Content uploaded by Lewechi Anyaogu

Author content

All content in this area was uploaded by Lewechi Anyaogu on Mar 16, 2019

Content may be subject to copyright.

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 137

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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, Scheffe’s simplex

model, Experimental Data.

- - - - - - - - - - - - - - - - - -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - -- - - - - -

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 138

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 139

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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 (q–1) 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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 140

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

A

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 141

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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 q–1,

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 142

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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)

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 143

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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)

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 144

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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)

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 145

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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)

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 146

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 147

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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.

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 148

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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)

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 149

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

+

16

(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

+

16

(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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 150

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 151

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 152

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 153

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 154

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 155

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

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.

REFERENCES

1. Anyaogu, L., Chijioke, C. and Okoye,

P. (2013): Prediction of compressive

strength of Pulverise fuel Ash-Cement

concrete. IOSR Journal of Mechanical

and Civil Engineering, Vol. 6, No. 1,

pp 01 – 09.

2. Anya C.U, (2015). Models for

Predicting the Structural

Characteristics of Sand-Quarry Dust

Blocks. A Thesis submitted to

Postgraduate School, University of

Nigeria Nsuka; Nigeria.

3. Akalin, O. Akay, K. U. Sennaroglu, B.

and Tez, M. (2008): Optimization of

chemical admixture for concrete on

mortar performance tests using mixture

experiments. 20th EURO Mini

International Conference on

Continuous Optimization and

Knowledge- Based Technologies, 266

– 272.

4. BS EN 1008 (2002). Mixing water for

concrete: - Specification for sampling,

testing and assessing the suitability of

water, including water recovered from

processes in the concrete industry, as

mixing water for concrete. British

Standard Institute London

5. BS EN 197 (2000). Cement.

Composition, specifications and

conformity criteria for common

cements. Part 1. British Standard

Institute London.

6. Ezeh, J.C. and Ibearugbulem, O. M.

(2009): Application of Scheffe’s Model

in Optimization of Compressive cube

Strength of River Stone Aggregate

IJSER

International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 156

ISSN 2229-5518

IJSER © 2017

http://www.ijser.org

Concrete. International Journal of

Natural and Applied Science, 5(4), 303

– 308.

7. Ezeh, J.C., Ibearugbulem, O. M. and

Anya, U. C. (2010): Optimization of

aggregate composition of laterite/sand

hollow block using Scheffe’s simplex

method. International Journal of

Engineering, 4(4), 471-478.

8. 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.

9. 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.

10. Neville, A.M. (2011). Properties of

Concrete, 5th ed. Pearson Education

Ltd., England.

11. 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.

12. 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.

13. Onwuka, D. O., Okere, C. E.,

Arimanwa, J. I. and Onwuka, S. U.

(2011): Prediction of concrete mix

ratios using modified regression theory.

Computational. Method in Civil

Engineering. Vol. 2, No 1. pp 95 – 107.

14. Obam, S. O. (2009): A mathematical

model for optimization of Strength of

concrete. A case for shear modulus of

Rice husk ash concrete. Journal of

Industrial Engineering International.

Vol. 5 (9), 76 – 84

15. Piepel, G. F. and Redgate, T. (1998): A

mixture experiment analysis of the

Hald cement data. The American

Statistician, Vol. 52, pp 23 – 30.

16. Simon, M. J. (2003). Concrete Mixture

Optimization Using Statistical

Methods: Final Report. FHWA Office

of Infrastructure Research and

Development, 6300.

17. Scheffe, H. (1958): Experiments with

mixtures. Journal of Royal Statistical

Society, Series B. Vol. 20, No 2, pp

344-360.

IJSER