Predicting Static Modulus of Elasticity of Laterite - Quarry Dust Blocks Using Osadebe's Regression Model

IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE)
e-ISSN: 2278-1684,p-ISSN: 2320-334X, Volume 14, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 21-26
www.iosrjournals.org
DOI: 10.9790/1684-1401042126 www.iosrjournals.org 21 | Page
Predicting Static Modulus of Elasticity of Laterite - Quarry Dust
Blocks Using Osadebe's Regression Model
F. O. Okafor1
and *E. A. Egbe2
1
Department of Civil Engineering, University of Nigeria, Nsukka
2
Operations Department, Cross River State Community & Social Dev. Agency, Calabar Nigeria
*egbesonandy@yahoo.com
Abstract: The use of alternative materials other than river sand in the production of sandcrete blocks is
gaining impetus in most major Nigerian cities. Manufacturers of these blocks are, however, faced with the
problem of mix proportion required for the production of the desired property. Static modulus of elasticity of
block is one key property required in the design of structures, which is usually neglected. In this work, a
mathematical model is formulated using Osadebe's regression theory for predicting the static modulus of
elasticity of laterite-quarry dust block. The model is tested for lack of fit using software and found adequate.
Keywords: Static modulus of elasticity, laterite, quarry dust, Osadebe regression model, mix proportion,
Fisher Test.
I. Introduction
Walls or masonry units are the vertical members of a building or structure which enclose the space
within it and which may also divide that space. Sandcrete blocks are still the most popularly and commonly used
material for the construction of walls unit in all developing countries including Nigeria. They are prismatic
precast units made from a combination or mix of well defined proportion of sand and cement. Sandcrete blocks
according to [1], are blocks made or moulded with sand, water and cement, which serve as a binder in the
matrix. As a result of the high cost and negative environmental impact caused by sand mining, one of the major
constituents used in the production of blocks, alternative materials are being sought for and utilized in block
production. Such alternative materials currently being used to partially or wholly replaced river sand includes:
quarry dust, laterite, recycled aggregate, etc. The use of these alternative materials in blocks production and
concrete works have been reported by several researchers. For instance, [2] carried out a study on the strength
and durability of concrete utilizing quarry dust as full replacement for natural sand. They found a 10%
improvement in the properties investigated over the conventional concrete made with natural sand. Also,[3] in
their work on effects of granite fines on some engineering properties of sandcrete blocks recommended 15%
optimum replacement of sand with quarry dust in the production of sandcrete blocks. Also, the effect of partial
replacement of sand with lateritic soil in sandcrete blocks was investigated by [4]. The study reveals that sand
can be replaced up to 20% with laterite in sandcrete blocks. While most of the studies focus on some strength
properties like compressive strength, properties like static modulus of elasticity (Ec) for blocks is scarcely
documented. Knowledge of the value of static modulus of material is needed in structural design to avoid
unrealistic assumptions.
There exist also the challenge of getting the appropriate mix proportion to attain a particular desired
property as several methods of mix proportions have limitations and are usually not cost effective. Thus,
researchers have worked on developing models for predicting properties of sand quarry dust and sand laterite
block. A model for static modulus of elasticity for sand- quarry dust blocks was developed by [5]. Similarly, [6]
developed a model for predicting the compressive strength and water absorption of sand - quarry dust blocks.
Others works in this direction include those by [7] and [8]. This paper presents a model for predicting the static
modulus of elasticity for laterite – quarry dust blocks. The model will help come up with the appropriate mix for
laterite – quarry dusts blocks as well as help reduce the cost and effort expended in conducting trial mix.
II. Materials And Method
The materials used for this work are:
Cement
Unicem brand of Ordinary Portland cement, grade 32.5 obtained from a major dealer in Calabar conforming to
BS 12 was used for all the tests.
Water
Potable pipe born water supplied by the Cross River State Water Board (CRSWB) Limited was used for both
specimen preparations and curing.

Predicting Static Modulus of Elasticity of Laterite –Quarry Dust Blocks Using Osadebe's ..
Laterite
Laterite was obtained from a borrow pit site at Akim - Akim in Odukpani Local Government Area of Cross
River State. The specific gravity for the laterite is 2.56.
Quarry dust
Quarry dust was obtained from the abundant deposits at Akamkpa quarry site in Akamkpa Local Government
area of Cross River State; located at a few minutes' drive from Calabar Metropolis. The quarry dust had a
specific gravity of 2.52.
III. Method
This study employs two methods: analytical and experimental. The analytical method deals with the
arrangement of points within the experimental region and selection of a second degree polynomial equation to
represent the response surface over the entire region. The response in this case is the static modulus of elasticity
of the laterite-quarry dust blocks. The response function is assumed to be multi-varied.
The response y is expressed as a function of the actual proportions of the constituents of the mixture, Zi by [9].
The sum of all the proportions as in all mixture experiments must add up to 1. That is:
𝑍1 + 𝑍2 + ⋯ + 𝑍 𝑞 = 𝑍𝑖
𝑞
𝐼=1
= 1 1
He assumed that y = F (Z), is continuous and differentiable with respect to its predictors, and can be expanded in
the neighbourhood of a chosen point, Z(0) using Taylor’s series.
𝑍 0 = 𝑍1
0
, 𝑍2
0
, … , 𝑍 𝑞
0 𝑇
2
Thus,
𝑦 𝑍 = 𝐹 𝑍 0
+
𝜕𝑓 𝑍(0)
𝜕𝑍𝑖
𝑞
𝑖=1
𝑍𝑖 − 𝑍 0
+
1
2!
𝜕2
𝑓 𝑍0
𝜕𝑍𝑖 𝜕𝑍𝑗
𝑞
𝑗 =1
𝑞−1
𝑖=1
𝑍𝑖 − 𝑍𝑖
0
𝑍𝑗 − 𝑍𝑗
0
+
1
2!
𝜕2
𝑓 𝑍(0)
𝜕𝑍𝑖
2
𝑞
𝑖=1
𝑍𝑖 − 𝑍 0
… ( 3)
For convenience, the point Z0
can be taken as the origin without loss in generality of the formulation.Thus
𝑍1
(0)
= 0, 𝑍2
(0)
= 0, . . , 𝑍 𝑞
(0)
= 0 (4)
Let
𝑏0 = 𝐹 0 , 𝑏𝑖 =
𝜕𝐹(0)
𝜕𝑍𝑖
, 𝑏𝑖𝑗 =
𝜕2
𝐹(0)
𝜕𝑍𝑖 𝜕𝑍𝑗
, 𝑏𝑖𝑖 =
𝜕2
𝐹(0)
𝜕𝑍𝑖
2 (5)
Substituting Equation (2.41 5) into Equation (2.39 3) gives:
𝑦 𝑍 = 𝑏0 + 𝑏𝑖 𝑍𝑖
𝑞
𝑖=1
+ 𝑏𝑖𝑗 𝑍𝑖 𝑍𝑗
𝑞
𝑖≤𝑗≤𝑞
+ 𝑏𝑖𝑖 𝑍𝑖
2
𝑞
𝑖=1
(6)
The number of terms in Equation (6) is 𝐶𝑛
(𝑞+𝑛)
Multiplying Equation (1) by b0 gives the expression:
𝑏0 = 𝑏0 𝑍1 + 𝑏0 𝑍2 + ⋯ … … + 𝑏0 𝑍 𝑞 (7)
Multiplying Equation (1) successively by Z1, Z2 … Zq and rearranging, gives respectively:
𝑍1
2
= 𝑍1 − 𝑍1 𝑍2 − … … … … . −𝑍1 𝑍 𝑞
𝑍2
2
= 𝑍2 − 𝑍1 𝑍2 − … … … . − 𝑍2 𝑍 𝑞
………………………………………………………
𝑍 𝑞
2
= 𝑍1 − 𝑍1 𝑍 𝑞 − … … … … . −𝑍(𝑞−1) 𝑍 𝑞 (8)
Substituting Equations (7) and (8) into Equation (6) and simplifying yields Equation (2.459)
𝑦 𝑍 = 𝛽𝑖 𝑍𝑖
𝑞
𝑖=1
+ 𝛽𝑖𝑗 𝑍𝑖 𝑍𝑗
𝑞
𝑖≤𝑗≤𝑞
(9)
Where
𝛽𝑖 = 𝑏0 + 𝑏𝑖 … … + 𝑏𝑖𝑖 (10)
𝛽𝑖𝑗 = 𝑏𝑖𝑗 − 𝑏𝑖𝑖 − 𝑏𝑖𝑗 (11)
Equation (9) is Osadebe’s regression model equation. It is defined if the unknown
constant coefficients 𝛽𝑖 and 𝛽𝑖𝑗 are uniquely determined.
If the number of constituents, q, is 4, and the degree of the polynomial n, is 2 then Osadebe's regression
equation is given as:

𝑦 = 𝛽1 𝑍1 + 𝛽2 𝑍2 + 𝛽3 𝑍3 + 𝛽4 𝑍4 + 𝛽12 𝑍1 𝑍2 + 𝛽13 𝑍1 𝑍3 + 𝛽14 𝑍1 𝑍4
+ 𝛽23 𝑍2 𝑍3 + 𝛽24 𝑍2 𝑍4 + 𝛽34 𝑍3 𝑍4 (12)
The number of coefficients, N is the same as that for the Scheffe’s {4, 2} model as provided by Equation 12.
That is:
𝑁 = 𝐶𝑛
(𝑞+𝑛−1)
= 𝑁 = 𝐶2
(4+2−1)
= 10
The response function is generally expressed as:
y Z = βiZi
q
i=1 + βijZiZj
q
i≤j≤q (13)
IV. Determination of the coefficients of the Osadebe’s regression equation
The least number of experimental runs or independent responses necessary to determine the coefficients of
Osadebe's regression coefficients is N.
Let y(k)
be the response at point k and the vector corresponding to the set of component proportions (predictors)
at point k be Z(k).
That is:
Z(k)
= { Z1
(k),
Z2
(k)
, ……., Zq
(k)
} (14)
Substituting the vector of Equation (3) into Equation (12) gives:
𝑦(𝑘)
= 𝛽𝑖 𝑍𝑖
(𝑘)
𝑞
𝑖=1
+ 𝛽𝑖𝑗 𝑍𝑖
(𝑘)
𝑍𝑗
(𝑘)
𝑞
𝑖≤𝑗≤𝑞
𝑘 = 1,2, … , 𝑁 (15)
Substituting the predictor vectors at each of the N observation points successively into Equation (9) gives a set
of N linear algebraic equations which can be written in matrix form as:
𝒁𝜷 = 𝒚 (16)
Where
𝜷 is a vector whose elements are the estimates of the regression coefficients.
𝒁 is an N xN matrix whose elements are the mixture component proportions
and functions of the component proportions.
𝒚 is a vector of the observations or responses at the various N observation points.
The solution to Equation (16) is given as:
𝜷 = 𝒁−1
𝒚 (17)
V. Experimental method
The pseudo proportion units were converted to actual mix proportion as indicated in table 1. The
actual mix proportions; water (Z1), cement (Z2), quarry dust (Z3), and laterite (Z4), were measured by weight
and used to produce machine vibrated laterite- quarry dust hollow blocks of size 450mm x 150mm x 225mm.
The blocks were cured for 28 days after 24 hours of demoulding by sprinkling with water in the morning and
evening. They were tested for compressive strength using the universal compression testing machine, in
accordance with BS EN 12390-4. The crushing load and net cross sectional area of the blocks were recorded.
The compressive strength was obtained from the following relation:
fc = P/A (18)
Where fc = the compressive strength, P = crushing load, and A = cross-sectional area of the specimen
The Static modulus of elasticity for the block was computed as a function of compressive strength and density
using the relation established by [10] represented as:
Ec =1.7ρ2
fc 0.33
*10-6
(19)
Where,
Ec = Static modulus of Elasticity, ρ = density and
fc = compressive strength
Table 1 below shows the design matrix in the pseudo and real ratios along with the experimental test results for
the compressive strength and static modulus of elasticity.
Table 1: Experimental test results
S/
N
Pseudo components units Actual mix ratio Average response (y)
Water
(X1)
Cement
(X2)
Quarry dust
(X3)
Laterite
(X4)
Water
(X1)
Cement
(X2)
Quarry
dust
(X3)
Laterite
(X4)
Compressive
strength, fc
(Nmm-2
)
Static modulus
of Elasticity
(GPa)
1 0 1 0 0 0.63 1 3.0 3.0 1.87 6.9476
2 0.25 0.25 0.25 0.25 0.72 1 5.6 2.4 2.56 8.287
3 1 0 0 0 0.54 1 5.4 0.6 2.50 8.8566

4 0.5 0 0.5 0 0.67 1 7.2 0.8 2.24 7.9930
5 0 0.5 0 0.5 0.77 1 4.0 4.0 2.37 8.2285
6 0.5 0 0 0.5 0.72 1 5.2 2.8 2.37 8.0286
7 0 0 1 0 0.80 1 9.0 1.0 1.81 7.9795
8 0.125 0.125 0.125 0.625 0.81 1 5.3 3.7 2.42 8.6075
9 0.5 0.5 0 0 0.58 1 4.2 1.8 2.56 8.3193
10 0 0 0.5 0.5 0.85 1 7.0 3.0 2.09 8.1875
11 0 1 0 0 0.63 1 3.0 3.0 1.89 7.0700
12 0 0 0 1 0.90 1 5.0 5.0 2.20 7.6358
13 1 0 0 0 0.54 1 5.4 0.6 2.45 8.4865
14 0 0.5 0.5 0 0.72 1 6.0 2.0 2.54 8.8676
15 0 0 0 1 0.90 1 5.0 5.0 2.20 8.2145
16 0.625 0.125 0.125 0.125 0.63 1 5.5 1.5 2.56 8.9084
17 0 0 1 0 0.80 1 9.0 1.0 1.90 8.0343
18 0.25 0.25 0.25 0.25 0.72 1 5.6 2.4 2.49 8.1386
19 0.125 0.625 0.125 0.125 0.67 1 4.3 2.7 2.50 8.9417
20 0.125 0.125 0.625 0.125 0.76 1 7.3 1.7 2.30 8.4656
A total of 15 mixes were considered. Out of this number,10 mixes were selected and used for the
formulation of the model, while the remaining were used for validation of the model as shown in table 2.
The table also contained the average experimental values for compressive strength and static modulus of
elasticity. Cells having two run order numbers indicate the replicate mixes and the response in this case is the
average response for the replicate mixes.
Table 2: Actual and fractional mix for Osadebe
VI. Results And Discussion
The test results of the compressive strength of the laterite-quarry dust blocks based on 28-day strength and
corresponding static modulus of elasticity are presented as part of Table 1.
Formulation of Model equation for static modulus of elasticity
The elements of the Z matrix with reference to equation 17 are as provided in Table 3
Table 3: Elements of the Z matrix for the Osadebe’s model
Z1 Z2 Z3 Z4 Z1Z2 Z1Z3 Z1Z4 Z2Z3 Z2Z4 Z3Z4
0.071618 0.132626 0.71618 0.079576 0.009498 0.051291 0.005699 0.094984 0.010554 0.05699
0.077126 0.131839 0.553724 0.23731 0.010168 0.042707 0.018303 0.073003 0.031287 0.131405
0.069286 0.103413 0.744571 0.08273 0.007165 0.051589 0.005732 0.076998 0.008555 0.061598
0.074074 0.102881 0.534979 0.288066 0.007621 0.039628 0.021338 0.055039 0.029636 0.154109
0.082569 0.131062 0.393185 0.393185 0.010822 0.032465 0.032465 0.051531 0.051531 0.154594
0.074074 0.102881 0.617284 0.205761 0.007621 0.045725 0.015242 0.063507 0.021169 0.127013
0.078813 0.102354 0.409417 0.409417 0.008067 0.032267 0.032267 0.041905 0.041905 0.167622
0.067797 0.084746 0.762712 0.084746 0.005745 0.051709 0.005745 0.064637 0.007182 0.064637
0.07173 0.084388 0.590717 0.253165 0.006053 0.042372 0.018159 0.04985 0.021364 0.149549
0.07563 0.084034 0.420168 0.420168 0.006355 0.031777 0.031777 0.035308 0.035308 0.176541

Substituting the numerical average replicate values of static modulus of elasticity observed at the ten design
points of the simplex into Equation (17) and solving simultaneously gives the following values of the
coefficients:
𝛽1 = - 72079.9872 𝛽2 = 527.8603 𝛽3 = - 406.1002 𝛽4 = - 736.4554
𝛽12 = 80706.0764, 𝛽13 = 83672.7022 𝛽14 = 87627.9165 𝛽23 = - 1058.1584
𝛽24 = - 982.6645 𝛽34 = 58.9751
The resulting second degree regression equation for Osadebe’s model is given below:
ŷ = - 72079.9872Z1 + 527.8603Z2 - 406.1002Z3 – 736.4554Z4 + 80706.0764Z1Z2
+ 83672.7022Z1Z3 + 87627.9165Z1Z4 - 1058.1584Z2Z3 – 982.6645Z2Z4
+ 58.9751Z3Z4 (20)
Table 4 presents the experimental and predicted values of static modulus of elasticity of the laterite quarry dust
blocks.
Table 4: Experimental and model predicted Static modulus of elasticity results
Run Order Actual mix ratios Experimental
result
(GPa)
Water Cement Quarry dust Laterite Model predicted result
(Osadebe) GPa
1 0.63 1 3 3 6.9476 6.95
2 0.72 1 5.6 2.4 8.2870 8.55
3 0.54 1 5.4 0.6 8.8566 8.49
4 0.67 1 7.2 0.8 7.9930 7.99
5 0.77 1 4 4 8.2285 8.23
6 0.72 1 5.2 2.8 8.0286 8.03
7 0.8 1 9 1 7.9795 7.98
8 0.81 1 5.3 3.7 8.6075 8.18
9 0.585 1 4.2 1.8 8.3193 8.32
10 0.85 1 7 3 8.1875 8.19
11 0.63 1 3 3 7.0700 6.95
12 0.9 1 5 5 7.6358 7.64
13 0.54 1 5.4 0.6 8.4865 8.49
14 0.72 1 6 2 8.8676 8.87
15 0.9 1 5 5 8.2145 7.64
16 0.63 1 5.5 1.5 8.9084 8.41
17 0.8 1 9 1 8.0343 7.98
18 0.72 1 5.6 2.4 8.1386 8.55
19 0.674 1 4.3 2.7 8.9417 8.34
20 0.76 1 7.3 1.7 8.4656 8.41
Test for adequacy of the model
The model was tested for adequacy against the controlled experimental results. The hypotheses for the model
are as follows:
Null Hypothesis (Ho): there is no significant difference between the experimental and the theoretical estimated
results at a 95% confidence level
Alternative Hypothesis (H1): There is a significant difference between the experimental and theoretically
expected result at a 95% confidence level. The Fisher Test was used to test for the adequacy of the model. The
analysis of variance for the Fisher test using [11] at the check point is as shown in Table 45 below. The
calculated F from the table is 3.72 which is less than the critical (tabulated) F value of 5.05, justifying the
adequacy of the model equation. Again the p-value of 0.088 which is greater than 0.05 further indicates the
adequacy of the model.
Table 5: Analysis of Variance table for Static modulus of Elasticity (Osadebe’s model)
Run order y(observed) y(predicted) y(observed) (GPa) Y(predicted)
(GPa) (GPa) (GPa)
2 8.287 8.548 Mean 8.625167 8.359667
18 8.139 8.548 Variance 0.118078 0.031698
16 8.908 8.13 Observations 6 6
19 8.942 8.341 df 5 5

20 8.868 8.41 F 3.725114
8 8.607 8.181 P(F<=f) one-tail 0.087631
F Critical one-tail 5.050329
Normal probability plot
Figure 1shows normal probability plot. The points in figure lie very close to the reference line with a p-value of
0.145 which is greater than 0.05. The data therefore follow a normal distribution, thereby justifying the
assumption required for use of analysis of variance
1.00.50.0-0.5-1.0
99
95
90
80
70
60
50
40
30
20
10
5
1
Residuals
Percent
Mean 0.1456
StDev 0.3059
N 14
AD 0.529
P-Value 0.145
Normal - 95% CI
Figure 1: Normal probability plot for Static modulus of elasticity residuals Osadebe’s model)
VII. Conclusion
A mathematical model for predicting the static modulus of elasticity of laterite-quarry dust block based
on actual proportions using Osadebe's theory was formulated. The model was tested for lack of fit and was
found to be adequate. There was no significant difference between the experimented and predicted values.
Reference
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dust as fine aggregate. ARPN Journal of Engineering and Applied Science. Vol. 3 No. 5 pp 20-26
[3]. Oyekan, G. L. and Kamiyo, O. M (2008) Effects of Granite fines on some engineering properties of sandcrete blocks. A paper
presented at 33rd
conference on our world in concrete and structure at Singapore.
[4]. Josphua, O., Amusan, L. M., Fagbenle, O. I. and Kukoyi, P. O. (2014): ''Effects of Partial Replacement of sand with Lateritic soil in
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sand- quarry dust blocks. The international Journalof Engineering and Science, pp 27-31
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inclusion. Nigerian Journal of Technology, Nsukka, Nigeria. 34(1) pp 1-11
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