Mathematical Regression Model for the Prediction of Concrete Strength

Mathematical Regression Model for the Prediction of Concrete Strength M. F. M. Zain 1, Suhad M. Abd 1, K. Sopian 2, M. Jamil 1, Che-Ani A.I 1 1 Faculty of Engineering and Built Environment, 2 Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 4600 UKM Bangi, Selangor Darul Ehsan, Malaysia Abstract: - In this study a new mathematical models were proposed and developed using non-linear regression equation for the prediction of concrete compressive strength at different ages. The variables used in the prediction models were from the knowledge of the mix itself, i.e. mix proportion elements. According to the analysis the models provide good estimation of compressive strength and yielded good correlations with the data used in this study. The correlation coefficients were 0.995 and 0.994 for the prediction of 7 and 28 days compressive strength respectively. Moreover, the proposed models proved to be significant tool in prediction compressive strength of different concretes in spite of variations in the results. Key-Words: - mathematical model, statistical analysis, compressive strength, strength prediction, concrete 1 Introduction Concrete is such a construction material that is widely used in the world. The advantages of concrete are low cost, availability of construction, workability, durability and convenient compressive strength that make it popular near engineers and builders. However, these advantages seriously depend on the correct mix, placing and curing [1]. In construction industry, strength is a primary criterion in selecting a concrete for a particular application. Concrete used for construction gains strength over a long period of time after pouring.the characteristic strength of concrete is defined as the compressive strength of a sample that has been aged for 28 days [2]. Neither waiting 28 days from such a test would serve the rapidity of construction, nor neglecting it, would serve the quality control process on concrete in large construction sites. Therefore, rapid and reliable prediction for the strength of concrete would be of great significance []. For example, it provide a chance to do the necessary adjustment on the mix proportion used to avoid situation where concrete does not reach the required design strength or by avoiding concrete that is unnecessarily strong, and also, for more economic use of raw materials and fewer construction failures, hence reducing construction cost. Prediction of concrete strength, therefore, has been an active area of research and a Considerable number of studies have been carried out. Many attempts have been made to obtain a suitable mathematical model which is capable of predicting strength of concrete at various ages with acceptable (high) accuracy [4]. 2 Statistical Analysis for Strength Prediction The strengthening of concrete is a complex process involving many external factors. A number of improved prediction techniques have been proposed by including empirical or computational modeling, statistical techniques and artificial intelligence approaches. Many attempts have been made for modeling this process through the used of computational techniques such as finite element analysis. While, a number of research efforts have concentrated on using multivariable regression models to improve the accuracy of predictions. Statistical models have the attraction that once fitted they can be used to perform predictions much more quickly than other modeling techniques, and are correspondingly simpler to implement in software. S. Popovics [5], augments Abrams model, a widely accepted equation relating the water cement ratio w/c of concrete to its strength with additional variables such as slump, and uses least square regression to determine equation coefficients. Using this approach improved strength prediction and insights into concrete compositions were achieved. M. Nagesha et al. [2], used multivariable ISSN: 1790-2769 96 ISBN: 978-960-474-012-

regression techniques on concrete composition data to predict 28 days compressive strength with reasonable accuracy, proposing a formula readily applicable for on-site use. Apart of its speed, statistical modeling has the advantage over other techniques that it is mathematically rigorous and can be used to define confidence interval for the predictions. This is especially true when comparing statistical modeling with artificial intelligence techniques. Statistical analysis can also provide insight into the key factors influencing 28 days compressive strength through correlation analysis. For these reasons statistical analysis was chosen to be technique for strength prediction of this paper [2]. Experimental Program Physical properties of the materials used in this study are shown in Table (1). Locally produced ordinary Portland cement (OPC) was used. It has a specific gravity of.1 and specific surface of 500 m2/kg. Fineness modulus was 2.82 for fine aggregate. The coarse aggregate was 20 mm maximum size crushed stone; its specific gravity was 2.7. No admixtures or additives were used in this study only the ordinary constituents of concrete (cement, sand, gravel, water) to study the effect of the ordinary mix proportion on the compressive strength of concrete. Since the aim of this study is studying the effect of mix proportions on the compressive strength of concrete,different mixes were used.the details of all mix proportions are shown in Table (). Compressive strength test was performed and evaluated in accordance to BS 1881: Part 116:198. Specimens were immersed in water until the day of testing at, 7, 28 days. Table () show the compressive strength test results. Table 4 shows the results of compressive strength test at the age of, 7 and 28 days Table 1: Physical Properties of Materials Materials Cement (C) Ordinary Portland Cement (OPC) Fine Aggregate (FA): Sand (S) Coarse Aggregate (CA): Crushed Stone Properties Specific Gravity:.1 Specific surface (by Blain) : 500 cm 2 \g Specific Gravity: 2.60 Fineness Modulus: 2.4 Specific Gravity: 2.7 Maximum particle size: 20 mm Mix No Table 2: Chemical composition of OPC Oxide (%) Silicon dioxide (SiO2) 22.1 Aluminum Trioxide (Al2O) 5.96 Ferric oxide (Fe2O).04 Calcium oxide (CaO) 61.5 Magnesium oxide (MgO) 2.5 Sodium oxide (Na2O) 0.16 Loss on ignition (L.O.I) 1.50 Insoluble residue (I.R) 1.10 Lime saturation factor (L.S.F) 0.85 Table : Details For Mix Proportions Water Kg/m Cem Kg/m Sand Kg/m Agg w/c Density Kg/m Kg/m 1 180 400 600 1200 0.45 2.2 2 195 90 588 1170 0.5 22.2 209 80 570 1140 0.55 210 4 222 70 555 1110 0.6 200 5 24 60 540 1080 0.65 229.6 6 245 50 525 1050 0.7 2275.5 7 146 25 650 100 0.45 2268 8 160 20 640 1280 0.5 2244 9 17 15 60 1260 0.55 224 10 186 10 620 1240 0.6 220 11 198 05 610 1220 0.65 2176 12 210 00 600 1200 0.7 2148 1 2 517 517 104 0.45 240 14 252 504 504 1008 0.5 2421 15 270 491 491 982 0.55 278 16 287 479 479 958 0.6 274 17 04 468 468 96 0.65 256 18 20 457 457 914 0.7 252 Table 4: Compressive Strength for the Concrete at, 7 and 28 Days Mix no. Density (Kg/m ) Compressive strength (MPa) days 7 days 28 days 1 2.25 17.9 24.5 4 2 22.2 17.4 22.5 2.5 210 16. 21.6 2.5 4 200 16.1 21.5 2. 5 229.6 15 21.1 0.5 6 2275.5 14.6 20.4 0. 7 2268 14.1 20. 29.2 8 2244 14.1 20 28.9 9 224 1.9 18.5 27.7 10 220 1.7 17.6 25.9 11 2176 1.2 17. 24.5 12 2148 12. 14.6 2.8 ISSN: 1790-2769 97 ISBN: 978-960-474-012-

1 240 26.1 1 44 14 2421 2 29.9 9.4 15 278 21.4 28. 7.5 16 274 19.6 26.7 6.1 17 256 19.5 25.8 5.2 18 252 18.4 25.7 4.6 4 Modelling the Prediction of Compressive Strength of Concrete The most popular regression equation used in prediction of compressive strength prediction is: f b 0 b1 w/ c...eq.1 where: : compressive strength of concrete w/c: water/cement ratio b 0,b 1 : coefficients The previous equation is the linear regression equation.the origin of this equation is Abram s Law [5] which relate compressive strength of concrete to the w/c ratio of the mix and according to this law, increasing w/c ratio will definitely lead to decrease in concrete strength. The original formula for Abram is: A =...Eq.2 f w / c B where: : compressive strength of concrete A, B: empirical constants Lyse [6] made a formula similar to Abram but he relate compressive strength to cement /water ratio and not water /cement ratio. According to Lyse strength of concrete increase linearly with increasing c/w ratio.the general form of this popular model was: f = A + Bc / w......eq. Where: : compressive strength of concrete c/w: cement /water ratio A, B: empirical constants The quantities of cement,fine aggregate and coarse aggreagate were not included in the model and not accounted for the prediction of concrete strength.so, for various concrete mixes were their w/c ratio is constant,the strength will be the same and this is not true.therefore, efforts should be concenerate on models taken into account the influence of mix constituents on the concrete strength in order to have more reliable and accurate results for the prediction of concrete strength. So, Eq. 1 which reffered to Abrams Law was extended to include other variables in the form of multiple linear regression equation and used widely to predict the compressive strength of various types of concrete as below: f = b0 + b1 w/ c...eq.1 linear least square regression (reffered to Abram) f = b0 + b1 w/ c + b2ca + bfa + C...Eq.4 multiple linear regession Where: : compressive strength of concrete w/c: water/cement ratio C: quantity of cement in the mix CA: quantity of coarse aggregate in the mix FA: quantity of fine aggregate in the mix According to Eq. 4 all the variables related to the compressive strength in a linear fashion, but this is not always true because the variables involved in a concrete mix and affecting its compressive strength are interrelated with each other and the additive action is not always true. Here, it appears that there is a need to another type of mathematical model can reliably predicts strength of concrete with acceptable high accuracy. So, if we took the general form of the multiple linear regressions as below; Y = a0 + a1 X 1 + a2 X 2 + a X +... a m X m...... multiple linear regression (Eq.4) For situations where the multiple dependency is curvilinear (non-linear) the logarithmic transformation can be applied to this type of regression [7]: log( y ) = log( a ) + a1 1) + a2 2 ) + a ) +... a m 0 m...eq.5 This equation could be transform back to a form that predicts the dependent variable (Y) by taking the antilogarithimto yeild an equation of the type: Y... a1 a a a = a X. X 2. X X m..eq.6 0 1 2 This eq. called the multivariable power eq. and in m ) ISSN: 1790-2769 98 ISBN: 978-960-474-012-

engineering, variables are often dependent on several independent variables, this functional dependency is best characterized by the equation mentioned earlier, and is said to give results that are more realistic too. In this study, the multivariable power equation was found to be very suitable for prediction strength of concrete (as a dependent variable). Factors affecting this strength were the elements of the concrete mix itself. 5 Results And Discussion It is very important to analyze the effect of mix constituents on strength of concrete. Mix design is a specific combination of raw materials that are used in a particular concrete to reach a given target strength. So the significant factor in 28 days compressive strength is the concrete composition. Concrete theory suggested that water to cement ratio (w/c) of concrete is a primary factor influencing the strengthening process, both the final strength and the rate of hardening are affected [2]. Also, it is well known that decreasing water content increases strength for the concrete. This explanation is well represented in Figure (1) which shows the relationship between the 28 days compressive strength and the water to cement ratio (w/c) for the concrete used in this study. Conventionally, strength is related to density and the denser the concrete the higher the strength as shown in figure (2). Furthermore, strength of concrete is highly affected by cement content and amount of fine and coarse aggregate used in the mix as well as any other aditional material added to the mix in order to improve specific property for the concrete like fly ash, silica fume and slag or admixture like superplasticizer. Fig. 2 Relationship Between 28 Days Compressive Strength and Density Table (4) shows the relatioship between the compressive strength at the age of 7 and 28 days with the variables provided from the experimental work and are going to be used in the proposed model. This relationship is represented by the correlation coefficient between each variable and each strength.from this table, it can be seen that some variables have significant correlation with the predicted strength at the specified age.the highest correlations were density followed by the cement content in the mix. Table 5. Correlations between 7&28 Days compressive Strength and Variables Used in the Proposed Model Variable 7 Days Compressive Strength 28 Days Compressive Strength Water/cement(w/c) 0.79 0.41 Water (W) 0.580 0.58 Cement (C) 0.970 0.95 Sand (FA) 0.72 0.680 Aggregate (CA) 0.72 0.68 Density (ρ) 0.98 0.986 After analyzing the influence of mix constituent on the strength at the age of 7 and 28 days, the proposed model was used to predict compressive strength at the specified ages comprises all the variables mentioned earlier.the final form of the proposed strength prediction model for both ages was: a 1 a 2 a a 4 a 5 a 6 7 a 0 C. W. FA. CA. ρ. w / c =..Eq. 7 a1 a 2 a a 4 a 5 a 6 28 a 0C. W. FA. CA. ρ. w / c =...Eq.8 Fig. 1 Relationship Between 28 Days Compressive Strength and (w/c) Ratio Table (6) gives the regression coefficients of the prediction model above, for the prediction of 7 and 28 days compressive strength respectively, ISSN: 1790-2769 99 ISBN: 978-960-474-012-

as well as the value of coeffient of correlation C.C (R). Figure () and Figure (4) show the relationship between the observed(actual) value of the compressive strength obtained from the experimental work, and the predicted values obtained using the proposed model for 7 and 28 days respectively. It is so obvious that almostly 99% of the data lacated on the line of equality which means that the actual and the predicted values for the concrete compressive strength are identical with each other.this is quite true because the correlation coefficients were 0.995 for 7 days prediction and 0.994 for 28 days prediction.moreover almostly 99.047% of the variance was explained for the 7 days prediction while 98.8% of the variance was explained for the 28 days predictions. Table 6: Regression coefficients for the 7 & 28 days compressive strength prediction models Coefficient 7 days prediction model 28 days prediction model A 0 0.25 0.4262 A 1-4.819-28.710 A 2 4.070 28.0856 A -4.168-28.02 A 4 -.9896-1.9259 A 5 2.5945 0.72819 A 6 1.4920 1.61814 C.C 0.995 0.994 Variance explaine 99.047% 98.8% Fig. Relationship Between the Observed and Predicted Values for 7 Days Compressive Strength Fig. 4 Relationship Between the Observed and Predicted Values for 28 Days Compressive Strength 6 Comparison with Other Data To test the proposed model obtained from this study,it was decided apply the model using data from other sources or data from other researchers.this comparison is very important to check the validity of the proposed model for the prediction of 7 and 28 days compressive strength of concrete for any set of data. These data were imported from literature belong to Jee Namyong et al. [6]. Table (7) shows full details of the data imported and used to check the proposed model. The data comprises on 59 different kind of mixture with specified compressive strength of 18-27 MPa, w/c ratio of 0.9-0.62, maximum aggregate size of 25 mm and slump of 12-15 cm. The reason to choose this set of data is, the large number of concrete mixes (which mean large number of sample) and these mixes were from different plants of ready mix concrete,and this is also a good prove that the proposed model could valid even for ready mix concrete. Another reason that these data were from Korea, so, this is also a good prove that the model could work for any type and any place inspite of variation of data. Variation in concrete strength of the test specimens depends on how well the materials, concrete manufacture and testing is controlled. Especially construction practices may cause variation in strength of in-situ concrete due to inadequate mixing, poor compaction, delay and improper curing [6]. The variables used in the model were these available from the data.the correlation coefficient for the prediction of 28 days compressive strength was 0.7579 and 0.7267 for 7 days prediction, these results consider to be good results concerning the variations in the data. Moreover, there are some relationships in previously published studies that can predict the 28 days compressive strength from 7 days values [1], or may be earlier values [, 4]. ISSN: 1790-2769 400 ISBN: 978-960-474-012-

So, if we use the concept of early age strength to predict later age strength in this case, i.e, the strength at 7 days ( 7 ) will be one of the variables used in the model. The coefficient of correlation in this case will improve significantly from 0.7579 to 0.866 which prove the importance of this concept. Table 7. Data for 7 & 28 days compressive strength of the proposed models w/c % Compressiv e strength Weight of unit volume (kg/m ) (MPa) S/a Age % W G S g nt 7 D 28 D 60.21 51.2 174 289 9 900 0.86 15.5 21.2 59.74 52.1 184 08 927 860 0.91 16. 24.2 60.6 52.1 18 02 926 860 0.91 16. 2 57.48 52.7 17 01 961 86 0.9 21.5 26.2 60.2 50.9 190 15 904 862 0.47 18.6 24 61.49 51.2 190 09 911 859 0.46 17.4 22.5 59.55 46.1 184 09 821 975 0.92 15.8 22.6 50 48.5 164 28 886 942 1.64 2.2 4.7 47.8 45.1 176 68 805 988 0.77 19.1 26.9 49.44 48.8 178 60 858 914 1.08 2. 0.7 52.5 47.9 178 40 89 91 0.51 22.6 28.8 44.47 44.9 165 71 810 1000 1.85 20.7 27.6 44.69 47.6 164 67 847 940 1.84 18.9 28.5 48.56 49.6 169 48 882 902 1.74 24.1 1.8 48.92 49.4 181 70 866 887 1.11 2 0.5 50 49.5 171 42 885 91 1.71 2 1.6 49.7 49.9 181 64 865 874 1.09 21.6 1.7 44.75 48.8 179 400 85 894 2.8 22 0.2 45.4 46.1 180 97 790 99 2.78 22.4 0.2 46.56 4.9 18 9 759 981 1.18 20.4 29.7 50 46 175 50 804 955 1.05 16 29.8 47.04 44.7 18 89 778 962 1.17 19.8 26.7 47. 44.7 184 89 778 962 1.17 18.1 25. 48.04 47.1 184 8 810 924 1.15 20.1 27.8 48.41 47.4 18 78 807 902 1.1 22 29.1 48.41 47.4 18 78 818 924 1.1 22.8 29.2 47.79 47.6 184 85 812 922 1.16 21.2 27.6 45.69 48. 175 8 846 905 0.77 22.8 28.9 46.76 50. 17 70 889 878 1.11 21. 27. 46.74 50.5 179 8 879 862 1.15 21.2 27.6 44.21 49.0 168 80 868 90 1.9 21.1 29.1 47.77 46.1 182 81 812 956 0.8 20 26.2 45.14 47.2 172 81 815 940 1.91 21.5 28. 48.41 47.4 18 78 807 92 1.1 20.7 27.5 45.89 4.5 184 401 754 978 2.01 2.7 1. 48.56 45.8 185 81 800 946 1.91 21.9 28.7 45.69 48. 175 8 86 888 1.92 20.6 28.2 47.78 45.1 172 60 794 965 1.08 21 27.7 45.87 44 189 412 70 947 1.65 25 0.9 45.99 4.9 189 411 72 951 1.44 20.7 0.7 42.76 46. 180 421 784 927 0.85 22.7 0 40.89 42.5 184 450 71 977 1.5 22.6 0.7 40.62 42.0 184 45 704 98 1.59 2.6 1 41.97 47.5 18 46 804 889 0.87 22.8 29.8 44.1 48. 187 424 812 882 1.27 22.4 0.2 4.57 46.8 18 420 785 920 1.26 22.8 0.7 44.1 46.6 18 41 780 921 1.24 2.9 0.1 44.1 46.6 18 41 791 92 1.24 2.6 0.9 40.98 46.4 168 410 811 96 2.05 22.6. 42.45 47.7 180 424 81 891 1.27 22.9 0.8 44.1 46.6 18 41 78 956 0.87 20.8 29. 48.4 47.6 196 405 786 879 0.61 22.4 0.7 45.17 48.5 187 414 818 88 1.24 22.8 29.5 44.1 46.6 18 41 780 900 1.24 25.9 4. 44.1 46.6 18 41 78 914 1.24 24.5 2.4 9.7 45.2 176 447 760 970 1.4 24.7 29.4 4.75 44.7 182 416 771 954 2.08 2.4 0.8 41.47 45.5 175 422 782 97 0.84 2.7 1.5 4.85 44.6 171 90 768 969 1.17 24.9 0.6 7 Conclusions From this study,a mathemaical reggretion model was developed. i) The importance of the influence of mix constituents on the strength of concrete was ii) approved Previouse models that deal with the prediction of concrete compressive strength lack of including other variables affecting strength gaining in concrete. iii) A mathematical models for the prediction of concrete compessive strength at the ages of 7 and 28 were proposed and developed (using non-linear regressions) from the knowledge of the mix constituents, i.e, the variables used are the mix proportions elements. The prediction models developed in this study are: a 1 a 2 a a 4 a 5 a 6 7 = a 0C. W. FA. CA. ρ. w / c a1 a 2 a a 4 a 5 a 6 28 = a 0C. W. FA. CA. ρ. w / c iv) These models prove to be used with any set of data inspite of variations in test results of the concrete in question. v) The concept of using early age strength to predict strngth at later ages proved to be valid and could be used sucussfuly. References: [1] N. Hamid-zadeh, A. Jamali, N. Nariman-zadeh, H., A Polynomial Model For Concrete Compressive Strength Prediction Using GMDHtype Neural Networks and Genetic Algorithem [2]Darren Williams, Concrete Strength Prediction From Early-Age Data-Technical Paper, Honour Project, Technical Paper, University of Adelaide []Suhad M.A., Mathematical model for the prediction of cement compressive strength at the ages of 7&28 days within 24 hours, MSc Thesis, Al-Mustansiriya University, college of engineering, civil engineering department, 2001. ISSN: 1790-2769 401 ISBN: 978-960-474-012-

[4] Kheder G.F.,Al-Gabban A.M. & Suhad M.A. Mathematical model for the prediction of cement compressive strength at the ages of 7&28 days within 24 hour materials and structures 200. 6: 69-701. [5] Sandor popovics, Analysis of Concrete Strength Versus Water-Cement Ratio Relationship, ACI Material Journal,Vol.87, No.5, September- October 1990, Pp.517-529 [6] Jee Namyong, Yoon Sangchun, Cho Hongbum, Prediction of Compressive Strength of In-Situ Concrete Based on Mixture Proportion, Journal of Asian Architecture and Building Engineering, 16, may 2004. [7]Steven, C. Chapra, Raymond & P. Canale. Numerical Methods for engineers with personal computer applications, 1989. ISSN: 1790-2769 402 ISBN: 978-960-474-012-