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This research work focused on the development of mathematical model for the effective replacement of Portland cement by sugar cane bagasse in mortar and concrete. The model techniques used here is Scheffe's Simplex Design. A total of ninety (90) cubes were cast, consisting of three cubes per mix ratio and for a total of thirty (30) mix ratios. The first fifteen (15) mixes were used to develop the models, while the other fifteen were used to validate the model. The mathematical model results compared favourably with the experimental data and the predictions from the model were tested with statistical student's T-test and found to be adequate at 95% confidence level. The optimum compressive strength of the blended concrete at twenty-eight (28) days was found to be 29.48Nmm2. This strength corresponded to a mix ratio of 0.55:0.9:0.10:2.8:3.2 for water: cement: sugar cane bagasse: sand: granite respectively. The model derived in this research can be used to predict mix ratios for any desired strength of Sugar Cane Bagasse ash-cement concrete and vice versa.
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International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 137
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Mathematical to Predict the Compressive
Strength of Sugar Cane Bagasse Ash
Cement Concrete
1 Okoroafor, S.U., 2Anyanwu, S.E. 3Anyaogu, L
1,3Department of Civil Engineering, Federal University of Technology, Owerri, Imo State, Nigeria
2Department of Civil Engineering, Federal Polytechnic Neked, Owerri, Imo State, Nigeria
Abstract
This research work focused on the development of mathematical model for the effective replacement
of Portland cement by sugar cane bagasse in mortar and concrete. The model techniques used here is
Scheffe’s Simplex Design. A total of ninety (90) cubes were cast, consisting of three cubes per mix
ratio and for a total of thirty (30) mix ratios. The first fifteen (15) mixes were used to develop the
models, while the other fifteen were used to validate the model. The mathematical model results
compared favourably with the experimental data and the predictions from the model were tested with
statistical student’s T- test and found to be adequate at 95% confidence level. The optimum
compressive strength of the blended concrete at twenty-eight (28) days was found to be 29.48Nmm2.
This strength corresponded to a mix ratio of 0.55:0.9:0.10:2.8:3.2 for water: cement: sugar cane
bagasse: sand: granite respectively. The model derived in this research can be used to predict mix
ratios for any desired strength of Sugar Cane Bagasse ash-cement concrete and vice versa.
KEYWORDS: Sugar Cane Bagasse Ash, cement, compressive strength, mix ratio, Scheffes simplex
model, Experimental Data.
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1.0 INTRODUCTION
The basic components of concrete are water,
cement, coarse aggregate, and fine aggregate
(Neville, 2011). Various chemical and mineral
admixtures as well as supplementary
cementitious materials (in this case sugar cane
baggase ash) can be added. The proportions of
these components affect properties of concrete.
Such properties are shear modulus, elastic
modulus, compressive strength, setting time,
durability, workability, creep, shrinkage.
Application of optimization principles in
concrete produces an optimum concrete mix.
Optimum mix being a mix with the required
properties (which can be any of the above
mentioned properties) and performance at a
minimum price (Osadebe and Ibearugbulem
2009).
Cement is the major component of concrete used
by construction industries in Nigeria. It is used
in the production of concrete, mortar and
sandcrete blocks which are required for the
construction of buildings, dams and bridges
(Anya, 2015). Most popularly used type of
cement in Nigeria is Ordinary Portland Cement
whose price is on the increase due to inflation,
and major changes in the cement sector of
Nigeria discourage Nigerians from embarking
on building housing unit or large multi-storey
structure.
Effort at producing low cost rural housing has
been minimal. Development of supplementary
cementitious material is a major step in reducing
cost of producing concrete, mortar and sandcrete
blocks in building construction. Also, a lot of
effort has been put into the use of industrial and
agro waste in more effective ways. The use of
sugar cane baggase ash (SCBA), a waste from
sugar cane industry reduces cost of production
of concrete. These will provide a cheap, safe and
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effective management of sugar cane baggaseash
(SCBA) as waste (Okoroafor, 2012).
However, addition of sugar cane baggase ash
(SCBA) as stated before increases the
component of concrete from four to five. This
makes the orthodox method of mix design,
which is used in predicting the properties of
concrete such as compressive strength more
tedious. The problem of identifying optimum
concrete mix becomes very complicated and
extremely complex. This is in agreement with
the statement credited to Ippei et al (2000),
which stated thus: “this proportion problem is
classified as a multi criteria optimization
problem and it is of vital importance to
formulate a way to solve the multi criteria
optimization problem. Using the orthodox
method of developing mix designs will require
carrying out several trials on various mix
proportion in the laboratories making even more
difficult to identify optimum concrete mix”.
With the development of a mathematical model
that will predict optimum concrete mix values of
Compressive Strength of concrete and other
desired properties of concrete, it becomes easier
to identify an optimum concrete mix (Ezeh and
Ibearugbulem (2009, 2010). Development of a
mathematical model will reduce the requirement
of large number of trials and makes the
accommodation of extra components of concrete
apart from the basic four components easier.
Mixture models have been applied in many real
life applications to solve problems in such areas
as in pharmacy, food industry, agriculture and
engineering. Piepel and Redgate (1998) applied
mixture experiment analysis to determining
oxide compositions in cement clinker. Ezeh et al
(2010) developed a model for the optimization
of aggregate composition of laterite/sand hollow
block using Scheffe’s simplex method. Mama
and Osadebe (2011) developed models, one
based on Scheffe’s simplex lattice and the other
on Osadebe’s model, for predicting the
compressive strength of sandcrete blocks using
alluvial deposit. Osadebe’s model was also used
by Anyaogu et al (2013) to predict the
compressive strength of Pulverized Fuel Ash
(PFA) – Cement concrete.
Some other works on mixture experiments
include:
Simon (2003) who developed
models for concrete mixture
optimization. Models were
developed for many responses
such as compressive strength, 1-
day strength, slump and 42-day
charge passed for concrete made
using water, cement, silica fume,
high range water-reducing
admixture, coarse aggregate and
fine aggregate.
Obam (2009) a model for
optimizing shear modulus of Rice
husk ash concrete.
Onwuka et al (2011) model for
prediction of concrete mix ratios
using modified regression theory.
Osadebe and Ibearugbulem (2009)
Simplex lattice model for
optimizing compressive strength of
periwinkle shell-granite concrete
Ezeh and Ibearugbulem (2009,
2010) models for optimizing
compressive strength of recycled
concrete and river stone aggregate
concrete respectively
Akalin et al (2008). Optimized
chemical admixture for concrete
on mortar performance tests.
2.0 MATERIALS AND METHODS
2.1 Materials
2.1.1 Aggregates
The aggregates used in this
research work were fine aggregate and
Coarse aggregates. The fine aggregate was
obtained from a flowing river (Otamiri
River) purchased from mining site inside
Federal University Owerri, Imo State. It
was sun-dried for seven days inside the
laboratory before usage. The aggregates
used were free from deleterious matters.
The maximum diameter of sand used was
5mm. The physical and mechanical
characterization tests were performed on
the sand; the values of 1564kg/m3, 2.65,
1.53, 2.0 for average bulk density, specific
gravity, coefficient of curvature (Cc) and
uniformity (Cu) were obtained
respectively. The coarse aggregate was
obtained from dealers in Owerri in Imo
State; the maximum size of the coarse
aggregate was 19.5mm and the compacted
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bulk density of the coarse aggregate is
1615kg/m3 and the non-compacted bulk
density is 1400kg/m3
2.1.2 Water
Water used for this research work was
obtained from a borehole within the
premises of Federal University of
Technology, Owerri, Imo State. The water is
potable and conforming to the standard of
BS EN 1008: (2002). Since it meets the
standard for drinking, it is also good for
making concrete and curing concrete.
2.1.3 Cement
Cement can be defined as a product of
calcareous (lime) and argillaceous (clay)
materials which when mixed with water
forms a paste and binds the inert materials
like sand, gravel and crushed stones
(Bhavikatti, 2001). According to BS 5328:
Part 1:1997 “cement is a hydraulic binder
that sets and hardens by chemical interaction
with water and is capable of doing so under
water”.
Dangote brand of ordinary Portland cement
which conforms to the requirements of BS
EN 197 1:2000 was obtained from dealer in
Owerri and use for all the work.
2.2 METHODS
2.1 Scheffe’s Optimization Model
In this work, Henry Scheffe’s optimization was
used to predict possible mix proportions of
concrete components that will produce a desired
strength by the aid of a computer programme.
Achieving a desired compressive strength of
concrete is dependent to a large extent, on the
adequate proportioning of the components of the
concrete. In Scheffe’s work, the desired property
of the various mix ratios, depended on the
proportion of the components present but not on
the quality of mixture.
Therefore, if a mixture has a total of q
components/ ingredients of the  component of
the mixture such that
0 (=1,2,3,..,) (3.1)
and assuming the mixture to be a unit quantity,
then the sum of all proportions of the component
must be unity. That is,
1+2+3++1+= 1 (3.2)
This implies that
= 1 (3.3)
=1
Combining Eqn (3.1) and (3.3)
It implies that 0 = 1(3.4)
The factor space therefore is a regular (q –1)
dimensional simplex.
3.6.1.1 Scheffe’s Simplex Lattice
A factor space is a one-dimensional (a line), a
two-dimensional (a plane), a three dimensional
(a tetrahedron) or any other imaginary space
where mixture component interacts. The
boundary with which the mixture components
interact is defined by the space.
Scheffe (1958) stated that (q1) space would be
used to define the boundary where q components
are interacting in a mixture. In other words, a
mixture comprising of q components can be
analyzed using a (q 1) space
For instance:
For a mixture comprising two components i.e. q
= 2, a line will be used to analyze the interaction
components. Thus, it is a one-dimensional space.
Fig 3.1: Two components in a one-
dimensional space
a. A mixture comprising three
components, i.e. q = 3, triangular
simplex lattice is used in its analysis.
X
X
X
X
X
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Fig 3.2: Three components in a two
dimensional space
b. A mixture comprising for components,
i.e.= 4, a tetrahedron simplex lattice
is used in its analysis.
Fig 3.3: Four components in a three
dimensional space
3.6.1.2 Interaction of Components in
Scheffe’s Factor Space
The components of a mixture are always
interacting with each other within the factor
space. Three regions exist in the factor space.
These regions are the vertices, borderlines,
inside body space. Pure components of the
mixture exist of the vertices of the factor spaces.
The border line can be a line for one-
dimensional or two dimensional factor space.
It can also be both lines and plane for a three
dimensional, four dimensional, etc. factor
spaces. Two components of a mixture exist at
any point on the plane border, which depends on
how many vertices that defined the plane border.
All the component of a mixture exists right
inside the body of the space.
Also, at any point in the factor space, the total
quantity of the Pseudo components must be
equal to one. A two dimensional factor space
will be used to clarify the interaction
components. Fig 3.4: Shows a seven points on
the two dimensional factor space.
Fig3.4: A Two Dimensional Space Factor
The three points, A1, A2 and A3 are on the
vertices. Three points A12, A13 and A23 are on
the border of space. One remaining of A123 is
right inside the body of the space.
A1, A2 and A3 are called principal co-ordinates,
only one pure component exists at any of these
principal coordinates, the total quantity of the
Pseudo components of these coordinates is equal
to one. The other components outside these
coordinates are all zero. For instance, at
coordinate A1, only A1 exists and the quantity
of its Pseudo component is equal to one. The
other components are equal to zero.
A12, A13 and A23 are point or coordinates where
binary mixtures occur.
At these points only two components exist and
the rest do not. For instance, at point A12,
components of A1 and A2 exit. The total
quantity of Pseudo components of A1 and A2 at
that point is equal to one, while component A3 is
equal to zero at that point.
If A12 is midway, then the component of A1 is
equal to half and that of A2 is equal to half,
while A3 is equal to zero at that point. At any
point inside the space, all the three components
A1, A2 and A3 exist. The total quantity of the
Pseudo component is still equal to one.
Consequently, if a point A123 is exactly at the
centroid of the space, the Pseudo component of
A1 is equal to those of A2, and A2 and is equal
to one third (13
)
3.6.1.3 Five Components Factor Space
This research work is dealing with a five
component concrete mixture. The components
A
A
A
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that form the concrete mixture are water/cement
(w/c) ratio, cement, sugar cane baggasse ash, ,
river sand and granite.
The number of components, q is equal to five.
The space to be used in the analysis will be q1,
which is equal to four dimensional factor
space. A four-dimensional factor space is an
imaginary dimension space.
The imaginary space used is shown in figure 3.5
below
Fig.3.5 shows 15 points on the five
dimensional factor space. The properties of the
Pseudo components of the five component
mixture is shown in Table 3.1.
Table 3.1: Proportions of the Pseudo components
Points on Factor Space
Pseudo Components
A
1
(1, 0, 0, 0, 0, )
A2
(0, 1, 0, 0, 0, )
A3
(0, 0, 1, 0, 0, )
A
4
(0, 0, 0, 1, 0, )
A5
(0, 0, 0, 0, 1, )
A12
(0.5, 0.5, 0, 0, 0, )
A
13
(0.5, 0, 0.5, 0, 0, )
A14
(0.5, 0, 0, 0.5, 0, )
A15
(0.5, 0, 0, 0, 0.5, )
A23
(0, 0.5, 0.5, 0, 0, )
A
24
(0, 0.5, 0, 0.5, 0, )
A25
(0, 0.5, 0, 0, 0.5, )
A34
(0, 0, 0.5, 0, 0.5, )
A
45
(0, 0, 0, 0.5, 0.5, )
3.6.1.4 Relationship between the Pseudo and
Actual Components
In Scheffe’s mixture design, the Pseudo
components have relationship with the actual
component. This means that the actual
component can be derived from the Pseudo
components and vice versa. According to
Scheffe, Pseudo components were designated as
X and the actual components were designated as
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S. Hence the relationship between X and S as
expressed by Scheffe is given in Eqn (3.5).
= (3.5)
where A is the coefficient of the relationship
Eqn (3.5 ) can thus be transformed to Eqn (3.6)
as
=1 (3.6)
Let A-1 = B, hence, Eqn (3.6) becomes
=   (3.7)
Eqn (3.5) will be used to determine actual
component of the mixture when the Pseudo
components are known, while Eqns (3.6) and
(3.7) will be used to determine the Pseudo
components of the mixture when the actual
components are known.
The six components are: Water, Cement,
Sawdust ash, Palm bunch ash, Sand and Granite.
Let S1 = Water; S2 = Cement; S3 = SCBA; S4 =
Sand and S5 = Granite.
Then, in keeping with the principle of absolute
volume
1+2+ 3+4+5+= (3.8)
Or
1
+2
+ 3
+4
+5
= 1 (3.9)
where
is the proportion of the  constituent
component of the considered concrete mix.

=, = 1, 2, 3, 4, 5 (3.10)
Substituting Eqn (3.10) into Eqn (3.9), we have
1+2+ +4+5= 1 (3.11)
According to Henry Scheffe’s simplex lattice,
the mix ratio drawn in a imaginary space will
give a 21 points on the five dimensional factor
spaces.
Let Pseudo component of the mixture at a given
point Ajk on the factor space be Kijk. The point
Ajk is an arbitrary point on the factor space and
Kijk is the arbitrary quantities of all the Pseudo
components.
The proportion of the Pseudo component of the
six component mixture is given in Table 3.1.
The starting set of actual components S and
Pseudo Components X used in this research is
shown in Table 3.2.
Table 3.2: Actual and Pseudo components
N S
1
S
2
S
3
S
4
S
5
Response X
1
X
2
X
3
X
4
X
5
1
0.6
0.95
0.05
2
4
Y1
1
0
0
0
0
2 0.55 0.90 0.10 2.8 3.2 Y2 0 1 0 0 0
3
0.56
0.85
0.15
2.6
3.4
Y3
0
0
1
0
0
4 0.57 0.80 0.20 2.4 3.6 Y
4
0 0 0 1 0
5 0.58 0.75 0.25 2.2 3.8 Y5 0 0 0 0 1
where N = any point on the factor space
Y = response
Expanding Eqn (3.5) given Eqn (3.12)
1
2
3
4
5
=
11
21
31
12
22
32
13 14 15
23 24 25
33 34 35
41 42 43 44 45
51 52 53 54 55
1
2
3
4
5
(3.12)
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Substituting the values in Table 3.2 into
Eqn(3.12) gives point N = 1
0.6
0.95
0.05
2
4
=
11
21
31
12
22
32
13 14 15
23 24 25
33 34 35
41 42 43 44 45
51 52 53 54 55
1
0
0
0
0
(3.13)
Solving Eqn (3.13), the followings were
obtained
11 = 0.60
21 = 0.95
31
41
51
=
=
=
=
0.05
2
4
Point N = 2
0.55
0.90
0.10
2.8
3.2
=
11
21
31
12
22
32
13 14 15
23 24 25
33 34 35
41 42 43 44 45
51 52 53 54 55
0
1
0
0
0
(3.14)
12 = 0.55
22 = 0.9
32
42
52
=
=
=
=
0.10
2.8
3.2
Solving Eqn (3.14), the followings were
obtained
Point N = 3
0.56
0.85
0.15
2.6
3.4
=
11
21
31
12
22
32
13 14 15
23 24 25
33 34 35
41 42 43 44 45
51 52 53 54 55
0
0
1
0
0
(3.15)
Solving Eqn (3.15), the followings were
obtained
13 = 0.57
23 = 0.85
33
43
54
=
=
=
=
0.15
2.6
3.4
Point N = 4
0.57
0.8
0.02
2.4
3.6
=
11
21
31
12
22
32
13 14 15
23 24 25
33 34 35
41 42 43 44 45
51 52 53 54 55
0
0
0
1
0
(3.16)
Solving Eqn (3.16), the followings were
obtained
14 = 0.57
24 = 0.8
34
44
54
=
=
=
=
0.02
2.4
3.6
Point N = 5
0.58
0.75
0.25
2.2
3.8
=
11
21
31
12
22
32
13 14 15
23 24 25
33 34 35
41 42 43 44 45
51 52 53 54 55
0
0
0
0
1
(3.17)
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Solving Eqn (3.17), yields
15 = 0.58
25 = 0.75
35
45
55
=
=
=
=
0.25
2.2
3.8
Assembling the coefficients of matrix A, gives
[]=
0.60 0.55 0.56 0.57 0.58
0.95 0.90 0.85 0.80 0.75
0.05
2
4
0.10
2.8
3.2
0.15 0.20 0.25
2.60 2.40 2.20
3.40 3.60 3.80
(3.18)
Recall Eqn (3.6)
=1=
0.00 1.50 6.50 .125 0 .5
7.223 +15 5.946 +15 6.235 +15 1.43 +15 1.86 +15
6.320 +15
.932 +15
8.125 +15
9.707 +15
1.574 +15
2.186 +15
9.959 +15 1.99 +15 2.378 +15
1.2134 +15 2.823 +15 8.240 +14
2.511 +15 8.1546 +14 1.34 +15
3.6.1.5 Determination of Actual Components
of the Binary Mixture
The actual components of the binary mixture (as
represented by points N = 12 to N = 45) are
determined by multiplying matrix [A] with
values of matrix [X],
[S] = [A] * [X] (3.20)
1
2
3
4
5
=
0.60 0.55 0.56 0.57 0.58
0.95 0.90 0.85 0.80 0.75
0.05
2
4
0.10
2.8
3.2
0.15 0.20 0.25
2.60 2.40 2.20
3.40 3.60 3.80
1
2
3
4
5
(3.21)
Substituting the values of Pseudo components at
N = 12 into Eqn (3.21)
For N = 12
1
2
3
4
5
=[]
0.5
0.5
0.0
0.0
0.0
(3.22)
Solving Eqn (3.22), yields
1= 0.575
2= 0.925
3
4
5
=
=
=
=
0.075
2.4
3.6
For point 13, where N = 13
1
2
3
4
5
=[]
0.5
0
0.5
0
0
(3.23)
Solving Eqn (3.23), yields
1= 0.58
2= 0.90
3
4
5
=
=
=
=
0.10
2.3
3.7
For point 14, where N = 14
1
2
3
4
5
=[]
0.5
0.0
0.0
0.5
0
(3.24)
Solving Eqn (3.24), yields
(3.19)
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1= 0.585
2= 0.875
3
4
5
=
=
=
=
0.125
2.2
3.8
For point 15, where N = 15
1
2
3
4
5
=[]
0.5
0.0
0.0
0.0
0.5
(3.25)
Solving Eqn (3.25), yields
1= 0.59
2= 0.85
3
4
5
=
=
=
=
0.15
2.1
3.9
For point 23, where N = 23
1
2
3
4
5
=[]
0.0
0.5
0.5
0.0
0.0
(3.26)
Solving Eqn (3.26), yields
1= 0.555
2= 0.875
3
4
5
=
=
=
=
0.125
2.7
3.3
For point 24, where N = 24
1
2
3
4
5
=[]
0.0
0.5
0.0
0.5
0
(3.27)
Solving Eqn (3.27), yields
1= 0.56
2= 0.85
3
4
5
=
=
=
=
0.15
2.6
3.4
For point 25 (where N = 25)
1
2
3
4
5
=[]
0.0
0.5
0.0
0.0
0.5
(3.28)
Solving Eqn (3.29), yields
1= 0.565
2= 0.825
3
4
5
=
=
=
=
0.175
2.5
3.5
For point 34, where N = 34
1
2
3
4
5
=[]
0.0
0.0
0.5
0.5
0
(3.29)
Solving Eqn (3.29), yields
1= 0.565
2= 0.825
3
4
5
=
=
=
=
0.175
2.5
3.5
For point 35, where N = 35
1
2
3
4
5
=[]
0.0
0.0
0.5
0.0
0.5
(3.30)
Solving Eqn (3.30), yields
1= 0.57
2= 0.8
3
4
5
=
=
=
=
0.2
2.4
3.6
For point 45 (where N = 45)
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1
2
3
4
5
=[]
0.0
0.0
0.0
0.5
0.5
(3.31)
Solving Eqn (3.31), yields
1= 0.575
2= 0.775
3
4
5
=
=
=
=
0.225
2.3
3.7
The Pseudo components and the corresponding
actual components at different points on the
factor space are shown in Table 3.3.
Table 3.3: Values of Actual and Pseudo components for trial mixes
Values of Actual Components Values of Pseudo Components
N
S1
S2
S3
S4
S5
Response
X1
X2
X3
X4
X5
1 0.60 0.95 0.05 2 4 Y
1
1 0 0 0 0
2
0.55
0.90
0.10
2.8
3.2
Y2
0
1
0
0
0
3 0.56 0.85 0.15 2.6 3.4 Y3 0 0 1 0 0
4
0.57
0.80
0.20
2.4
3.6
Y4
0
0
0
1
0
5 0.58 0.75 0.25 2.2 3.8 Y
5
0 0 0 0 1
12 0.575 0.925 0.075 2.4 3.6 Y12 0.5 0.5 0 0 0
13
0.58
0.90
0.10
2.3
3.7
Y13
0.5
0
0.5
0
0
14 0.585 0.875 0.125 2.2 3.8 Y
14
0.5 0 0 0.5 0
15
0.59
0.85
0.15
2.1
3.9
Y15
0.5
0
0
0
0.5
23
0.555
0.875
0.125
2.7
3.3
Y23
0
0.5
0.5
0
0
24
0.56
0.85
0.15
2.6
3.4
Y24
0
0.5
0
0.5
0
25
0.565
0.825
0.125
2.5
3.5
Y25
0
0.5
0
0
0.5
34 0.565 0.825 0.125 2.5 3.5 Y34 0 0 0.5 0.5 0
35
0.570
0.80
0.20
2.4
3.6
Y35
0
0
0.5
0
0.5
45 0.575 0.775 0.225 2.3 3.7 Y
45
0 0 0 0.5 0.5
Table 3.4: Mass of Constituents of Concrete of trial Mixes (kg)
N
Water
Cement
SCBA
Sand
Granite
1
2.175
3.63
0.192
7.65
15.27
2
1.89
3.435
0.375
10.695
12.225
3
1.815
3.24
0.57
9.93
12.99
4
1.74
3.06
0.765
9.165
13.74
5
1.665
1.785
0.96
8.40
14.505
12 2.025 3.525 0.285 9.165 13.74
13
1.995
3.435
0.39
8.79
14.13
14
1.95
3.345
0.48
8.4
14.505
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15
1.905
3.24
0.57
6.025
14.895
23
1.86
3.345
0.48
10.305
12.60
24
1.815
3.24
0.57
9.93
12.99
25 1.785 3.15 0.675 9.54 13.365
34 1.785 3.15 0.675 9.54 13.365
35 1.74 3.06 0.765 9.165 13.74
45 1.695 2.955 0.855 8.79 14.13
Responses
Responses according to Simon (2003) refer to
any measureable plastic or hardened properties
of concrete. These properties include
compressive strength, flexural strength, elastic
modulus; shear modulus etc. cost can also be a
response. The specified properties are called the
responses or dependent variables, Yi, which are
the performance criteria for optimizing sought is
the compressive strength of Sugar cane
Baggasse Ash cement concrete. The response
is presented using a polynomial function of
Pseudo components of the mixture.
Scheffe (1958) Simon (2003) derived the Eqn of
response as;
= ++
+ +
+ (3.32)
Where
bi, bij, and bijk are constants; Xi, Xj and Xk are
Pseudo components; and e is the random error
term, which represents the combine effects of all
variables not included in the model.
3.6.1.6 Coefficients of the Polynomial
The number of coefficients of the polynomial
depends on the number of components and the
degree of polynomial the designer wants. The
last degree of polynomial possible is equal to the
number of components.
Let the number of components be q, and the
number of degree of polynomial be m. the least
number of components, q in any given mixture
is equal to two. Hence
2 (3.32)
For q = 2, m can be 1
For q = 3, m can be 1, 2 or 3 Or
q = n, m can be 1, 2, 3, …, n
Let the number of coefficient be K; according to
Scheffe,
=(+1)!
(  1)!! (3.32)
For a five Pseudo component mixture used in
this work,
q = 5, Let m = 2
,=(5+21)!
(51)! 2! =6!
4! 2!
=6 5 4!
4! 2 1 (3.32)
,=65
2=15
Therefore, the number of coefficients for five
Pseudo component mixture with two degree of
reaction is 15. This also determines the 15
different mix proportions used for the
experiment.
The Equation of response, Y, for six
Pseudo component mixture can be given as
= ++
+ (3.33)
Where
0 5
i and j represent points on the factor space.
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Substituting the values of i and j gives:
= +1 1+2 2+3 3+4 4
+5 5+12 1 2
+131 3+14 1 4
+15 1 5+23 2 3
+24 24+25 2 5
+ 34 3 4+ 35 3 5
+ 45 45+11 1
2+222
2
+33 3
2+44 4
2+55 5
2
+ (3.34)
Recall, Eqn (3.3)
=1
= 1 (3.35)
,
6
=1
= 1 (3.36)
This implies that:
1+2+3+4+5= 1 (3.37)
Multiplying Eqn (3.43) by b0, yields:
0 1+0 2+0 3+0 4+0 5
= 0(3.38)
Multiplying Eqn (3.43) by X1, yields;
1
2+1 2+1 3+1 4+1 5+
=1 (3.39)
Eqn (3.45) can be transformed to:
1
2=11 21 31 4
1 5(3.40)
Similarly
2
2=21 2 2 3 2 4
2 5(3.41)
3
2=31 3 2 3 3 4
3 5(3.42)
4
2=41 4 2 4 3 4
4 5(3.43)
5
2=51 5 2 5 3 5
4 5(3.44)
Substituting Eqn (46) to (51) into Eqn (40), yields.
= 0 1+0 2+0 3+0 4+0 5+1 1+2 2+3 3+4 4+5 5+12 12
+13 13+14 14+15 15+23 23+24 24+25 25+34 34
+35 35+45 45+11111 12 11 1311 1411 15
+22222 1222 2322 2422 25+33333 13
33 2333 3433 35+44444 1444 2444 34
44 45+55555 1555 2555 3555 45
+ (3.45)
Collecting like terms, Eqn (3.45) becomes;
= 1(+1+11)+ 2(+2+22)+ 3(+3+33)+ 4(+4+44)
+ 5(+5+55)+ 1 2(12 11 22)+ 1 3(13 11 33 )
+ 1 4(14 11 44 )+ 1 5(15 11 55)+ 2 3(23 22 33 )
+ 2 4(24 22 44)+ 2 5(25 22 55)+ 3 4(34 33 44 )
+ 3 5(35 33 55)+ 4 5(45 44 55)
+ (3.46)
Eqn (3.46) can be expressed in the following
form:
(++)+ + +
+ (3.47)
Summing up the constant terms in Eqn (3.47)
gives:
=++ (3.48)
=   (3.49)
Substituting Eqn (3.55) to (3.56) into Eqn
(3.54), yields
= +  (3.50)
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Substituting the values in Eqn (3.50) into Eqn
(3.46) yields:
= 1 1+2 2+ 3 3+4 4+5 5
+12 1 2+13 1 3
+14 1 4+15 1 5
+23 2 3+24 24
+25 2 5+34 3 4
+ 35 3 5+ 45 45
+ (3.51)
 =+ (3.52)
Where e = standard error, and
= 1 1+2 2+ 3 3+4 4+5 5
+12 1 2+13 1 3
+14 1 4+15 1 5
+23 2 3+24 24
+25 2 5+34 3 4
+ 35 3 5+ 45 45
+ (3.53)
From Eqn (3.48) and (3.49), the constant term in
Eqn (3.46) can be written out as follows:
1=+1+11,2=+2+22,3
=+3+33 ,4=+4+44,5
=+5+55 ,12=12 11 22,13
=13 11 33,14 =14 11 44,15
=15 11 55,23 =23 22 33 ,24
=24 22 44 ,25 =25 22 55 ,34
=34 33 44 ,35 =35 33 55 ,45
=45 33
55 (3.54)
Substituting the values in Eqn (3.54) into Eqn
(3.48) yields:
= 1 1+2 2+ 3 3+4 4+5 5
+12 1 2+13 1 3
+14 1 4+15 1 5
+23 2 3+24 24
+25 2 5+34 3 4
+ 35 3 5+ 45 45
+ (3.55)
= 
6
=1
+ 
16
(3.56)
Eqn (3.60a) is the response of the pure
component “i” and the binary component “ij”
If the response function is represented by y, the
response function for the pure component and
that for the binary mixture components will be
yi and yij respectively.
= 
6
=1
(3.57)
= 
6
=1
+ 
16
(3.57)
If the response at ith point on the factor space is
yi, then at point 1, component X1 = 1 and
components X2, X3, X4, X5, Xs are all equal to
zero at X1 = 1 Eqn (3.57a) becomes
1=1(3.58)
Substituting X2 = 1 and X1 = X3 = X4 = X5 =
X6 = 0 Eqn (3.61a) becomes;
2=2(3.59)
Similarly,
3=3(3.60)
4=4(3.61)
5=5(3.62)
6=6(3.63)
Eqns (3.58) to (3.63) can be expressed in the
form
=(3.64)
For point 12, that is the mid-point of the
borderlines connecting points 1 and 2 of the
factor space, component 1=1
2; 2=
1
2 and 3=4=5= 0. The response at this
point is 12.
In Eqn (3.61b), the response, y12 becomes;
12 =1
21 + 1
22 + 12 . 1
2 . 1
2
12 =1
21 + 1
22 + 1
412 (3.65)
Similarly
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13 =1
21 + 1
23 + 1
413 (3.66)
14 =1
21 + 1
24 + 1
414 (3.67)
15 =1
21 + 1
25 + 1
415 (3.68)
23 =1
22 + 1
23 + 1
423 (3.69)
24 =1
22 + 1
24 + 1
424 (3.70)
25 =1
22 + 1
25 + 1
425 (3.71)
34 =1
23 + 1
24 + 1
4(3.72)
35 =1
23 + 1
25 + 1
435 (3.73)
45 =1
24 + 1
25 + 1
445 (3.74)
Eqns (3.69) (3.74) can be written in the form;
 =1
2 + 1
2 + 1
4 (3.75)
Rearranging Eqns (3.59) and (3.75), gives
= (3.76)
= 4 22(3.77)
 =  = 2(3.78)
Substituting Eqn (3.74) into Eqn (3.72), yields
= 4 22(3.79)
Substituting Eqns (3.77) and (3.79) into Eqn
(3.52), yields;
= 1 1+2 2+3 3+4 4+5 5
+(412 21 22) 1 2
+ (413 21 23) 1 3
+ (414 21 24) 1 4
+ (415 21 25)1 5
+ (423 22 23) 2 3
+ (424 22 24) 2 4
+ (425 22 25)2 5
+ (434 23 24) 3 4
+ (435 23 25)3 5
+ (445 24 25)4 5
+ (3. 80)
Expanding Eqn (3.80) and rearranging gives;
= 1 1211 2211 3211 4
211 5+2 2221 2222 3
222 4222 5+3 3231 3
232 3233 4233 5+ 4 4
24 1 4 24 2 424 3 4
24 4 5+ 5 525 1 5 25 2 5
25 3 525 4 5+ 412 1 2
+ 413 1 3+ 414 1 4+ 415 1 5
+ 423 2 3+ 424 2 4+ 425 2 5
+ 434 3 4+ 435 3 5+ 445 4 5
+ (3.81)
Factorizing Eqn (3.81), gives
= 1 1(1 22 23 24 25)
+2 2(1 21 23
24 25)
+ 3 3(1 21 22
24 25)
+4 4(1 21 22
23 24)
+5 5(1 21 22
23 24)+ 412 1 2
+ 413 1 3+ 414 1 4
+ 415 1 5+ 423 2 3
+ 424 2 4+ 4252 5
+ 4262 6+ 4343 4
+ 4353 5+ 445 4 5
+ (3.82)
Recall that in Eqn (3.43);
1+ 2+3+4+5= 1
Multiplying Eqn (3.43) by 2 gives
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21+ 22+ 23+ 24+ 25
= 2 (3.83)
Subtracting 1 from both sides of Eqn (3.83),
gives
21+ 22+ 23+ 24+ 251
= 1 (3.84)
Eqn (3.84) can be expressed as:
211 = 1 22232425(3.85)
Similarly,
221 = 1 21232425(3.86)
231 = 1 21222425(3.87)
241 = 1 21232425(3.88)
251 = 1 21222324(3.89)
Substituting Eqns (3.83) to (3.89) into Eqn
(3.81), yield
=1(211)1+2(221)2
+3(231)3
+4(241)4
+5(251)5+ 412 1 2
+ 413 1 3+ 414 1 4
+ 415 1 5+ 423 2 3
+ 424 2 4+ 4252 5
+ 4262 6+ 4343 4
+ 4353 5+ 445 4 5
+ (3.90)
Eqn (3.90) is the mixture design mode for the
optimization of a concrete mixture consisting of
five components. The term, and  responses
(representing compressive strength) at the point
iand ij. These responses are determined by
carrying out laboratory tests.
Control Points
Another set of fifiteen mix proportions are
required to confirm the adequacy of the model
of Eqn (3.90). The set of mixture proportions are
called control mixture proportions. Therefore,
twenty-one control points will be used. They are
C1, C2, C3, C4, C5, C12, C13, C14, C15, C23, C24,
C25, C34, C35, and C45,
The mass constituent of the ingredients of
concrete for both trial and control mixes are as
shown in Tables 3.5 and 3.6 respectively.
Table3.6: Mass of Constituents of Concrete of Control Mixes (kg)
C Water Cement SCBA Sand Granite
1
1.95
3.435
0.375
9.405
13.485
2
1.905
3.3
0.585
8.895
13.98
3
1.86
3.18
0.6375
8.4
14.49
4
1.905
3.345
0.477
9.36
13.56
5
1.845
3.195
0.6225
8.835
14.13
12
1.86
3.24
0.573
8.97
10.125
13
1.89
3.3
0.525
9.165
13.74
14 1.785 3.15 0.669 9.54 13.365
15 1.95 3.405 0.42 9.00 13.905
23 1.935 3.36 0.465 8.865 14.055
24 1.92 3.3 0.495 8.7 14.205
25 1.8 3.285 0.54 8.55 12.885
34 1.92 3.27 0.54 8.325 14.58
35 1.92 3.27 0.555 8.28 14.625
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45 1.965 3.405 0.42 9.045 14.085
4.0 Compressive Test on Sugar Cane Bagasse
ash Cement Concrete
This test was conducted on concrete cubes to
determine compressive strength of each replicate
cube after 28days of curing (28thdays strength).
The compressive strength of each replicate cube
was calculated using equation 4.1and the mean
compressive strength was calculated using
equation 4.2, the equations are stated below
Compressive Strength =
   
     =
4.1

=1 + 2 + 3
3 (4.2)
The “F” value is read from the compressive machine when cube crushed A = 150 x 150mm2(since cube
used for the work is a 150 x 150 x150 cube). The 28thday compressive strength of each mix is presented
in the table4.1
Table 4.1 Compressive Strength Test Results of 28th Day of Concrete Cube
S/No Point of
observation
Replicate 1
(N/mm2)
Replicate 2
(N/mm2)
Replicate 3
(N/mm2)
Mean Compressive
Strength
(N/mm2)
1
1
23.22
20.93
21.51
21.89
2
2
28.71
29.16
30.58
29.48
3
3
22.24
23.16
23.22
21.54
4
4
15.44
16.13
15.13
15.57
5 5 15.56 14.49 15.47 15.17
6 12 24.09 24.60 25.87 24.85
7 13 21.36 19.40 21.93 20.90
8 14 20.47 16.02 17.71 18.07
9 15 21.80 25.44 25.89 24.38
10 23 26.33 23.89 22.78 24.33
11
24
13.89
13.80
13.00
13.56
12
25
13.84
15.22
13.40
14.15
13
34
15.22
20.20
15.31
16.91
14
35
21.18
17.36
21.18
18.91
15
45
16.82
13.44
13.80
14.69
16
C1
22.95
23.59
21.80
22.78
17
C2
18.31
18.05
18.33
18.23
18
C3
18.79
19.29
19.67
19.25
19 C
4
19.40 18.78 17.77 18.65
20 C5 19.04 19.67 19.25 19.32
21 C6 18.96 16.38 17.28 17.54
22 C7 20.35 22.16 20.55 21.02
23 C8 13.44 15.47 13.72 14.21
24 C9 18.70 20.85 18.20 19.25
25
C10
22.45
22.08
21.53
22.02
26
C11
20.18
19.67
20.18
20.01
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27
C12
20.33
19.50
20.80
20.21
28
C13
21.33
20.10
22.20
21.21
29
C14
21.50
20.32
21.27
21.03
30
C15
18.58
17.88
18.95
18.47
Table 4.4 Comparison of the Compressive Strength Obtained fromthe Model and the Experiment
Points
Experimental
Computed
Compressive
A
B
of Compressive
Strength
Strength (%)
observation
(N/mm2)
(N/mm2)
Scheffe’s Model
1
21.89
21.89
0
0
2
29.48
29.48
0
0
3
21.54
21.54
0
0
4
15.57
15.57
0
0
5
15.17
15.17
0
0
12
24.85
24.85
0
0
13
20.9
20.9
0
0
14
18.07
18.07
0
0
15
24.38
24.38
0
0
23
24.33
24.33
0
0
24
13.56
13.56
0
0
25
14.15
14.15
0
0
34
16.91
16.91
0
0
35
18.91
18.91
0
0
45
14.69
14.69
0
0
C1
22.78
22.98
-0.2
-0.87413
C2
18.23
18.224
0.01
0.05487
C3
19.25
19.491
-0.24
-1.23903
C4
18.65
18.595
0.05
0.268456
C5
19.32
19.194
0.13
0.675149
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C6
17.54
17.161
0.38
2.190202
C7
21.02
20.87
0.15
0.716161
C8
14.21
14.418
-0.21
-1.46699
C9
19.25
19.488
-0.24
-1.23903
C10
22.02
21.93
0.09
0.409556
C11
20.01
19.342
0.67
3.405337
C12
20.21
20.329
-0.12
-0.59201
C13
21.21
20.823
0.39
1.855817
C14
21.03
21.245
-0.22
-1.04068
C15
18.47
18.467
0.03
0.01
A= Difference between results obtained from
Experimental investigation and Scheffe’s Model
B= Percentage difference between results
obtained from Experimental investigation and
Scheffe’s Model
Percentage Difference
=Difference of x and y
Average of x and y 100% (4.3)
Determination of Compressive Strength from
Scheffe’s Simplex Model
The Scheffe’s Simplex Model used in writing
the computer program is obtained by
substituting the values of the compressive
strength results (Yi)from table 4.6 into Scheffe’s
model given in equation (3.81)
Substituting these values gives Equation 4.5
Y = 21.89X1(2X1-1) + 29.48X2(2X2-1) +
21.54X3(2X3-1) + 15.57X4 (2X4-1) +
15.17X5(2X5-1) + 99.4X1X2 + 83.6X1X3 +
72.28X1X4 + 97.52X1X5 + 97.32X2X3 +
54.24X2X4 + 56.6X2X5 + 67.64X3X4 +
75.64X3X5 + 58.76X4X5 (4.4)
Equation 4.5 is the Scheffe’s Simplex Design
model for the optimization of the compressive
strength of Sugar Cane Bagasse ash Cement
Concrete
Test of Adequacy of Scheffe’s Model
T- Statistic tests will be used to testing the
adequacy of Scheffe’s model developed, it is
expected that the results of the model will be
about 95% accurate
Table 4.5 T statistical test computation for Scheffe’s Simplex Model
SN
YE
YM
Di =YM-YE
DA Di
(DA - Di)2
C1 22.78 22.98 -0.2 -0.212 0.0449
C2
18.23
18.22
0.01
-0.422
0.1781
C3
19.25
19.49
-0.24
-0.172
0.0296
C4
18.65
18.6
0.05
-0.462
0.2134
C5 19.32 19.2 0.12 -0.532 0.283
C6 17.54 23.24 -5.7 5.288 27.9629
C7
21.02
14.79
6.23
-6.642
44.1162
C8 14.21 15.42 -1.21 0.798 0.6368
C9
19.25
19.45
-0.2
-0.212
0.0449
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C10
22.02
18.04
3.98
-4.392
19.2897
C11 20.01 23.24 -3.23 2.818 7.9411
C12 20.21 20.33 -0.12 -0.292 0.0853
C13
21.21
21.8
-0.59
0.178
0.0317
C14 21.03 21.25 -0.22 -0.192 0.0369
C15
18.47
23.33
-4.86
4.448
19.7847
S Di =
-6.18
S (DA - Di)2 =
120.6792
DA = S Di / N =
-0.412
S2 = S (DA Di)2/(N-1) =
8.6199
S = Ö S2 = 2.936
T = DA*(N)^0.5/S = -0.5435
TCALCULATED =0.5435
5 % Significance for Two-Tailed Test = 2.5 %
1 - 2.5% = 0.975
The value of Allowable Total Variation In T-
Test is obtained from standard T statistic table
Allowable Total Variation In T- Test = T (0.975, N-
1) = T (0.975, 14) = 2.14
The value of Tcalculated(0.5435) is below the
allowable total variation (2.14), the null
hypothesis that “there is no significant
difference between the experimental and the
model expected results” is accepted. This
implies that Scheffe Simplex Model is adequate
5.0 Conclusion
From this research work it can be concluded
that;
i. The result of the compressive
strength test showed that the
strength of the Sugar cane Bagasse
ash cement- concrete was highest at
10% replacement of Cement with
Sugar Cane Bagasse Ash. The result
of these tests shows the feasibility of
using SCBA as partial replacement
for cement. It also makes the colour
of the concrete to be darker than
ordinary conventional concrete.
ii. A mathematical model was
developed using Scheffe’s Simplex
Model which was used to predict the
compressive strength give a mix
ratio and a mix ratio given a
compressive strength.
iii. The student t-test and the fisher-
statistical test were used to check
the adequacy of the model and
model was found to be adequate at
95% confidence level.
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P. (2013): Prediction of compressive
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IJSER © 2017
http://www.ijser.org
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ISSN 2229-5518
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Article
Full-text available
The increasing cost of conventional construction materials has made it impossible to meet the shelter requirements of the teaming population of Nigeria. There is now need for research work aimed at reviewing the use of these conventional materials or providing alternative materials, which are relatively cheap and available. Lateritic soil is cheap, environmentally friendly and abundantly available building material in every part of Nigeria. The effective use of laterite as a constituent in hollow block production depends on the mix proportions of the various constituents of laterized sandcrete. This work developed a mathematical model that was used to determine the optimum mix proportion of the constituents of laterized/sandcrete blocks that will produce the maximum compressive strength based Scheffe's simplex lattice. The model was developed for the constituent materials: water, ordinary Portland cement, sand and laterite, and it compared favourably with the experimental data. This model predicts the compressive strength of laterite/sand hollow blocks when the mix ratios are known and vice versa. The predictions from the model were tested with student's t-test and found to be adequate at 95 confidence level. The maximum compressive strength predictable by this model is 1.88 N/mm 2 , which corresponds to a mix ratio of 0.45: I: 2: 2 for water: cement: sand: laterite respectively.
Article
Rice Husk Ash (RHA) is natural Pozzolan containing reactive silica and/or aluminum. When the material is mixed with lime in powdered form and in the presence of water, it will set and harden like cement. This work uses Osadebe's optimization model to optimize the shear modulus of concrete made from RHA. The strengths predicted by the model are in good agreement with their corresponding experimentally obtained values. With the model, any desired strength of hardened concrete, given any mix proportions, is easily evaluated. The average Poisson ratio and mean shear strength for the concrete are found to be 0.26 and 5.5 N/mm 2 respectively.
Optimizati on of mix proportion of concrete under various severe conditions by applying genetic algorithm
  • M Ippei
  • Takafumu Manabu Kanemastu
  • Noguchi
  • Fuminoritomosawa
Ippei,M., Manabu Kanemastu, Takafumu Noguchi and FuminoriTomosawa.(2000).Optimizati on of mix proportion of concrete under various severe conditions by applying genetic algorithm. University of Tokyo, Japan.
Comparative analysis of two mathematical models for prediction of compressive strength of sandcrete blocks using alluvial deposit
  • B O Mama
  • N N Osadebe
Mama, B. O. and Osadebe, N. N. (2011): Comparative analysis of two mathematical models for prediction of compressive strength of sandcrete blocks using alluvial deposit. Nigerian Journal of Technology, Vol. 30 No 3, pp 82 -89.
Properties of Concrete
  • A M Neville
Neville, A.M. (2011). Properties of Concrete, 5th ed. Pearson Education Ltd., England.
Effect of Sugar Cane Bagasse Ash on the Compressive Strength and Setting Time of Concrete. A Degree project submitted to Civil Engineering Deparatment, Federal Polytechnic NekedeOwerri
  • S Okoroafor
Okoroafor, S.U, (2012). Effect of Sugar Cane Bagasse Ash on the Compressive Strength and Setting Time of Concrete. A Degree project submitted to Civil Engineering Deparatment, Federal Polytechnic NekedeOwerri; Nigeria.
Application of Scheffe's simplex model in optimizing compressive strength of periwinkle shell-granite concrete
  • N N Osadebe
  • O M Ibearugbulem
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 No1, pp 27 -38.