing of the aggregates [
Attempts have been made in the past to optimize the concrete mixture design using either the fully experimental methods or fully analytical methods or semiexperimental (halfanalytical) methods or statistical methods. Fully experimental methods involve an extensive series of tests, sometimes conducted on a trialanderror basis, and the optimization results are often applicable only to a narrow range of local materials [
Fully analytical methods are less expensive and less time consuming but they have the disadvantage of being less precise because of the variations in the materials characteristics of the aggregates and cements. Fully experimental or semiexperimental (i.e., halfanalytical) methods are reliable and accurate; however, they involve comprehensive laboratory works [
In the present work, an effort has been made to exhibit the application of a statistical approach proposed to obtain optimum proportioning of concrete mixtures using the data obtained through an experiment design considering watercementitious materials ratio, cementitious materials content, and fine aggregate to total aggregate ratio as design factors. The experimental data were analyzed statistically and mathematical polynomials regression was developed for concrete strength as a function of mixture variables. The utility of the developed compressive strength model in optimizing the mixture designs was illustrated considering different possible options.
The proposed approach to optimizing the proportions of concrete mixtures is based on the planned experimental works (within the domain of required characteristic performance of concrete) and statistical analysis of the data generated, which would reduce the number of trial batches needed. The proposed approach consists of the following steps.
The information pertaining to required workability, strength, and exposure conditions (for durability requirements) should be first collected. The workability requirements depend on the mode of transportation, handling, and placing and also on type of construction [
Selection of the levels of the three key mixture design factors, namely, cementitious materials content, water/cementitious materials ratio, and fine/total aggregate ratio, which mainly affect the quality of concrete will be made to ensure that enough experimental data are generated for obtaining a regression model for compressive strength which can be used to optimize the mixture proportions meeting the specified characteristic performance of concrete.
The minimum level of cementitious materials content should not be less than 335 kg/m^{3} which is the minimum value to satisfy the durability requirements for aggressive exposure conditions. The maximum level of cementitious materials content should be selected considering the risk of shrinkage. The minimum level of water/cementitious materials ratio should be selected considering the strength requirements. In case of choosing a very low level of the water/cementitious materials ratio, the difficulty in transporting, handling, and placing concrete and extra cost of superplasticizer to meet the workability requirements should be considered. The maximum level of the water/cementitious materials ratio should be within the maximum permissible limit for the water/cementitious materials ratio for the given exposure condition. The minimum and maximum levels of the fine/total aggregate ratio should be selected within the optimum range for achieving maximum packing of aggregates. For example, Soudki et al. [
An experimental work should be conducted involving designing, preparing, and testing various trial mixtures according to the full factorial experiment design considering the various possible combinations of the levels of the mixture variables within their selected ranges of variation. The workability of each trial mixture should be equal to or more than the specified value. In case if superplasticizer is needed to achieve the intended workability, the cost of superplasticizer should be added to the cost of cement. After finalizing the dosage of superplasticizer based on the required workability for each of the trial mixtures, the cubical or cylindrical specimens should be prepared, cured for 28 days, and then tested for compressive strength for generating data to obtain statistical model for strength to be used for optimization.
Analysis of variance (ANOVA) can be used for examining the significance of the factors considered for developing the strength model and subsequently fitting an empirical model for compressive strength in terms of the significant mixture factors using polynomial regression. In ANOVA, the statistical terminologies used are as follows.
The statistical model for the compressive strength derived utilizing the experimental model can be used to obtain the optimal mixture proportions satisfying the specified characteristic performance of concrete as required constraints. The mixture satisfying all the constraints and having the lowest requirements of cement and superplasticizer would be considered as optimum mixture.
For illustrating the utilization of the proposed approach to optimizing concrete mixture design, an experimental program was considered. A full factorial experiment with
Test program.
Factor  1  2  3  Level 

Cementitious materials content ( 
350  375  400  3 


Water/cementitious materials ratio ( 
0.38  0.43  0.48  3 


Fine/total aggregate ratio ( 
0.35  0.40  0.45  3 
Trial mixtures.
Mix number 
Water/cementitious materials ratio ( 
Cementitious materials content, 
Fine/total aggregate ratio ( 

1  0.38  350  0.35 
2  0.38  350  0.40 
3  0.38  350  0.45 
4  0.38  375  0.35 
5  0.38  375  0.40 
6  0.38  375  0.45 
7  0.38  400  0.35 
8  0.38  400  0.40 
9  0.38  400  0.45 
10  0.43  350  0.35 
11  0.43  350  0.40 
12  0.43  350  0.45 
13  0.43  375  0.35 
14  0.43  375  0.40 
15  0.43  375  0.45 
16  0.43  400  0.35 
17  0.43  400  0.40 
18  0.43  400  0.45 
19  0.48  350  0.35 
20  0.48  350  0.40 
21  0.48  350  0.45 
22  0.48  375  0.35 
23  0.48  375  0.40 
24  0.48  375  0.45 
25  0.48  400  0.35 
26  0.48  400  0.40 
27  0.48  400  0.45 
The cementitious materials used in this study consisted of 92% Type I Portland cement conforming to ASTM C 150 [
Specific gravity, water absorption, and sieve analysis of aggregates.
Sieve size  Cumulative percentage retained 

(a) Coarse aggregate  


19 mm  5 
12.5 mm  60 
9.5 mm  95 
4.75 mm  100 


(b) Fine aggregate  


1.18 mm  0 
0.60 mm  24.42 
0.30 mm  90.49 
0.15 mm  96.59 
The proportioning of all 27 trial mixtures was carried out in terms of absolute volume using the specific gravities of the concrete ingredients and the values of water/cementitious materials ratio, cementitious materials content, and fine/total aggregate ratio for each of 27 mixtures, as given in Table
Considering three replicates for each of the 27 mixtures, a total number of 81 cylindrical concrete specimens (size: 75 mm diameter and 150 mm high) were cast for determining compressive strength. After casting, the concrete specimens were cured for 28 days in a curing tank under laboratory conditions and then tested for compressive strength in accordance with ASTM C 39 [
Average 28day compressive strength test results for all 27 concrete mixtures along with the standard deviation of three replicates of each mixture are presented in Table
Compressive strength test results.
Mix number  28day average compressive strength, 
Standard deviation of three replicates of each mixture (MPa) 

1  39.7  1.9 
2  38.8  1.0 
3  39.1  0.8 
4  34.1  1.2 
5  38.2  1.9 
6  40.6  2.0 
7  34.2  1.1 
8  39.3  1.1 
9  39.8  1.6 
10  27.9  1.1 
11  37.4  1.8 
12  38.5  1.1 
13  31.9  0.8 
14  37.1  1.3 
15  33.9  0.2 
16  26.5  1.4 
17  30.7  1.7 
18  36.5  1.6 
19  30.0  1.5 
20  32.1  1.3 
21  30.5  0.8 
22  20.7  1.8 
23  27.5  0.8 
24  29.9  0.3 
25  25.4  1.1 
26  31.0  0.2 
27  25.3  0.2 
Analysis of variance (ANOVA) was carried out to pinpoint the individual and interactive effects of variable factors on the dependent variable. ANOVA of the test results in the present study was done with the software named MINITAB [
The results of ANOVA for compressive strength are presented in Table
ANOVA for compressive strength test results.
Factors  Type  Level  Scale values  


Fixed  3  0.875  0.938  1.000  

Fixed  3  0.792  0.896  1.000  

Fixed  3  0.778  0.889  1.000  


Source  DF  SS  MS 


Significance 



2  39.672  19.380  1.990  0.199  No 

2  464.501  232.250  23.260  0.000  Yes 

2  135.281  67.640  6.770  0.019  Yes 

4  23.686  5.921  0.590  0.678  No 

4  4.993  1.248  0.120  0.969  No 

4  22.437  5.609  0.560  0.697  No 
Error  8  79.890  9.986  


Total  26  770.456 
The polynomial regression model obtained for compressive strength using the data presented in Table
The upper and lower bounds of each of the three variables are given in Table
The empirical model obtained for compressive strength can be used for optimization of concrete mixture proportions using any suitable optimization method/tool. The developed compressive strength model was utilized for optimization of concrete mixture design corresponding to the following options (i.e., constraints) typically using the
optimizing the levels of the
optimizing the levels of the
The optimization results, presented in Table
Optimization of concrete mixture design.
Optimization option 

Optimum levels of the mixture variables  Cost level  





(i) Optimizing the levels of 
42.1  350  0.38  0.45  Low 
40.7  375  0.38  0.45  Medium  
39.3  400  0.38  0.45  High  


(ii) Optimizing the levels of 
30.0  0.48  0.40  
35.0  350  0.44  0.42  Low  
40.0  0.40  0.44  
30.0  0.47  0.41  
35.0  375  0.43  0.42  Medium  
40.0  0.38  0.44  
30.0  0.46  0.41  
35.0  400  0.41  0.43  High  
40.0  0.38  0.45 
From Table
The data obtained from the optimization option (ii), as presented in Table
Variation of compressive strength with
Variation of compressive strength
A simplified stepbystep approach is proposed for optimizing the concrete mixture design based on the analysis of the data obtained through a statistically planned experimental program. The proposed approach consists of five steps, as follows: (i) specification of the characteristic performance of concrete, (ii) selection of the levels of key mix design factors, (iii) experimental work considering trial mixtures using full factorial experiment design for generating data to obtain statistical model for optimization, (iv) statistical analysis of experimental data and fitting of the strength model, and (v) optimization of mixture proportions using the fitted strength model.
The results of the experimental works conducted in the present study for demonstrating the utility of the proposed statistical approach have indicated the significant effects of water/cementitious materials ratio, cementitious materials content, and fine/total aggregate ratio on compressive strength. The optimum values of water/cementitious materials ratio and fine/total aggregate ratios have resulted in a higher compressive strength at a lower cementitious materials content resulting in significant cost saving in the concrete production.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support received from King Abdulaziz City for Science & Technology (KACST), Saudi Arabia, for conducting this research work (Project no. KACST AT2321). The support of the Department of Civil Engineering, King Fahd University of Petroleum and Minerals (KFUPM), Saudi Arabia, is also acknowledged.