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Bagasse is a valuable byproduct in sugar milling and is often used as a primary fuel source due to its high calorific value required to supply energy for plant operations. Many researchers have been able to determine suitability of sugarcane bagasse ash (SCBA) as a cement replacement in concrete, but there is no evidence of regression model development to predict compressive strengths of SCBA concrete. Results from laboratory experiments performed to characterize the compressive strength of concrete blended with different SCBA were used to develop regression mathematical models. Four models; the least square quadratic (LSQP), Simultaneous Equation (SEP), Newton Interpolation (NIP) and Legrange interpolation (LIP) polynomials were developed using these statistical techniques to predict the compressive strength of SCBA concrete. Water – cement ratio was kept constant at 0.45 and cement replacement level ranged from 0 to 30%. Curing period varied at age of 7, 14, and 28 days. The best model was identified by adopting leave-one-out (LOO) cross-validation. The developed models provided a closed form estimate of compressive strength of SCBA concrete. The models would serve as useful guidelines for proportioning concrete mixes incorporating SCBA as a partial cement replacement material. Such concrete could be used successfully in structural applications with economic and environmental advantages.
Key words: Concrete, Sugar Cane Baggase Ash, Mathematical model, Compressive strength
Introduction
Bagasse is a valuable byproduct in sugar milling and is often used as a primary fuel source due to its high calorific value required to supply energy for plant operations [1]. The burning of bagasse leaves solid black particles known as sugar cane bagasse ash (SCBA), see Plate 1. Since the early 1980’s, there has been an enormous demand for a mineral admixture and in future this demand is expected to increase even more [2]. Due to this, requirements for more economical and environmental-friendly cementing materials have led interest for partial cement replacement materials [3].
Statistical procedures provide tools of considerable value when evaluating the results of strength tests. Prediction of concrete strength, therefore, has been an active area of research and a considerable number of studies have been carried out [4]. Many researchers have been able to determine suitability of SCBA as a cement replacement in concrete, but there has been no evidence of regression model development [1]; [2]; [5]; [6]; [7]; [8]; [9]. Scientists employ these models mainly because they are less expensive in terms of time or cost to collect information required to make predictions about events [10].
Regression model development has been investigated in this research based on results of experimental work done on characterization of SCBA concrete. Statistical techniques were used to provide mathematical models to estimate the compressive strength of concrete when cement was replaced partially by SCBA. Two model development techniques; least square and interpolation were applied. The best model was identified by adopting leave-one-out (LOO) cross-validation. The philosophy of this paper was based on the fact that models are not complex but judicious tradeoffs between realism and simplicity [11].
Model Development
Regressions models are powerful tools frequently used to predict a dependent variable from a set of predictors and are widely used in a number of different contexts [12]. Statistical techniques were used to provide mathematical models to estimate the compressive strength of concrete when cement was replaced partially by SCBA. Mathematical models were developed in form of equations through polynomial regression analysis of the data showing the variation of concrete compressive strengths with curing age and percentage replacement of OPC with SCBA. Regression analysis was used to quantitatively correlate the compressive strength of concrete at 7, 14 and 28 days curing period to selected range SCBA content.
The methods of least squares and interpolation that leads to the best fitting line of a postulated form to a set of data were used to form regression models between predicted compressive strengths and SCBA content at constant water – binder (cement + SCBA) ratio of 0.45. Various studies showed that when the water/binder ratio is used instead of water/cement ratio as basis for mix design, strength prediction becomes more accurate [13]. Suitable analytical tools in Matlab R2013a were used to plot appropriate polynomial curves, generate the corresponding mathematical equations, and obtain regression models using values of compressive strengths of SCBA (size < 0.075mm) concrete laboratory values shown in table 1. The compressive strength of SCBA blended concrete for curing periods 7, 14 and 28 days was denoted as f_{cu7d }, f_{cu14d} and f_{cu28d }respectively. _{ }Each of the former was determined from the average of 3 compressive strengths; f_{cu1}, f_{cu2} and f_{cu3}. C was the control mix or 0% SCBA replacement of cement in concrete.
Table 1 Compressive strength of blended OPC – D_{<0.075} SCDA concrete.
SCBA % Replacement | Compressive Strength
(N/mm^{2}) |
|||||||||||
7d | 14d | 28d | ||||||||||
f_{cu1} | f_{cu2} | f_{cu3} | f_{cu7d} | f_{cu1} | f_{cu2} | f_{cu3} | f_{cu14d} | f_{cu1} | f_{cu2} | f_{cu3} | f_{cu284d} | |
C | 15.2 | 15.1 | 15.1 | 15.1 | 17.2 | 17.1 | 17.0 | 17.1 | 22.9 | 23.2 | 23.5 | 23.2 |
5 | 15.3 | 15.2 | 15.3 | 15.3 | 17.4 | 17.6 | 17.5 | 17.5 | 23.3 | 23.4 | 23.3 | 23.3 |
10 | 15.6 | 15.9 | 15.5 | 15.7 | 18.4 | 18.2 | 18.0 | 18.2 | 24.2 | 24.1 | 24.0 | 24.1 |
15 | 9.0 | 9.0 | 9.1 | 9.0 | 12.2 | 12.3 | 12.2 | 12.2 | 16.6 | 16.5 | 16.4 | 16.4 |
20 | 7.9 | 7.9 | 8.0 | 7.9 | 9.1 | 9.1 | 9.1 | 9.1 | 13.2 | 13.2 | 13.1 | 13.2 |
25 | 4.6 | 4.4 | 4.5 | 4.5 | 6.9 | 7.0 | 7.1 | 7.0 | 8.7 | 8.6 | 8.5 | 8.6 |
30 | 3.6 | 3.4 | 3.5 | 3.4 | 5.3 | 5.3 | 5.3 | 5.3 | 7.1 | 7.1 | 7.0 | 7.1 |
Curve Fittings
The data in table 1 was analyzed by fitting curves using polynomial regression and applying Mat-lab software. Polynomial regression is normally used to create a generic curve through the data points; making application of the mathematical form of the equation irrelevant [14]. There are two general approaches for curve fitting; least-squares regression and interpolation [15]. This research attempted to test both these curve techniques to ultimately come up with the most appropriate model. These techniques were represented by four models:
Least Square regression
Least Square Quadratic Polynomial (LSQP)
Interpolation.
Simultaneous Equation Polynomial (SEP)
Newton Interpolation Polynomial (NIP)
Lagrange Interpolating Polynomial (LIP)
Least Square Quadratic Polynomial
One method to accomplish curve fitting was to fit polynomials to the data using LSQP also referred as Non – Linear or Second Order Regression. MATLAB was applied to determine polynomial coefficients (a_{0}; a_{1} and a_{2}) using decreasing powers. The least-squares procedure can be readily extended to fit the data to a higher-order polynomial [16]. In this research a second-order polynomial or quadratic mathematical model was fitted to predict the compressive strength (f) with different SCBA content (s):
f = a_{0 }+ a_{1}s + a_{2 }s^{2 }+ e…….…………1.0
Simultaneous Equation Polynomial
SEP is a way of computing polynomial coefficients (p_{1}, p_{2 }and p of a parabola that passes through 3 predetermined values of s [16]. This enables generation of 3 linear algebraic equations that can be solved simultaneously for determination the 3 coefficients for compressive strengths (f).
f = p_{1 }s^{2} + p_{2 }s + p_{3}…………2.0
(c) Newton Interpolation Polynomial
NIP is one of the most popular and useful alternative form of expressing an interpolating polynomial [14]. If 3 data points are available, a 2^{nd} order polynomial will be of the form:
f = p_{1 +} p_{2}(s – s_{1}) + p_{3 }(s – s_{1}) (s – s_{2}) ..…3.0
The procedure was used to determine the value of the coefficients:
p_{1} = f (s_{1})
p_{2} = [f (s_{2}) – f (s_{1})]/ (s_{2 }– s_{1})
p_{3} = [f (s_{3}) – f (s_{2})]/ (s_{3 }– s_{2}) – [f (s_{2}) – f (s_{1})]/ (s_{2 }– s_{1})
(s_{3 }– s_{1})
(d) Lagrange Interpolating Polynomial
Basing on the table 1, the compressive strength – bagasse ash relationship has maxima (s_{2}) and two minimums (s_{1 }and s_{3}) for the three curing periods, 7, 14 and 28 days. Three parabolas were used with each one passing through one of the points and equaling zero at the other two. Their sum would then represent the unique parabola that connects the three points [15]. Such a second-order Lagrange interpolating polynomial can be written as:
f = (s – s_{2})(s – s_{3}) f(s_{1}) + (s – s_{1})(s – s_{3}) f(s_{2}) +
(s_{1}–s_{2})(s_{1}–s_{3}) (s_{2}–s_{1})(s_{2}–s_{3})
(s – s_{1})(s – s_{2}) f(s_{3})……..……………………3.0
(s_{3}– s_{1})(s_{3}– s_{2})
Model Validation
Model validation is possibly the most important step in the model building sequence. It is also one of the most overlooked. Often the validation of a model seems to consist of nothing more than quoting the r^{2} statistic from the fit (which measures the fraction of the total variability in the response that is accounted for by the model). The parameter estimation procedure picks out the best model. Model validation techniques include simulating the model under known input conditions and comparing model output with system output [16]. This paper was concerned with cross validation criterion for choice of the model which could be considered as an approximation for compressive strength determination of SCBA concrete. Used to perform model selection, cross validation is a widespread strategy because of its simplicity and its apparent universality in application [17] and is a model evaluation method that is better than simply looking at the residuals [20]. Residual evaluation does not indicate how well a model can make new predictions on cases it has not already seen.
With cross validation application, the quality of fit is evaluated on new data points that are not used to fit the parameters of the model. Specifically, this research adopted leave-one-out (LOO) cross-validation, a method which leaves a single data point out, fit the model on the remaining data point, predicts the left-out data point, and repeat this whole process for every single data point. r^{2} statistic was then be applied to quantify how close the model predictions match the data. LOO has many advantages over other cross validation techniques because it limits the number of splits [19], is a most exhaustive procedure [20] and will most certainly give unbiased estimate of true error and variance [21], [22].
Statistical analysis of the results of the three series of tests (at 7, 14 and 14 days curing periods) produced 4 models with their corresponding coefficient of determination (r^{2}) and standard error of estimate (s_{e}), the most commonly used measure of the spread or dispersion of data around the mean.
The summary of compressive strengths of SCBA concrete at various cement contents and curing periods are shown in Table 2. Where c is the % of SCBA replacing cement in concrete and f_{cu} is the actual compressive strengths at various curing periods.
Table 2. Compressive strengths of SCBA concrete for various models
c | Compressive Strength
(N/mm^{2}) |
|||||||||||||||
7d | 14d | 28d | ||||||||||||||
f_{cu} | LSQP | SEP | NIP | LIP | f_{cu} | LSQP | SEP | NIP | LIP | f_{cu} | LSQP | SEP | NIP | LIP | ||
0 | 15.1 | 16.23 | 15.10 | 15.10 | 15.10 | 17.1 | 16.20 | 18.33 | 17.10 | 17.10 | 15.1 | 23.2 | 24.46 | 23.20 | 23.20 | |
5 | 15.5 | 14.80 | 16.00 | 15.96 | 15.97 | 17.5 | 14.82 | 17.19 | 18.23 | 18.28 | 15.5 | 3.3 | 22.98 | 24.43 | 24.44 | |
10 | 15.7 | 13.00 | 15.70 | 15.70 | 15.71 | 18.2 | 13.08 | 13.47 | 18.20 | 18.20 | 15.7 | 24.1 | 20.83 | 24.10 | 24.11 | |
15 | 9.0 | 10.84 | 14.31 | 14.31 | 14.33 | 12.2 | 11.00 | 10.90 | 16.85 | 16.86 | 9.0 | 16.4 | 18.03 | 22.20 | 22.21 | |
20 | 7.9 | 8.35 | 11.80 | 11.80 | 11.81 | 9.1 | 8.56 | 10.90 | 14.24 | 14.27 | 7.9 | 3.2 | 14.56 | 18.74 | 18.75 | |
25 | 4.5 | 5.53 | 8.163 | 8.163 | 8.17 | 7.0 | 5.77 | 7.86 | 10.36 | 10.41 | 4.5 | 8.6 | 10.44 | 13.71 | 13.71 | |
30 | 3.4 | 2.36 | 3.400 | 3.400 | 3.41 | 5.3 | 2.64 | 14.34 | 5.22 | 5.30 | 3.4 | 7.1 | 5.66 | 7.12 | 7.11 |
From the tabular presentations above, all the interpolation models capture similar trends in the data but LSQP model seems to characterize the data better although it initially underestimate the data at middle values of % SCBA cement replacement in concrete. r^{2 }statistic is widely used as a goodness – of – fit measure, however a high r^{2 }does not imply a good model [23]. Focusing only on r^{2} can lead to selecting models that do not have inferior predictive capability. Instead researchers should evaluate a model according to a combination of r^{2} and cross validation metrics [24].
Cross Validation of the Models
Cross validation of the models will produce more robust models with better predictive powers. To establish the model that has the best fit quality this research adopted the LOO cross – validation approach. Cross validation of LSQP, SEP, NIP and LIP models gave new predictive compressive strengths data with their corresponding r^{2}. The compressive strength of BA at 7, 14, and 28days respectively, were determined as given in table 3
Table 3. Summary of r2 for all the validated models.
r^{2} | |||
7d | 14d | 28d | |
LSQP | 0.8508 | 0.8408 | 0.8408 |
SEP | 0.6502 | 0.5374 | 0.6444 |
NIP | 0.6657 | 0.6257 | 0.6665 |
LIP | 0.6662 | 0.7125 | 0.6803 |
Results and Discussions
There is a noticeable difference after validation; this reflects the fact that each predicted data point comes from different model estimate. This shows that the LSQP model achieves a higher cross – validated r^{2 }value than the other three Interpolating models. Further, the validated r^{2} among the interpolation models shows gradual increase in value to a maximum attained by LIP. This results can be collaborated by similar conclusion that solutions attributed to SEP are very sensitive to round-off errors hence need to apply NIP or LIP which do not manifest such short-coming [16].
It was also observed that the main challenge with application of interpolation models in determination of goodness of fit depends on the number of approximation points and their locations. Thus, LSQP model produces reliable and accurate predication values and could be used conveniently to estimate the compressive strength of concrete with SCBA as a partial replacement of cement. The strength of conventional concrete can generally be estimated with good accuracy based on strength charts or by experience if mix proportions, age, and curing conditions are known. However, when SCBA is used as cement replacement material the exact nature of strength development becomes complicated as such charts or experience would be no longer applicable, hence the need to adopt the LSQP model.
Conclusions
The LSQP model developed in this research is the ‘best’ as indicated by the validated r^{2} statistic which is near 0.85. ‘best’, a term frequently used in model simulation statistics, is a balance between model complexity and goodness of fit [23]. With these models, compressive strength of SCBA concrete at 7, 14 and 28 days can be effectively predicted over a range of cement replacement levels. The suggested models would be of value in the design process of such concrete mixes where specific target strength needs to be achieved at a certain age or can be used to modify any basic concrete mix so that the concretes with and without SCBA have acceptable compressive strengths.
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