CHAPTER 4 STATISTICAL MODELS FOR STRENGTH USING SPSS

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1 69 CHAPTER 4 STATISTICAL MODELS FOR STRENGTH USING SPSS 4.1 INTRODUCTION Mix design for concrete is a process of search for a mixture satisfying the required performance of concrete, such as workability, strength and durability. The basic ingredients of concrete can be classified into two groups: cementitious materials and aggregates. Although the quality of cementitious materials is governed mainly by w/c ratio, the quantity of cementitious materials required to achieve a targeted quality of concrete depends on the characteristics of aggregates. Concrete mixture design can be optimized by adjusting the levels of the key mixture factors such as water to cementitious materials ratio, coarse aggregate to total aggregate ratio, and cementitious material content or aggregate to cementitious materials ratio. A statistical model is a formalization of relationships between variables in the form of mathematical equations. It describes how one or more random variables are related to one more other variables. Statistical methods were used to improve the experimental methods, in which, instead of selecting one starting mix proportion and then adjusting by trial and error for achieving the optimum solution. (Padmanaban 2009). A set of trial batches covering a chosen range of proportions for each mixture component is defined according to established statistical procedures. Trial batches are then carried out, test specimens are fabricated and tested, and experimental results are analyzed using standard statistical methods. These methods include fitting empirical models to the data for each performance criterion. In these models, each response (resultant concrete

2 70 property) such as strength, slump or cost is expressed as an algebraic function of factors (individual component proportions) such as w/c ratio, cement content, chemical admixture dosage and percent pozzolana replacement. After a response can be characterized by an equation (model), several analyses are possible. 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 cementitious materials. Fully experimental or semi experimental (i.e. half-analytical) methods are reliable and accurate; however, they involve comprehensive laboratory works. Statistical methods also require a certain amount of experimental works but they have an additional advantage in a sense that the expected properties (responses) can be characterized by an uncertainty (variability). This has important implications for specifications and for production of the costeffective concrete mixture. Statistical package for the social sciences (SPSS) is a window based program that can be used to perform data entry and analysis and to create tables and graphs. It is capable of handling large amount of data and can perform all of the analyses covered in the text and more. In the present work, an effort has been made to exhibit the application of a statistical approach proposed to obtain the compressive strength. 4.2 REGRESSION ANALYSIS Regression analysis is a statistical tool for the investigation of historical relationship between an independent and a dependent variable to predict the future values of the dependent variables. The relationship is expressed through a statistical model equation that predicts a response

3 71 (dependent) variable from a function of regression (independent) variables. It is used when a continuous dependent variable is to be predicted from a number of independent variables. Regression helps to estimate one variable or the independent variable from the other variables or the independent variables. In other words this method can estimate the value of one variable, provided the values of the other variable given. The parameters are estimated so that a measure of fit is optimized Linear Regression Analysis The most commonly used form of regression is linear regression and the most common type of linear regressions called ordinary least squares regression. Linear regression was the first type of regression analysis and to be used extensively in practical applications. Linear regression uses the values from an existing data set consisting of measurements of the values of two variables, X and Y, to develop a model that is useful for predicting the value of the dependent variable, Y for given value of X. Linear regression is an approach for modeling the relationship between a dependent variable (Y) and one or more independent variables (X). R 2 is a measure of association; it represents the percentage of the variance in the value of Y that can be explained by knowing the value of X. The value R 2 various from a low of 0.0 (none of the variance explained), to a high of +1.0 (all the variance explained). It represents the percent of the variance in the values of Y that can be explained by knowing the values of X. The regression equation is written as, Y = a + bx +e (4.1) Where, in Equation 4.1,

4 72 Y is the value of dependent variable, what is being predicted. a or Alpha, a constant; equals the value of Y when the value of X = 0 b or Beta, the coefficient of X; the slope of the regression line; how much Y changes for each one -unit change in X. X is the value of the independent variable, what is predicting the value of Y. e is the error term ;the error in predicting the value of Y, given the value of X Multiple Regression Analysis Multiple regressions are a technique that allows additional factors to enter the analysis separately so that the effect of each can be estimated. It is valuable for quantifying the impact of various simultaneous influences upon a single dependent variable. Multiple regressions are an extension of simple (bi-aviate) regression. The goal of multiple regressions is to enable a researcher to assess the relationship between a dependent (predicted) variable and more independent (predictor) variables. The end result of multiple regressions is the development of a regression equation (line of best fit) between the dependent variable and several independent variables. It is used to predict the value of a dependent variable based on the value of two or more other variables. When there are more than two variables available say Y on X 1, X 2, X 3, X 4,..in which Y is the dependent variable and the remaining variables are independent the equation can be written as in the given equation,

5 73 Y = b 0 + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X b 5 X b 6 X b 7 X 1 X 2 + b 8 X 1 X 3 + b 9 X 2 X 3 + (4.2) Where, in Equation 4.2, b 0 is regression constant and b 1, b2, b 3, b 4,... are regressions coefficient (Sunilaa George 2012). The independent variables can be selected by the forward and stepwise regression methods. Once the multiple regression equation has been constructed, one can check by examining the coefficient of determination R 2. R 2 always lies between 0 and PROPOSED APPROACH The proposed approach to arrive the strength of concrete is based on the planned experimental works and statistical analysis of the data generated, which would reduce the number of trial batches needed. Selection of the levels of three key mixture design factors namely, cementitious materials content, water/cementitious materials ratio and age which mainly affect the quality of concrete. It will be made to ensure that enough experimental data are generated for obtaining a regression model for compressive strength which can be used to obtain the specified characteristic strength of concrete Experimental Work for Generating Data to Obtain Statistical Model An experimental work was conducted involving designing, preparing, and testing various trial mixtures for three different grades of concrete with various percentage replacements of fly ash and super plasticizer. The standard cubes were prepared, cured for 7, 14, 28, 56 and 90days, and then tested for compressive strength for generating data to obtain

6 74 statistical model for strength. Data analysis is carried out in this research based on the experimental results from 585 cubes in order to develop a generic and custom regression models. Data analysis is also carried out with experimental results considering percentage of fly ash and percentage of super plasticizers from 144 cylinders and prisms in each Statistical Analysis of Experimental Data and Fitting of the Strength Model The collected data were analyzed by first creating tables for the various parameters (Indelicato 1999) of conventional and HVFA concrete. The tables are formed for the various parameters of concrete such as percentage of fly ash, percentage of super plasticizer and curing period. The tables are formed for the water binder ratios of 0.4 having replacement of cement by fly ash in the range of 0%, 50%, 55% and 60% and cured for time period of 7,14,28,56 and 90 days. The dependent variables are considered as compressive strength (f ck ), flexural strength (f cr ) and split tensile strength (f t ).The following are considered as the independent variables. 1. Fly ash content in percentage(x 1 ) 2. Dosage of super plasticizer in percentage (X 2 ) 3. Age in days (X 3 ) 4.4 STATISTICAL MODELS FOR MECHANICAL PROPERTIES A statistical model is a formalization of relationships between variables in the form of mathematical equations. It describes how one or more random variables are related to one or more variables. Statistical models used for prediction are often called regression models, of which linear regression and logistic regression are two examples.

7 75 The regression analysis was carried out using SPSS (Statistical Package for the Social Sciences) software and the regression curve was drawn (Field 2005). This analysis was calculated based on the experimental results of various concrete mixes with cement replacement by fly ash of 0.4 W/b ratios (Simon et al 1997).The following independent variables such as percentage of FA (X 1 ), percentage of Sp (X 2 ) and age in days (X 3 ) were considered to prepare the regression fit with compressive strength (Y) as dependent variable Linear Regression Analysis Considering the percentage of FA (X 1 ) and percentage of Sp (X 2 ) for 28 days curing period the linear regression model of cube compressive strength (f ck ), flexural strength (f cr ) and split tensile strength (f t ) are obtained Linear regression model for compressive strength (f ck ) The results of regression analysis of cube compressive strength (f ck ) considering percentage of FA (X 1 ) and percentage of Sp (X 2 ) for 28 days curing period from statistical modeling is presented in Table 4.1 below. Table 4.1 Comparison of experimental and statistical results for compressive strength (f ck )of HVFA concrete % of FA Sl.No replaced(x 1 ) % of Sp(X 2 ) Compressive strength of HVFA concrete (MPa) M 20 Grade M 25 Grade M 30 Grade Expt SPSS Expt SPSS Expt SPSS

8 Mean The linear regression model has been obtained for M20 grade concrete for compressive strength and mentioned in Equation 4.3. f ck = X X 2 and (R 2 =0.921) (4.3) Similarly, the model was obtained for M25 grade concrete (Equation 4.4) and for M30 grade concrete (Equation 4.5). f ck = X X 2 and (R 2 =0.850) (4.4) f ck = X X 2 and (R 2 = 0.813) (4.5) Combining all three Equations namely 4.3, 4.4 and 4.5, the generalized linear regression model has been obtained and mentioned in Equation 4.6 for compressive strength. f ck = X X 2 and(r 2 = 0.699) (4.6)

9 Linear regression model for flexural strength (f cr ) The results of regression analysis of flexural strength (f cr ) considering percentage of FA (X 1 ) and percentage of Sp (X 2 ) for 28 days curing period from statistical modeling is presented in Table 4.2 below. Table 4.2 Comparison of experimental and statistical results for flexural strength (f cr ) of HVFA concrete Sl.No % of FA replaced(x 1 ) % of Sp(X 2 ) Flexural strength (f cr ) (Mpa) M 20 Grade M25 Grade M30 Grade Expt SPSS Expt SPSS Expt SPSS Mean The linear regression model has been obtained for M20 grade concrete for flexural strength and mentioned in Equation 4.7. f cr = X X 2 and (R 2 = 0.943) (4.7) Similarly, the model was obtained for M25 grade concrete (Equation 4.8) and for M30 grade concrete (Equation 4.9). f cr = X X 2 and (R 2 = 0.956) (4.8) f cr = X X 2 and (R 2 = 0.946) (4.9)

10 78 Combining all three Equations namely 4.7, 4.8 and 4.9, the generalized linear regression model has been obtained and mentioned in Equation 4.10 for flexural strength. f cr = X X 2 and (R 2 = 0.797) (4.10) Linear regression model for split tensile strength (f t ) The results of regression analysis of split tensile strength (f t ), considering percentage of FA (X 1 ) and percentage of Sp (X 2 ) for 28 days curing period from statistical modeling is presented in Table 4.3 below Table 4.3 Comparison of experimental and statistical results for split tensile strength (f t ) of HVFA concrete Split tensile strength (f t ) (MPa) % of FA % of Sl.No M20 Grade M25 Grade M30 Grade replaced (X 1 ) Sp(X 2 ) Expt SPSS Expt SPSS Expt SPSS Mean

11 79 The linear regression model has been obtained for M20 grade concrete for flexural strength and mentioned in Equation f t = X X 2 and (R 2 = 0.925) (4.11) Similarly, the model was obtained for M25 grade concrete (Equation 4.12) and for M30 grade concrete (Equation 4.13). f t = X X 2 and (R 2 = 0.983) (4.12) f t = X X 2 and (R 2 = 0.929) (4.13) Combining all three Equations namely 4.11, 4.12 and 4.13, the generalized linear regression model has been obtained and mentioned in Equation 4.14 for split tensile strength. f t = X X 2 and(r 2 = 0.951) (4.14) Comparison of statistical and experimental results The ultimate strength of statistical and experimental results of compressive strength, flexural strength and split tensile strength for all the grades of HVFA concrete are shown in Table 4.1, 4.2 and 4.3. The average ratios of statistical and experimental results of compressive strength, flexural strength and split tensile strength were found to be 0.86, 0.94, and 0.95 respectively for all the specimens. Figure 4.1, Figure 4.2 and Figure 4.3 shows the agreement between statistical results and experimental results of compressive strength, flexural strength and split tensile strength of HVFA concrete of all grades on 45 graph with plus or minus 15% boundary lines. The points on the 45 line correspond to give full agreement. From the above

12 80 test results, out of all specimens, the results of only few specimen falls outside the boundary lines. +15 %+15 % -15 %-15 % Figure 4.1 Experimental vs statistical results for f ck +15 % -15 % Figure 4.2 Experimental vs statistical results for f cr

13 % -15 % Figure 4.3 Experimental vs statistical results for f t Multiple Regression Analysis for Compressive Strength (f ck ) The multiple regression analysis of cube compressive strength (f ck ) by considering percentage of FA(X 1 ), percentage of Sp (X 2 ) and age of concrete (X 3 ) from statistical modeling are also obtained and mentioned in Equation f ck = X X X X 1 *X X 2 *X X 3 *X X 1 *X X 1 *X X 2 *X 3 (4.15) The step wise multiple regression model indicated that out of the explanatory variables, two variables namely, X 1, X 3,X 1 *X 1 and X 3 *X 3 have significantly contributing to f ck. The analysis of variance of multiple regression models for f ck indicates the overall significance of the model fitted.

14 PREDICTION OF f cr FROM f ck The statistical analysis was carried out using SPSS software and the regression curve was drawn. This analysis was done based on the experimental results of various concrete mixes with cement replacement by fly ash. The relationship between f cr and f ck was obtained as follows. The test results of all three grades of concrete namely M20, M25 and M30 for three different percentage of replacement of fly ash were used for modeling. The ratio between f cr and f ck is kept as K 1 exp. for experimental values, and K 1 cal. for SPSS results (Equation 4.16).The graphical representation of f cr versus K 1 cal. is presented in Figures 4.4 (a-c) for all the three grades of HVFA concrete respectively. K 1 cal = f cr / f ck (4.16) The calculated values of statistical and experimental results of K 1 with f cr and f ck for M20 grade, M25 grade and M30 grade HVFA concrete are shown in Table 4.4, Table 4.5 and Table 4.6 respectively. K 1 Cal y = x R² = Flexural strength (MPa) (a) M20 grade HVFA concrete

15 83 K 1 Cal y = x R² = Flexural strength (MPa) (b) M25 grade HVFA concrete K 1 Cal y = x R² = Flexural strength (MPa) (c) M30 grade HVFA concrete Figure 4.4 (a- c) Relation between f cr and f ck Table 4.4 Comparison between calculated and experimental results of K 1 with f cr and f ck for M20grade HVFA concrete Specimen % of fly ash f cr f ck K 1 Exp K 1 Cal Ratio of ID and Sp (Mpa) (Mpa) f cr / f ck K 1 Exp / K 1 Cal A0 0%- 0% AA0 50% -0% AA1 50% -1% AA1.5 50% -1.5% AA2 50% -2% AB0 55% -0% AB1 55% -1% AB1.5 55% -1.5%

16 84 AB2 55% -2% AC0 60% -0% AC1 60% -1% AC1.5 60% -1.5% AC2 60% -2% Mean value of HVFA concrete Table 4.5 Comparison between calculated and experimental results of K 1 with f cr and f ck for M25 grade HVFA concrete Specimen % of fly ash f cr f ck K 1 Exp K 1 Cal Ratio of ID and Sp (Mpa) (Mpa) f cr / f ck K 1 Exp / K 1 Cal B0 0%- 0% BA0 50% -0% BA1 50% -1% BA1.5 50% -1.5% BA2 50% -2% BB0 55% -0% BB1 55% -1% BB1.5 55% -1.5% BB2 55% -2% BC0 60% -0% BC1 60% -1% BC1.5 60% -1.5% BC2 60% -2% Mean value of HVFA concrete Table 4.6 Comparison between calculated and experimental results of K 1 with f cr and f ck for M30grade HVFA concrete Specim en ID % of fly ash and Sp f cr (Mpa) f ck (Mpa) K 1 Exp f cr / f ck K 1 Cal Ratio of K 1 Exp / K 1 Cal C0 0%- 0% CA0 50%-0% CA1 50% -1% CA1.5 50%-1.5% CA2 50% -2% CB0 55% -0%

17 85 CB1 55% -1% CB1.5 55%-1.5% CB2 55% -2% CC0 60% -0% CC1 60% -1% CC1.5 60%-1.5% CC2 60% -2% Mean value of HVFA concrete Table 4.7Abstract of statistical modeling for f cr with f ck of HVFA concrete Sl. No. Grade of HVFA concrete % of fly ash and Plasticizer f ck (Mpa) Mean values of f cr with f ck (Mpa) f cr value by experimentally f cr value by SPSS modeling 1 M20 50% & 1.5% f ck f ck 2 M25 50% & 1.5% f ck f ck 3 M30 50% & 1.5% f ck f ck It was observed and verified that there exists a good correlation between the experimental and theoretical results for f cr with f ck. It is also found that the mean value coincides with the relationship for OPC as per IS: PREDICTION OF E c FROM f ck The test results of all the three grades of concrete namely M20, M25 and M30 for three different percentage of replacement of fly ash were used in SPSS modeling. The following equations have been arrived. The ratio between E c and f ck is kept as C 1 exp. for experimental values, and C 1 cal. for SPSS results (Equation 4.17).The graphical

18 86 representation of f cr versus C 1 cal. is presented in Figures 4.5 (a-c) for all the three grades of HVFA concrete respectively. C 1 cal = E c / f ck (4.17) 2500 C 1 cal y = x R² = fck (a) M20 grade HVFA concrete C 1 cal y = x R² = fck (b) M25 grade HVFA concrete

19 87 C 1 cal y = x R² = fck (c) M30 grade HVFA concrete Figure 4.5 (a- c) Relation between E c and f ck The abstract of predicted C 1 cal. values and R 2 for various HVFA concrete grades are given in Table 4.8. Table 4.8 Abstract of predicted values of C 1 cal. and R 2 Sl.No Grade of HVFA Predicted value of C 1 concrete cal. R 2 1 M (f ck ) R 2 = M (f ck ) R 2 = M (f ck ) R 2 = The calculated values of statistical and experimental results of C 1 with E c and f ck for M20, M25 and M30 grades HVFA concrete are shown in Table 4.9, Table 4.10 and Table 4.11 respectively.

20 88 Table 4.9 Comparison between statistical and experimental results of C 1 for M20 HVFA concrete Specimen ID % of fly ash and Sp E c (Mpa) f ck (Mpa) C 1 Exp C 1 Cal Ratio of E c / f ck C 1 Exp / C 1 Cal A0 0%- 0% AA0 50% -0% AA1 50% -1% AA1.5 50% -1.5% AA2 50% -2% AB0 55% -0% AB1 55% -1% AB1.5 55% -1.5% AB2 55% -2% AC0 60% -0% AC1 60% -1% AC1.5 60% -1.5% AC2 60% -2% Mean value of HVFA concrete

21 89 Table 4.10 Comparison between statistical and experimental results of C 1 for M25 HVFA concrete Specimen ID % of fly ash and Sp E c (Mpa) f ck (Mpa) C 1 Exp C 1 Cal Ratio of C 1 Exp / C 1 Cal E c / f ck B0 0%- 0% BA0 50% -0% BA1 50% -1% BA1.5 50% -1.5% BA2 50% -2% BB0 55% -0% BB1 55% -1% BB1.5 55% -1.5% BB2 55% -2% BC0 60% -0% BC1 60% -1% BC1.5 60% -1.5% BC2 60% -2% Mean value of HVFA concrete Table 4.11 Comparison between statistical and experimental results of C 1 for M30 HVFA concrete Specimen ID % of fly ash and Sp E c (Mpa) f ck (Mpa) C 1 Exp C 1 Cal Ratio of C 1 Exp / C 1 Cal E c / f ck C0 0%- 0% CA0 50% -0% CA1 50% -1% CA1.5 50% -1.5% CA2 50% -2% CB0 55% -0% CB1 55% -1% CB1.5 55% -1.5% CB2 55% -2% CC0 60% -0% CC1 60% -1% CC1.5 60% -1.5% CC2 60% -2% Mean value of HVFA concrete

22 90 The abstract of statistical modeling for E c with f ck of M20, M25 and M30 grades of concrete (Prabir Sarker 2008) are presented in Table Table 4.12 Abstract of statistical modeling for E c with f ck of HVFA concrete Sl. No. Grade of HVFA concrete % of fly ash and Plasticizer f ck (Mpa) Mean values of E c (Mpa) E c value by experimentally E c value by SPSS modeling 1 M20 50% & 1.5% f ck 4505 f ck 2 M25 50% & 1.5% f ck 5007 f ck 3 M30 50% & 1.5% f ck 4782 f ck Based on the above tables of results, it is concluded that Modulus of elasticity of HVFA concrete can be taken as E c = 4800 f ck. It was observed and verified that, it exists a good correlation between the experimental and statistical values of E c and f ck for M20, M25 and M30 grade HVFA concrete and is almost comparable with experimental values to SPSS values. 4.7 KEY FINDINGS This chapter describes for getting empirical mathematical expressions using the experimental values obtained for predicting the value of compressive and flexural strength of specimens of similar concrete composites. These mathematical relationships are obtained by regression analysis. The equations may be used to avoid casting of further specimens whose strength may be predicted using the obtained equations.

23 91 Statistical models were developed by linear and multiple regression analysis for M20, M25 and M30 grade HVFA concrete, considering % of FA and % of Sp for 28 days curing period. The generalized statistical model using multiple regression analysis was performed for M20, M25 and M30 grade HVFA concrete, considering % of FA, % of Sp and curing period in days for compressive strength, flexure strength and split tensile strength. The coefficients of determination (R 2 ) are satisfied and significant at 1% level. Statistical models for f cr and f ck were arrived for HVFA concrete, based on the regression analysis of experimental results. There exist a good correlation between experimental results and calculated values. It is also found that the mean value coincides with the relationship for OPC as per IS: It was observed and verified that, it exists a good correlation between the experimental and statistical values of E c and f ck for M20, M25 and M30 grade HVFA concrete and is almost comparable with experimental values to SPSS values. Based on the experimental results, a statistical model for E c for HVFA concrete was also developed and is differed by 4% from IS: OPC concrete values.