Application of ibearugbulem’s model for optimizing granite concrete mix
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This document presents a mathematical model for predicting the compressive strength of granitic concrete using Ibearugbulem's regression model. 45 concrete cubes were cast using different mix ratios and tested to determine compressive strength. The first 11 mixes were used to determine the regression model coefficients, and the full 15 mixes were used to validate the model. A t-test found the model results adequately predicted the experimental compressive strengths at a 95% confidence level. The developed model can predict compressive strength or mix proportions for granitic concrete given one is known.
![IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 405
APPLICATION OF IBEARUGBULEM’S MODEL FOR OPTIMIZING
GRANITE CONCRETE MIX
Ibearugbulem O.M1
, Amanambu, I.K2
, Elogu T.I3
1
Civil Engineering Department, Federal University of Technology, Owerri, Nigeria
2
Civil Engineering Department, Madonna University, Elele, Nigeria
3
Civil Engineering Department, Madonna University, Elele, Nigeria
Abstract
This work aims at developing a mathematical model for predicting the compressive strength of granitic concrete using
Ibearugbulem’s regression model. A total of 45 cubes were cast, comprising of 3 cubes for each mix ratio of a total of 15 mix
ratios. The first 11 mix ratios were used to determine the coefficients of the regression while the whole 15 was used to validate the
model. The results from the model compared favourably with the experimental results. The model was tested with the statistical
student t-test and found to be adequate at 95% confidence level. With the model developed in this work, any desired compressive
made from granite concrete can be predicted if the mix proportions are known and vice versa.
Keywords: Ibearugbulem’s regression, compressive strength.
--------------------------------------------------------------------***------------------------------------------------------------------
1. INTRODUCTION
Ibearugbulem et al (2013) observed that Scheffe’s and
Osadebe’s models are statistical methods of concrete mix
design most frequently used in Civil Engineering. These
methods are quite suitable for concrete mix optimization but
they are greatly limited in that a predetermined number of
experiments must be carried out in order to formulate them
and they can only be applied for mix ratios that fall within
the predetermined observation points (Scheffe, 1958 &
1963, Obam 1998 & 2006, Ibearugbulem, 2008, Osadebe
and Ibearugbulem, 2008 & 2009). It is based on this that led
to the search of an alternative method. Ibearugbulem’s
regression was formulated to take care of the inherent
problems of the previous models (Ibearugbulem 2013).
2. IBEARUGBULEM’S REGRESSION
EQUATION
Ibearugbulem et al (2013), gave the final model equation as
----------1
Solving the above matrix will give the values of the
coefficient of regression function equation. This equation (1)
can be written in a short form as
[F(z).Z] = [CC] [α] ----------2
CC is always a symmetric matrix
In this study we are considering a mixture of three
components whose CC is a 7 x 7 matrix shown in equation
(3)](https://image.slidesharecdn.com/applicationofibearugbulemsmodelforoptimizinggraniteconcretemix-141116015336-conversion-gate01/85/Application-of-ibearugbulem-s-model-for-optimizing-granite-concrete-mix-1-320.jpg)
![IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 407
Four materials were used, but the components of the mix are
three. They are water/cement ratio (S1), sand/cement ratio
(S2) and granite/cement ratio (S3).
Note: S = S1 + S2 + S3 and Zi = Si / S. It will observed that
the actual number of elements in the concrete mix used
herein is four, but this regression is keeping cement
constant, thereby reducing the components for use in the
regression to three. It is an improvement the present
Ibearugbulem’s regression has over Scheffe’s and Osadebe’s
regression.
Substituting the values of Zi in table1 into CC matrix of
equation (3) gave:
CC Matrix =
0.053741 0.245248 0.4682 0.017174 0.032787 0.146846 0.01028
0.245248 1.165412 2.106868 0.081227 0.146846 0.685551 0.047766
0.4682 2.106868 4.140213 0.146846 0.288567 1.274471 0.0888
0.017174 0.081227 0.146846 0.005686 0.01028 0.047766 0.003343
0.032787 0.146846 0.288567 0.01028 0.020201 0.0888 0.006214
0.146846 0.685551 1.274471 0.047766 0.0888 0.407111 0.028357
0.01028 0.047766 0.0888 0.003343 0.006214 0.028357 0.001984
Inverse of CC Matrix =
-979881 -53279.7 -325.924 1794966 1052040 62844.19 -843216
-53279.7 -11780.8 -6.32644 237724.2 60304.17 18514.26 -294078
-325.924 -6.32644 839.9567 6954.5 -9898.7 -2479.33 18968.07
1794966 237724.2 6954.5 -5183080 -1987541 -369773 4908433
1052040 60304.17 -9898.7 -1987541 -986691 -41758.2 576277
62844.19 18514.26 -2479.33 -369773 -41758.2 -21778.2 404725.2
-843216 -294078 18968.07 4908433 576277 404725.2 -5259490
Using the values Zi from table 1 and the laboratory
compressive strength the RZ vector was obtained as;
∑(Z1.F(z)) = 18.79579 ------6
∑(Z2.F(z)) = 85.71375 ------7
∑(Z3.F(z)) = 164.9504 ------8
∑(Z1Z2.F(z)) = 5.976329 ------9
∑(Z1Z3.F(z)) = 11.50265 -----10
∑(Z2Z3.F(z)) = 51.47409 -----11
∑(Z1Z2Z3.F(z)) = 3.588031 -----12
Substituting the values of [F(z).Z] and the inverse of CC
matrix into Equation (2), the coefficient [α] of the regression
which upon substituting into equation 4 gave;
F(z) = -290.299Z1 - 37.3228Z2 + 20.5639Z3 + 1278.404Z1Z2
+ 491.0673Z1Z3 + 105.4005Z2Z3 - 1905.23Z1Z2Z3
Using the above final model equation, the model
compressive strength is obtained thus; (see table 2)
Student t- test on the compressive cube strength was carried
out to determine whether there exist significant difference
between the compressive strengths from the laboratory and
those from the model. The results are shown on table 2](https://image.slidesharecdn.com/applicationofibearugbulemsmodelforoptimizinggraniteconcretemix-141116015336-conversion-gate01/85/Application-of-ibearugbulem-s-model-for-optimizing-granite-concrete-mix-3-320.jpg)
![IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 408
Table 2: LEGEND Di = (YM - YE) DA = ∑ Di / N
S/N YE YM Di DA - Di (DA - Di)2
1 25.48 25.61 0.13 -0.263333 0.069344
2 24.44 24.78 0.34 -0.473333 0.224044
3 22.22 24.06 1.84 -1.973333 3.894043
4 25.33 23.46 -1.87 1.736667 3.016012
5 23.85 25.16 1.31 -1.443333 2.083210
6 29.78 24.43 -5.35 5.216667 27.213615
7 20.44 23.88 3.44 -3.573333 12.768709
8 27.11 25.57 -1.54 1.406667 1.978712
9 20.89 24.76 3.78 -3.913333 15.314175
10 26.81 24.20 -2.61 2.476667 6.133879
11 23.11 23.90 0.79 -0.923333 0.852544
12 25.33 24.23 -1.1 0.966667 0.934445
13 24.89 24.96 0.07 -0.203333 0.041344
14 24.74 23.78 -0.96 0.826667 0.683378
15 24.59 24.32 -0.27 0.136667 0.018678
Total -2 75.226132
YE = Experimental Result
YM = Model Result
DA = ∑ Di / N = -2/15 = -0.133333
S2
= ∑ (DA - Di)2
/ (N - 1) = 75.226132/(15 – 1) = 5.373295
S = √5.373295 = 2.318
t = DA*√N / S = -0.133333*√15 / 2.318 = -0.2227
/t/ = 0.2227
t value from table is given as tα/N, (V).
This is equal to t 0.05 / 15, (14) = t0.0033, (14)
From Statistics table, t0.0033, (14) = 3.06 (by interpolation),
whereas calculated t is 0.2227
The t from statistical table is greater than the calculated t
value of 0.2227 i.e. (t(table) > t(calculated)), hence the Null
Hypothesis is accepted and Alternative Hypothesis rejected.
This means that there is no significant difference between
the results predicted by the model and the laboratory.
5. CONCLUSIONS
The Student t-test revealed that the compressive cube
strength predicted by the regression is very close to those
from the experiment. The calculated t (t(calculated) = 0.2227) is
less than the allowable t or t from statistic table (t(table) =
3.06) at 95% confidence level. Thus, within 95% confidence
level, one can predict the compressive cube strength of
concrete made with water, cement (OPC), river sand and
granite by using this model. Based on this result, it will be
concluded that the “IBEARUGBULEM’S Regression” as
used herein is working well and can be used to optimize
granitic concrete mixes at 95% confidence level.
REFERENCES
[1] Ibearugbulem, O.M. (2006) “Mathematical models
for optimization of compressive strength of
periwinkle shell-granite concrete”. A master’s thesis
submitted to postgraduate school Federal University
of Technology, Owerri.
[2] Ibearugbulem, O.M. (2013) “A New Regression
Model for Optimizing Concrete Mix” International
Journal of Engineering Sciences & Research
Technology, vol.2 No.7, ISSN: 2277-9655,
(Published).
[3] Osadebe, N.N., (2003). “Generalized Mathematical
Modelling of Compressive Strength of Normal
Concrete as Multi-variant function of the Properties
of its Constituents components”. A paper delivered
at college of Engineering, University of Nigeria
Nsukka.
[4] Osadebe, N. N. and Ibearugbulem, O. M. (2008)
“Application of Osadebe’s alternative regression
model in optimizing compressive strength of
periwinkle shell-granite concrete”. NSE Technical
transaction, vol.43, no.1, Jan. – Mar, pp.47-59.
[5] Osadebe, N. N. and Ibearugbulem, O. M. (2009),
“Application of Scheffe’s simplex model in
optimizing compressive strength of periwinkle shell
granite concrete,” The Heartland Engineer, vol.4,
No. 1, pp. 27 – 38.](https://image.slidesharecdn.com/applicationofibearugbulemsmodelforoptimizinggraniteconcretemix-141116015336-conversion-gate01/85/Application-of-ibearugbulem-s-model-for-optimizing-granite-concrete-mix-4-320.jpg)
Application of ibearugbulem’s model for optimizing granite concrete mix
- 1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _______________________________________________________________________________________ Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 405 APPLICATION OF IBEARUGBULEM’S MODEL FOR OPTIMIZING GRANITE CONCRETE MIX Ibearugbulem O.M1 , Amanambu, I.K2 , Elogu T.I3 1 Civil Engineering Department, Federal University of Technology, Owerri, Nigeria 2 Civil Engineering Department, Madonna University, Elele, Nigeria 3 Civil Engineering Department, Madonna University, Elele, Nigeria Abstract This work aims at developing a mathematical model for predicting the compressive strength of granitic concrete using Ibearugbulem’s regression model. A total of 45 cubes were cast, comprising of 3 cubes for each mix ratio of a total of 15 mix ratios. The first 11 mix ratios were used to determine the coefficients of the regression while the whole 15 was used to validate the model. The results from the model compared favourably with the experimental results. The model was tested with the statistical student t-test and found to be adequate at 95% confidence level. With the model developed in this work, any desired compressive made from granite concrete can be predicted if the mix proportions are known and vice versa. Keywords: Ibearugbulem’s regression, compressive strength. --------------------------------------------------------------------***------------------------------------------------------------------ 1. INTRODUCTION Ibearugbulem et al (2013) observed that Scheffe’s and Osadebe’s models are statistical methods of concrete mix design most frequently used in Civil Engineering. These methods are quite suitable for concrete mix optimization but they are greatly limited in that a predetermined number of experiments must be carried out in order to formulate them and they can only be applied for mix ratios that fall within the predetermined observation points (Scheffe, 1958 & 1963, Obam 1998 & 2006, Ibearugbulem, 2008, Osadebe and Ibearugbulem, 2008 & 2009). It is based on this that led to the search of an alternative method. Ibearugbulem’s regression was formulated to take care of the inherent problems of the previous models (Ibearugbulem 2013). 2. IBEARUGBULEM’S REGRESSION EQUATION Ibearugbulem et al (2013), gave the final model equation as ----------1 Solving the above matrix will give the values of the coefficient of regression function equation. This equation (1) can be written in a short form as [F(z).Z] = [CC] [α] ----------2 CC is always a symmetric matrix In this study we are considering a mixture of three components whose CC is a 7 x 7 matrix shown in equation (3)
- 2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _______________________________________________________________________________________ Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 406 -----------3 For a mixture of three components, the final model equation is given as F(z) = α1Z1 + α2Z2 + α3Z3 + α12Z1Z2 + α13Z1Z3 +α23Z2Z3 + α123Z1Z2Z3 ------------4 3. MATERIALS USED FOR THE COMPRESSIVE STRENGTH TEST Four materials were used for the experimental (laboratory) test. They include: i. Water that is fit for drinking and taken from borehole tap in Madonna University, Akpugo Campus in Enugu State, Nigeria was used. ii. The cement used for this test was Dangote cement, a brand of ordinary Portland cement that conforms to BS 12 (1978). iii. River sand that was free from deleterious matters, and was well graded in the size range of 0.15mm ≤ x ≤ 4.75mm was used. iv. Granite aggregate that was free from deleterious matters and conforms to BS 882 (1992) was used. Its size range is 4.75 mm ≤ x ≤ 19 mm. 4. METHOD The materials were batched by mass in their dried state. Cement and sand were mixed thoroughly together first and granite aggregate was introduced and the three materials were mixed thoroughly to a consistent state using manual means. After mixing properly to a consistent state the concrete was cast into the moulds and de-moulded after 24hrs. The cubes were cured for 28 days after which they were crushed using the analogue okhard compression machine and the compressive strength was determined using equation (6) where the cross-sectional area is 22500mm2 . Compressive Strength = Failure Load (N) / Cross sectional Area (mm2 ) -------5 Table 1 values for S and Z S/N S1 S2 S3 S Z1 Z2 Z3 Z1Z2 Z1Z3 Z2Z3 Z1Z2Z3 N1 0.55 2 6 8.55 0.064327 0.233918 0.701754 0.015047 0.045142 0.164153 0.01056 N2 0.55 2.5 5.5 8.55 0.064327 0.292398 0.643275 0.018809 0.04138 0.188092 0.012099 N3 0.55 3 5 8.55 0.064327 0.350877 0.584795 0.022571 0.037618 0.205191 0.013199 N4 0.55 3.5 4.5 8.55 0.064327 0.409357 0.526316 0.026333 0.033857 0.215451 0.013859 N5 0.6 2.25 5.75 8.6 0.069767 0.261628 0.668605 0.018253 0.046647 0.174926 0.012204 N6 0.6 2.75 5.25 8.6 0.069767 0.319767 0.610465 0.022309 0.042591 0.195207 0.013619 N7 0.6 3.25 4.75 8.6 0.069767 0.377907 0.552326 0.026366 0.038534 0.208728 0.014562 N8 0.65 2 6 8.65 0.075145 0.231214 0.693642 0.017374 0.052123 0.16038 0.012052 N9 0.65 2.5 5.5 8.65 0.075145 0.289017 0.635838 0.021718 0.04778 0.183768 0.013809 N10 0.65 3 5 8.65 0.075145 0.346821 0.578035 0.026062 0.043436 0.200474 0.015065 N11 0.65 3.5 4.5 8.65 0.075145 0.404624 0.520231 0.030405 0.039093 0.210498 0.015818 C1 0.55 2.875 5.125 8.55 0.064327 0.336257 0.599415 0.021631 0.038559 0.201558 0.012966 C2 0.6 2.375 5.625 8.6 0.069767 0.276163 0.65407 0.019267 0.045633 0.18063 0.012602 C3 0.6 3.375 4.625 8.6 0.069767 0.392442 0.537791 0.02738 0.03752 0.211052 0.014725 C4 0.65 2.875 5.125 8.65 0.075145 0.33237 0.592486 0.024976 0.044522 0.196924 0.014798
- 3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _______________________________________________________________________________________ Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 407 Four materials were used, but the components of the mix are three. They are water/cement ratio (S1), sand/cement ratio (S2) and granite/cement ratio (S3). Note: S = S1 + S2 + S3 and Zi = Si / S. It will observed that the actual number of elements in the concrete mix used herein is four, but this regression is keeping cement constant, thereby reducing the components for use in the regression to three. It is an improvement the present Ibearugbulem’s regression has over Scheffe’s and Osadebe’s regression. Substituting the values of Zi in table1 into CC matrix of equation (3) gave: CC Matrix = 0.053741 0.245248 0.4682 0.017174 0.032787 0.146846 0.01028 0.245248 1.165412 2.106868 0.081227 0.146846 0.685551 0.047766 0.4682 2.106868 4.140213 0.146846 0.288567 1.274471 0.0888 0.017174 0.081227 0.146846 0.005686 0.01028 0.047766 0.003343 0.032787 0.146846 0.288567 0.01028 0.020201 0.0888 0.006214 0.146846 0.685551 1.274471 0.047766 0.0888 0.407111 0.028357 0.01028 0.047766 0.0888 0.003343 0.006214 0.028357 0.001984 Inverse of CC Matrix = -979881 -53279.7 -325.924 1794966 1052040 62844.19 -843216 -53279.7 -11780.8 -6.32644 237724.2 60304.17 18514.26 -294078 -325.924 -6.32644 839.9567 6954.5 -9898.7 -2479.33 18968.07 1794966 237724.2 6954.5 -5183080 -1987541 -369773 4908433 1052040 60304.17 -9898.7 -1987541 -986691 -41758.2 576277 62844.19 18514.26 -2479.33 -369773 -41758.2 -21778.2 404725.2 -843216 -294078 18968.07 4908433 576277 404725.2 -5259490 Using the values Zi from table 1 and the laboratory compressive strength the RZ vector was obtained as; ∑(Z1.F(z)) = 18.79579 ------6 ∑(Z2.F(z)) = 85.71375 ------7 ∑(Z3.F(z)) = 164.9504 ------8 ∑(Z1Z2.F(z)) = 5.976329 ------9 ∑(Z1Z3.F(z)) = 11.50265 -----10 ∑(Z2Z3.F(z)) = 51.47409 -----11 ∑(Z1Z2Z3.F(z)) = 3.588031 -----12 Substituting the values of [F(z).Z] and the inverse of CC matrix into Equation (2), the coefficient [α] of the regression which upon substituting into equation 4 gave; F(z) = -290.299Z1 - 37.3228Z2 + 20.5639Z3 + 1278.404Z1Z2 + 491.0673Z1Z3 + 105.4005Z2Z3 - 1905.23Z1Z2Z3 Using the above final model equation, the model compressive strength is obtained thus; (see table 2) Student t- test on the compressive cube strength was carried out to determine whether there exist significant difference between the compressive strengths from the laboratory and those from the model. The results are shown on table 2
- 4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _______________________________________________________________________________________ Volume: 03 Issue: 08 | Aug-2014, Available @ http://www.ijret.org 408 Table 2: LEGEND Di = (YM - YE) DA = ∑ Di / N S/N YE YM Di DA - Di (DA - Di)2 1 25.48 25.61 0.13 -0.263333 0.069344 2 24.44 24.78 0.34 -0.473333 0.224044 3 22.22 24.06 1.84 -1.973333 3.894043 4 25.33 23.46 -1.87 1.736667 3.016012 5 23.85 25.16 1.31 -1.443333 2.083210 6 29.78 24.43 -5.35 5.216667 27.213615 7 20.44 23.88 3.44 -3.573333 12.768709 8 27.11 25.57 -1.54 1.406667 1.978712 9 20.89 24.76 3.78 -3.913333 15.314175 10 26.81 24.20 -2.61 2.476667 6.133879 11 23.11 23.90 0.79 -0.923333 0.852544 12 25.33 24.23 -1.1 0.966667 0.934445 13 24.89 24.96 0.07 -0.203333 0.041344 14 24.74 23.78 -0.96 0.826667 0.683378 15 24.59 24.32 -0.27 0.136667 0.018678 Total -2 75.226132 YE = Experimental Result YM = Model Result DA = ∑ Di / N = -2/15 = -0.133333 S2 = ∑ (DA - Di)2 / (N - 1) = 75.226132/(15 – 1) = 5.373295 S = √5.373295 = 2.318 t = DA*√N / S = -0.133333*√15 / 2.318 = -0.2227 /t/ = 0.2227 t value from table is given as tα/N, (V). This is equal to t 0.05 / 15, (14) = t0.0033, (14) From Statistics table, t0.0033, (14) = 3.06 (by interpolation), whereas calculated t is 0.2227 The t from statistical table is greater than the calculated t value of 0.2227 i.e. (t(table) > t(calculated)), hence the Null Hypothesis is accepted and Alternative Hypothesis rejected. This means that there is no significant difference between the results predicted by the model and the laboratory. 5. CONCLUSIONS The Student t-test revealed that the compressive cube strength predicted by the regression is very close to those from the experiment. The calculated t (t(calculated) = 0.2227) is less than the allowable t or t from statistic table (t(table) = 3.06) at 95% confidence level. Thus, within 95% confidence level, one can predict the compressive cube strength of concrete made with water, cement (OPC), river sand and granite by using this model. Based on this result, it will be concluded that the “IBEARUGBULEM’S Regression” as used herein is working well and can be used to optimize granitic concrete mixes at 95% confidence level. REFERENCES [1] Ibearugbulem, O.M. (2006) “Mathematical models for optimization of compressive strength of periwinkle shell-granite concrete”. A master’s thesis submitted to postgraduate school Federal University of Technology, Owerri. [2] Ibearugbulem, O.M. (2013) “A New Regression Model for Optimizing Concrete Mix” International Journal of Engineering Sciences & Research Technology, vol.2 No.7, ISSN: 2277-9655, (Published). [3] Osadebe, N.N., (2003). “Generalized Mathematical Modelling of Compressive Strength of Normal Concrete as Multi-variant function of the Properties of its Constituents components”. A paper delivered at college of Engineering, University of Nigeria Nsukka. [4] Osadebe, N. N. and Ibearugbulem, O. M. (2008) “Application of Osadebe’s alternative regression model in optimizing compressive strength of periwinkle shell-granite concrete”. NSE Technical transaction, vol.43, no.1, Jan. – Mar, pp.47-59. [5] Osadebe, N. N. and Ibearugbulem, O. M. (2009), “Application of Scheffe’s simplex model in optimizing compressive strength of periwinkle shell granite concrete,” The Heartland Engineer, vol.4, No. 1, pp. 27 – 38.