Model development for optimizing compressive strength of ...

Model development for optimizing compressive

strength of remixed concrete

K.L.Bidkar Research Scholar, Department of Civil Engineering, K.K.W.I.E.E.&R.Nasik (India)

Dr.P.D.Jadhao

Head and Professor ,Department of Civil Engineering, K.K.W.I.E.E.&R.Nasik (India)

Abstract- This work is performed to improve Scheffe’s Second Degree Polynomial based model that can be helpful

to optimize the compressive strength of remixed concrete. Using Scheffe’s Simplex method, the compressive strength

of remixed concrete was determined for variation of different parameters like cement, CA, FA, W/C ratio, blend ratio

and time lag (t). Control experiments were also carried out and the compressive strength determined. After the test has

been conducted, the satisfactoriness of the model was tested using student’s t-test and Fisher’s test (ANOVA); the result

of the test shows a good correlation between the model and control results. An optimum compressive strength of

33.04275 N/mm2 corresponding to mix ratio of 0.7443, 0.7668, 1.0838, 1.1063, 1.425, 1.447 and the minimum strength

was found to be 21.547 N/mm2 corresponding to mix ratio of 1.375: 1.375: 1.475: 1.475: 1.475: 1.475 for cement,

CA, FA, W/C ratio, blend ratio and time lag (t). The maximum strength value was greater than the minimum value

specified by the IS code for the compressive strength of good quality concrete. Using the model, compressive strength

of all points in the simplex can be derived.

Keywords – Compressive strength, Scheffe’s Simplex method, ratio and time lag (t)

I. INTRODUCTION

Compressive strength of concrete cube test provides an idea about all the characteristics of concrete. By this single test we can judge that whether concreting has been done properly or not. Compressive strength is the ability of material or structure to carry the loads on its surface without any crack or deflection. The compressive strength of concrete is determined in batching plant laboratories for every batch in order to maintain the desired quality of concrete during casting. The strength of concrete is required to calculate the strength of the members. Concrete specimens are a cast and tested under the action of compressive loads to determine the strength of concrete. In very simple words, compressive strength is calculated by dividing the failure load with the area of application of load, usually after 28 days of curing. The strength of concrete is controlled by the proportioning of cement, coarse and fine aggregates, water, and various admixtures. Blend ratio (Qo/Qf = ratio of partially set old mass to fresh mass of concrete) and time of placing concrete (time lag) are also important parameters that affect strength of concrete. In order to test the compressive strength of a concrete cube, 150mm*150mm*150mm cube were cast considering the variation of six parameters (cement, CA, FA, W/C ratio, blend ratio and time lag(t) for remixed concrete. [1-3]. The basic constituents of concrete are cement, fine aggregate (sand), coarse aggregate and water, but still two additional parameters blend ratio and time lag affect the strength of remixed concrete. Fig.1

The compressive strength is by far the most important strength property used to judge the overall quality of

concrete. It may often be the only strength property of the concrete that may be determined since with a few exceptions

almost all the properties of concrete can be related to its compressive strength. Compressive strength is usually

determined by subjecting the hardened concrete, after appropriate curing, usually 28 days, to increasing compressive

load until it fails by crushing, and determining the crushing force. Mathematically, it is given as:

Fc=F/A

Fc = Compressive Strength

F = Crushing load

A = Cross sectional area of the test specimen

Journal of Information and Computational Science

Volume 9 Issue 11 - 2019

ISSN: 1548-7741

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https://civildigital.com/various-strands-mix-designing-significance-specific-principles/

https://civildigital.com/admixtures-in-concrete/

Fig. 1. (6,2)space lattice design

II. METHODOLOGY

2.1 Mathematical modelling and formulation

Mathematical modeling has various applications in concrete technology, the general being the application of predictive

models, derived from Henry Scheffe’s mixture models for predicting compressive strength of concrete. Scheffe’s

mixture model is a single step multiple comparison procedure which applies to a set of estimates of all possible contrasts

among the factor level means [4]. The prediction model are useful to reduce the time required by using data from

previous experimentations given the parameters mentioned.

2.1.1 Introduction to factor space in simplex design

Scheffe [4] considered experiments with mixtures of which the property studied on the proportions of the components

but not their quantities in the mixture. An obvious example of such a study is the relationship between the compressive

strength of concrete and the proportion of w/c (water- cement), cement, sand and coarse aggregate. A simplex is

defined as a convex polyhedron with (k + 1) vertices produced by k intersecting hyper planes in k-dimensional space.

Any co-ordinate system above 3- dimensions are referred to as hyper planes, such planes are not orthogonal. A 2-

dimensional regular simplex is, therefore, an equilateral triangle, while a 3-dimensional regular simplex

is a regular tetrahedron. Scheffe [5] used a regular (q-1) simplex to represent a factor space needed

to describe a response surface for mixtures consisting of several components. If the number of

components is denoted by q, then for binary system (q = 2) the required simplex is a straight line;

for q = 3, the required simplex is an equilateral triangle; and for q = 4, the simplex is a regular

tetrahedron. Notations

q= number of components

k= degree of dimensional space

Xi =proportion of ith components of mixtures

m =degree of the Scheffe polynomial

X1 =fraction of water cement ratio

X2 =fraction of cement

X3 =fraction of fine aggregate

X4 =fraction of coarse aggregate

X5 =fraction of blend ratio

X6 =fraction of time lag

n =degree of polynomial regression

Z =actual components

X =pseudo components

Y1, Y2, Y3, Y4, Y5, Y6, Y12, Y13, Y14, Y15, Y16,Y23, Y24, Y25, Y26,Y34, Y35, Y36, Y45 , Y46, Y56= responses from treatment

mixture proportions

C1, C2, C3, C4, C5, C12, C13, C14,C15, Y16,C23, C24, C25, C26,C34, C35, C36,C45 , C46, C56=responses from control mixture

proportions

α1, α2, α3, α4, α5, α12, α13, α14, α15, α16,α23, α24, α25, α26,α34, α35, α36, α45, α46. α56 = model coefficients

Yp= optimized compressive strength of remixed concrete.



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2.1.2 Scheffe’s Simplex lattice design and Scheffe’s factor space

A simplex is a geometric figure with the number of vertices being one more than the number of variable factor space,

q. It is a projection of n-dimensional space onto an n-1 dimensional coordinate system. A lattice is an ordered

arrangement of points in a regular pattern. Claringbold first introduced simplex lattice design in his study of joint

action on related hormones. Scheffe [5], however, expanded and generalized the simplex lattice design. His work is

often seen as a pioneering work in simplex lattice mixture design. Lattice designs are presently often referred to as

Scheffe’s simplex lattice designs. He assumed that each components of the mixture resides on a vertex of a regular

simplex-lattice with q-1 factor space. If the degree of the polynomial to be fitted to the design is n and the number of

components is q then the simplex lattice, also called a {q,n} simplex will consist of uniformly spaced points whose

coordinates are defined by the following combinations of the components: the proportions assumed by each

component take the n+1 equally spaced values from 0 to 1, that is;

Xi = 0, 1/n, 2/n- - - - -1 (1)

and the simplex lattice consists of all possible combinations of the components where the proportions of Equation (1)

for each component are used. Thus, for the quadratic lattice {q,n} approximating the response surface with second-

degree polynomials, (n = 2) the following levels of every factor must be used; 0, 1/2 , and 1; for a cubic polynomial

(n = 3): 0,1/3 ,2/3 , and 1, and for a fourth degree polynomial (n = 4): 0, 1/4 , 2/4 , 3/4 and 1

Consider a six component mixture. The factor space is a hexahedron. If a second degree polynomial is to be used to

define the response over the factor space then each component (X1, X2….X6) must assume the proportions Xi = 0, 1/2,

and 1. The {6, 2} simplex lattice consists of the twenty one points at the boundaries and the vertices of the hexahedron.

Strength of concrete depends on the adequate proportioning of its ingredients. Scheffe [5], developed an optimization

theory that is used to optimize the strength of concrete, considered experiments with mixtures of which the property

studied depends on the proportions of the components but not their quantities in the mixture. He introduced polynomial

regression to model the response, called ‘‘{q, n} polynomials”. These polynomials have to be of low degree (n),

otherwise the polynomial contains a large number of coefficients, making interpretation difficult and requiring a large

number of design points.

2.1.3 Number of coefficients

P = 6, M = 2

N= (p+m-1)! /(m!(p-1)!

N= (6+2-1)! /(2!(6-1)!

N=21

2.1.4 Six component factor space

The first six pseudo components are located at the vertices of the tetrahedron simplex.

A1 [1:0:0: 0:0:0], A2 [0:1:0:0:0:0], A3 [0:0:1:0:0:0], A4 [0:0:0:1:0:0], A5 [0:0:0:0:1:0], A6 [0:0:0: 0:0:1],

Fifteen other pseudo mix ratios located at mid points of the lines joining the vertices of the simplex are

A12[0.5:0.5:0:0:0:0], A13[0.5:0:0.5:0:0:0], A14[0.5:0:0:0.5:0:0], A15[0.5:0:0:0:0.5:0],

A16[0.5:0:0:0:0:0:0.5],A23[0:0.5:05:0:0:0], A24[0:0.5:0:0.5:0:0], A25[0:0.5:0:0:0.5:0], A26[0:0.5:0:0:0:0.5],A34

[0:0:0.5:05:0:0], A35 [0:0:0.5:0:0.5:0], A36 [0:0:0.5:0:0:0.5],

A45 [0:0:0:0.5:0.5:0], A46 [0:0:0:0.5:0:0.5], A56 [0:0:0:0:0.5:0.5]

2.1.5. Responses

Responses are the properties of fresh and hardened concrete. A simplex lattice is described as a structural

representation of lines joining the atoms of a mixture. The atoms are constituent components of the mixture [6]. For a

normal concrete mixture, the constituent elements are water, cement, fine and coarse aggregates .For remixed concrete

two additional components are bled ratio and time lag. And so, it gives a simplex of a mixture of four components.

Hence the simplex lattice of this six-component mixture is a three- dimensional solid equilateral hexahedron. Mixture

components are subject to the constraint that the sum of all the components must be equal to one. As a rule, the

response surfaces in multi-component systems are very intricate. To describe such surfaces adequately, high degree

polynomials are required, and hence a great many experimental trials. A polynomial of degree n in q variable has Cq+n n coefficients. If a mixture has a total of q components and x1 be the proportion of the ith component in the mixture

such that,

Xi ≥ 0(1, 2, ----q)

Then the sum of the component proportion is a whole unity i.e.

X1 + X2 + X3+ X4+X5 + X6 =1 (2)



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n = b0+∑biXi +∑bjjXiXj +∑bijkXiXjXk+ ∑bi1, i2 . . . . . . inXj; Xi2___; Xjn (3)

where l ≤ i ≤q, l ≤i ≤j ≤q, l≤ i ≤j ≤ k ≤q and 1≤ i1≤i1 ≤ in ≤q respectively [12,13].

Y = b0 + b1X1 + b2X2 + b3X3 + b4X4 + b5X5+ b6X6 + b11X12+b12X1X2 + b13X1X3 + b14X1X4 +b15X1X5 +

b16X1X6+b22X22+ b23X2X3 + b24X2X4 + b25X2X5 + b26X2X6+ b33X3

2+ b34X3X4+ b35X3X5 + b36X3X6+ b44X42+ b45X4X5

+ b46X4X6+b55X52 +b56X5X6 + b66X6

2b66X62 ,where b is a constant coefficient. (4)

From Eqn: (2) we can write b0X1 + b0X2 + b0X3 + b0X4 + b0X5 + b0X6= b0 (5)

b0 = b0 (X1 + X2 + X3 + X4 +X5 + X6)

Multiplying Eq. (5) by X1, X2, X3, X4, and X5 in succession gives

X21=X1 __X1X2 _ X1X3 _ X1X4 _ X1X5_ X1X6

X22=X2 __X1X2 _ X2X3 _ X2X4 _ X2X5_ X2X6

X23 =X3 _ X1X3 _ X2X3 _ X3X4 _ X3X5_ X3X6

X24=X4 _ X1X4 _ X2X4 _ X3X4 _ X4X5_ X4X6

X25 =X5 __X1X5 _ X2X5 _ X3X5 _ X4X5_ X5X6

X26=X6 __X1X6 _ X2X6 _ X3X6 _ X4X6_ X5X6 (6)

Substituting Eq. (5) into Eq. (6), we obtain after necessary transformation that

Y = (b0+b1+b11) X1 + (b0+b2 +b22) X2 + (b0+ b3+b33) X3 + (b0+b4 +b44)X4 +(b0 +b5 + b55)X5

+(b0+b6+b66)X6+(b12-b11-b22)X1X2+(b13-b11-b33)X1X3+(b14-b11-b44)X1X4+(b15-b11-b55)X1X5

+(b16-b11–b66)X1X6 +(b23-b22 -b33)X2X3 +(b24-b22-b44) X2X4+(b25-b11-b55)X2X5+(b26-b11-b66) X2X6 +(b34-b33-

b44)X3X4+(b35-b33-b55)X3X5+(b36-b33–b66)X3X6+(b45-b44-b55)X4X5

+ (b46-b44-b55) X4X6+ (b56-b44-b55) X5X6

If we denote

αi = b0 + bi + bii

And αij = bij - bii - bjj,

Then we arrive at the reduced second-degree polynomial:

Y =α1X1 + α2X2 + α3X3 + α4X4 + α5X5+ α6X6 + α12X1X2+ α13X1X3 +α14X1X4+α15X1X5 + α16X1X6+ α23X2X3+

α24X2X4 + α25X2X5 + α26X2X6+ α34X3X4 + α35X3X5+ α36X3X6+ α45X4X5+ α46X4X6+ α56X5X6 (7)

Y1 = α1; Y2 =α2; Y3 =α3; Y4 = α4; Y5 = α5; Y6 = α6

α12 = 4Y12 -2Y1 -2Y2; α13 = 4Y13-2Y1 -2Y3; α14 = 4Y14 -2Y1 -2Y4; α15 = 4Y15 -2Y1-2Y5;

α16 = 4Y16 -2Y1-2Y6; α23 = 4Y23 -2Y2 -2Y3; α24= 4Y24 -2Y2 -2Y4; α25 = 4Y25 -2Y2 -2Y5;

α26 = 4Y26 -2Y2 -2Y6; α34 = 4Y34 -2Y3-2Y4; α35 = 4Y35 -2Y3 -2Y5; α36 = 4Y36 -2Y3 -2Y6;

α45 = 4Y45 -2Y4 -2Y5, α46 = 4Y46 -2Y4 -2Y6; α56 = 4Y56 -2Y5 -2Y6 (8)

2.1.6. Actual components and pseudo components

AZ = AX (9)

Z represents the actual components while X represents the pseudo components, where A is the constant; a five by

five matrix. The value of matrix A will be obtained from the first five mix ratios.

The mix ratios are

Z1 [0.45:1.0:1.45:1.75:0.33:0.75], Z2 [0.55:1.0:1.45:1.75:0.33:0.75],

Z3 [0.45:1.0:1.45:1.95:1.0:1.5], Z4 [0.55:1.0:1.45:1.95:1.0:1.5],

Z5 [0.45:1.0:1.95:2.55:1.5:2.25], Z6 [0.55:1.0:1.95:2.55:1.5:2.25]

The corresponding pseudo mix ratios are

X1 [1:0:0:0:0:0], X2 [0: l: 0:0:0:0], X3 [0:0:1:0:0:0], X4 [0:0:0:1:0:0],

X5[0:0:0:0: l: 0].X6 [0:0:0:0:0:1].

Substitution of Xi and Zi into Eq. (8) use the corresponding pseudo components to determine the corresponding

actual mixture components.

X1 = fraction of water cement ratio

X2 = fraction of cement

X3 = fraction of fine aggregate

X4 = fraction of coarse aggregate

X5 = blend ratio r

X6=time lag



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Z1

a11 a12 a13 a14 a15 a16

X1

Z2

a21 a22 a23 a24 a25 a26

X2

Z3 = a31 a32 a33 a34 a35 a36

X3

Z4

a41 a42 a43 a44 a45 a46

X4

Z5

a51 a52 a53 a54 a55 a56

X5

Z6

a61 a62 a63 a64 a65 a66

X6

For the first run

0.45

a11 a12 a13 a14 a15 a16

1

1

a21 a22 a23 a24 a25 a26

0

1.45 = a31 a32 a33 a34 a35 a36

0

1.75

a41 a42 a43 a44 a45 a46

0

0.33

a51 a52 a53 a54 a55 a56

0

0.75

a61 a62 a63 a64 a65 a66

0

a11=0.45 a21=1 a31=1.45 a41=1.75 a51=0.33 a61=0.75

For the second run

0.55

a11 a12 a13 a14 a15 a16

0

1

a21 a22 a23 a24 a25 a26

1

1.45

a31 a32 a33 a34 a35 a36

0

1.75

a41 a42 a43 a44 a45 a46

0

0.33 = a51 a52 a53 a54 a55 a56

0

0.75

a61 a62 a63 a64 a65 a66

0

a21=0.55 a22=1 a23=1.45 a24=1.75 a25=0.33 a26=0.75

For the third run

0.45

a11 a12 a13 a14 a15 a16

0

1

a21 a22 a23 a24 a25 a26

0

1.45

a31 a32 a33 a34 a35 a36

1

1.95 = a41 a42 a43 a44 a45 a46

0

1

a51 a52 a53 a54 a55 a56

0

1.5

a61 a62 a63 a64 a65 a66

0

a31=0.45 a32=1 a33=1.45 a34=1.95 a35=1 a36=1.5

For the fourth run

0.55

a11 a12 a13 a14 a15 a16

0

1

a21 a22 a23 a24 a25 a26

0

1.45 = a31 a32 a33 a34 a35 a36

0

1.95

a41 a42 a43 a44 a45 a46

1

1

a51 a52 a53 a54 a55 a56

0

1.5

a61 a62 a63 a64 a65 a66

0

a41=0.55 a42=1 a43=1.45 a44=1.95 a45=1 a46=1.5

For the fifth run



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0.45

a11 a12 a13 a14 a15 a16

0

1

a21 a22 a23 a24 a25 a26

0

1.95 = a31 a32 a33 a34 a35 a36

0

2.55

a41 a42 a43 a44 a45 a46

0

1.5

a51 a52 a53 a54 a55 a56

1

2.25

a61 a62 a63 a64 a65 a66

0

a51=0.45 a52=1 a53=1.95 a54=2.55 a55=1.5 a56=2.25

For the sixth run

0.55

a11 a12 a13 a14 a15 a16

0

1

a21 a22 a23 a24 a25 a26

0

1.95 = a31 a32 a33 a34 a35 a36

0

2.55

a41 a42 a43 a44 a45 a46

0

1.5

a51 a52 a53 a54 a55 a56

0

2.25

a61 a62 a63 a64 a65 a66

1

a61=0.55 a62=1 a63=1.95 a64=2.55 a65=1.5 a66=2.25

Substituting the values of the constants, we have [A] matrix

0.45 1 1.45 1.75 0.33 0.75

0.55 1 1.45 1.75 0.33 0.75

0.45 1 1.45 1.95 1 1.5

0.55 1 1.45 1.95 1 1.5

0.45 1 1.95 2.55 1.5 2.25

0.55 1 1.95 2.55 1.5 2.25

Therefore, for A12

Z1

0.45 1 1.45 1.75 0.33 0.75

0.5

Z2

0.55 1 1.45 1.75 0.33 0.75

0.5

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=0.725 Z2=0.775 Z3=0.725 Z4=0.775 Z5=0.725 Z6=0.775

For A13

Z1

0.45 1 1.45 1.75 0.33 0.75

0.5

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0.5

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=0.95 Z2=1 Z3=0.95 Z4=1 Z5=1.2 Z6=1.25

For A14



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Z1

0.45 1 1.45 1.75 0.33 0.75

0.5

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.5

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.1 Z2=1.15 Z3=1.2 Z4=1.25 Z5=1.5 Z6=1.55

For A15

Z1

0.45 1 1.45 1.75 0.33 0.75

0.5

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.5

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=0.39 Z2=0.44 Z3=0.725 Z4=0.775 Z5=0.975 Z6=1.025

For A16

Z1

0.45 1 1.45 1.75 0.33 0.75

0.5

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.5

Z1=0.6 Z2=0.65 Z3=0.975 Z4=1.025 Z5=1.35 Z6=1.4

For A23

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0.5

Z3 = 0.45 1 1.45 1.95 1 1.5

0.5

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.225 Z2=1.225 Z3=1.225 Z4=1.225 Z5=1.475 Z6=1.475

For A24

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0.5

Z3

0.45 1 1.45 1.95 1 1.5

0

Z4 = 0.55 1 1.45 1.95 1 1.5

0.5

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.375 Z2=1.375 Z3=1.475 Z4=1.475 Z5=1.475 Z6=1.475

For A25



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Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0.5

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.5

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=0.665 Z2=0.665 Z3=1 Z4=1 Z5=1.25 Z6=1.25

For A26

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0.5

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.5

Z1=0.875 Z2=0.875 Z3=1.25 Z4=1.25 Z5=1.625 Z6=1.625

For A34

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0.5

Z4

0.55 1 1.45 1.95 1 1.5

0.5

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.6 Z2=1.6 Z3=1.7 Z4=1.7 Z5=2.25 Z6=2.25

For A35

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3

0.45 1 1.45 1.95 1 1.5

0.5

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.5

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=0.89 Z2=0.89 Z3=1.225 Z4=1.225 Z5=1.725 Z6=1.725

For A36

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3

0.45 1 1.45 1.95 1 1.5

0.5

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.5

Z1=1.1 Z2=1.1 Z3=1.475 Z4=1.475 Z5=2.1 Z6=2.1

For A45



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Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.5

Z5

0.45 1 1.95 2.55 1.5 2.25

0.5

Z6

0.55 1 1.95 2.55 1.5 2.25

0

For A46

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3

0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.5

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.5

For A56

Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3

0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.5

Z6

0.55 1 1.95 2.55 1.5 2.25

0.5

Z1=0.54 Z2=0.54 Z3=1.25 Z4=1.25 Z5=1.875 Z6=1.875

Table 1-Matrix Table for Scheffe’s (6, 2) - Lattice Polynomial

Sr.No. Z1 Z2 Z3 Z4 Z5 Z6 Response X1 X2 X3 X4 X5 X6

A1 1 0.45 1 1.45 1.75 0.33 0.75 Y1 1 0 0 0 0 0

A2 2 0.55 1 1.45 1.75 0.33 0.75 Y2 0 1 0 0 0 0

A3 3 0.45 1 1.45 1.95 1 1.5 Y3 0 0 1 0 0 0

A4 4 0.55 1 1.45 1.95 1 1.5 Y4 0 0 0 1 0 0

A5 5 0.45 1 1.95 2.55 1.5 2.25 Y5 0 0 0 0 1 0

A6 6 0.55 1 1.95 2.55 1.5 2.25 Y6 0 0 0 0 0 1

A12 7 0.725 0.775 0.725 0.775 0.725 0.775 Y12 0.5 0.5 0 0 0 0

A13 8 0.95 1 0.95 1 1.2 1.25 Y13 0.5 0 0.5 0 0 0

A14 9 1.1 1.15 1.2 1.25 1.5 1.55 Y14 0.5 0 0 0.5 0 0

A15 10 0.39 0.44 0.725 0.775 0.975 1.025 Y15 0.5 0 0 0 0.5 0

A16 11 0.6 0.65 0.975 1.025 1.35 1.4 Y16 0.5 0 0 0 0 0.5

A23 12 1.225 1.225 1.225 1.225 1.475 1.475 Y23 0 0.5 0.5 0 0 0

A24 13 1.375 1.375 1.475 1.475 1.775 1.775 Y24 0 0.5 0 0.5 0 0

A25 14 0.665 0.665 1 1 1.25 1.25 Y25 0 0.5 0 0 0.5 0

A26 15 0.875 0.875 1.25 1.25 1.625 1.625 Y26 0 0.5 0 0 0 0.5

A34 16 1.6 1.6 1.7 1.7 2.25 2.25 Y34 0 0 0.5 0.5 0 0

A35 17 0.89 0.89 1.225 1.225 1.725 1.725 Y35 0 0 0.5 0 0.5 0

A36 18 1.1 1.1 1.475 1.475 2.1 2.1 Y36 0 0 0.5 0 0 0.5

A45 19 1.04 1.04 1.475 1.475 2.025 2.025 Y45 0 0 0 0.5 0.5 0

A46 20 1.25 1.25 1.725 1.725 2.4 2.4 Y46 0 0 0 0.5 0.5 0

A56 21 0.54 0.54 1.25 1.25 1.875 1.875 Y56 0 0 0 0 0.5 0.5

Z1=1.04 Z2=1.04 Z3=1.475 Z4=1.475 Z5=2.025 Z6=2.025

Z1=1.25 Z2=1.25 Z3==1.725 Z4=1.725 Z5=2.4 Z6=2.4



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2.1.7. Mixture proportion of control points showing actual and pseudo components

Control Points for A1

Z1

0.45 1 1.45 1.75 0.33 0.75

0.25

Z2

0.55 1 1.45 1.75 0.33 0.75

0.25

Z3 = 0.45 1 1.45 1.95 1 1.5

0.25

Z4

0.55 1 1.45 1.95 1 1.5

0.25

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.1625 Z2=1.1875 Z3=1.2125 Z4=1.2375 Z5=1.4875 Z6=1.5125


Z1

0.45 1 1.45 1.75 0.33 0.75

0.25

Z2

0.55 1 1.45 1.75 0.33 0.75

0.25

Z3

0.45 1 1.45 1.95 1 1.5

0.25

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5 = 0.45 1 1.95 2.55 1.5 2.25

0.25

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=0.8075 Z2=0.8325 Z3=0.975 Z4=1 Z5=1.225 Z6=1.25


Z1

0.45 1 1.45 1.75 0.33 0.75

0.25

Z2

0.55 1 1.45 1.75 0.33 0.75

0.25

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.25

Z6

0.55 1 1.95 2.55 1.5 2.25

0.25

Z1=0.6325 Z2=0.6575 Z3=0.9875 Z4=1.0125 Z5=1.3 Z6=1.325


Z1

0.45 1 1.45 1.75 0.33 0.75

0.25

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3

0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.25

Z5

0.45 1 1.95 2.55 1.5 2.25

0.25

Z6

0.55 1 1.95 2.55 1.5 2.25

0.25

Z1=0.82 Z2=0.845 Z3=1.225 Z4=1.25 Z5=1.687 Z6=1.7125


Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3

0.45 1 1.45 1.95 1 1.5

0.25

Z4

0.55 1 1.45 1.95 1 1.5

0.25

Z5

0.45 1 1.95 2.55 1.5 2.25

0.25

Z6

0.55 1 1.95 2.55 1.5 2.25

0.25

Z1=1.07 Z2=1.07 Z3=1.47 Z4=1.475 Z5=2.0625 Z6=2.0625



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Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0.25

Z3

0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.25

Z5

0.45 1 1.95 2.55 1.5 2.25

0.25

Z6

0.55 1 1.95 2.55 1.5 2.25

0.25

Z1=0.9575

Z2=0.9575

Z3=1.3625

Z4=1.3625

Z5=1.825

Z6=1.825


Z1

0.45 1 1.45 1.75 0.33 0.75

0.166667

Z2

0.55 1 1.45 1.75 0.33 0.75

0.166667

Z3 = 0.45 1 1.45 1.95 1 1.5

0.166667

Z4

0.55 1 1.45 1.95 1 1.5

0.166667

Z5

0.45 1 1.95 2.55 1.5 2.25

0.166667

Z6

0.55 1 1.95 2.55 1.5 2.25

0.166667

Z1=0.955 Z2=0.971667 Z3=1.225 Z4=1.241667 Z5=1.616667 Z6=1.633333


Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.1

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.07925 Z2=1.10175 Z3=1.19125 Z4=1.21375 Z5=1.48875 Z6=1.51125


Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.1

Z1=1.12125 Z2=1.14375 Z3=1.24125 Z4=1.26375 Z5=1.56375 Z6=1.58625



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Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.1

Z1=0.80175 Z2=0.82425 Z3=1.0275 Z4=1.05 Z5=1.3275 Z6=1.35


Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.1

Z1=0.86925 Z2=0.89175 Z3=1.14 Z4=1.1625 Z5=1.4625 Z6=1.485


Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.1

Z1=0.9705 Z2=0.993 Z3=1.24125 Z4=1.26375 Z5=1.67625 Z6=1.69875


Z1

0.45 1 1.45 1.75 0.33 0.75

0

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.1

Z1=1.09425 Z2=1.09425 Z3=1.365 Z4=1.365 Z5=1.8 Z6=1.8



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Z1

0.45 1 1.45 1.75 0.33 0.75

0.1

Z2

0.55 1 1.45 1.75 0.33 0.75

0

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=1.008 Z2=1.018 Z3=1.3725 Z4=1.3825 Z5=1.90125 Z6=1.91125


Z1

0.45 1 1.45 1.75 0.33 0.75

0.1

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=0.90675 Z2=0.91675 Z3=1.27125 Z4=1.28125 Z5=1.6875 Z6=1.6975


Z1

0.45 1 1.45 1.75 0.33 0.75

0.1

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=0.83925 Z2=0.84925 Z3=1.15875 Z4=1.16875 Z5=1.5525 Z6=1.5625


Z1

0.45 1 1.45 1.75 0.33 0.75

0.1

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=1.15875 Z2=1.16875 Z3=1.3725 Z4=1.3825 Z5=1.78875 Z6=1.79875



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Z1

0.45 1 1.45 1.75 0.33 0.75

0.1

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0

Z1=1.06425 Z2=1.07425 Z3=1.26 Z4=1.27 Z5=1.62 Z6=1.63


Z1

0.45 1 1.45 1.75 0.33 0.75

0.1

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0.225

Z5

0.45 1 1.95 2.55 1.5 2.25

0

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=1.15875 Z2=1.16875 Z3=1.3725 Z4=1.3825 Z5=1.78875 Z6=1.79875


Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0.225

Z4

0.55 1 1.45 1.95 1 1.5

0

Z5

0.45 1 1.95 2.55 1.5 2.25

0.1

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=0.85425 Z2=0.87675 Z3=1.09 Z4=1.1125 Z5=1.42125 Z6=1.44375


Z1

0.45 1 1.45 1.75 0.33 0.75

0.225

Z2

0.55 1 1.45 1.75 0.33 0.75

0.225

Z3 = 0.45 1 1.45 1.95 1 1.5

0

Z4

0.55 1 1.45 1.95 1 1.5

0.1

Z5

0.45 1 1.95 2.55 1.5 2.25

0.225

Z6

0.55 1 1.95 2.55 1.5 2.25

0.225

Z1=0.74425 Z2=0.76675 Z3=1.08375 Z4=1.10625 Z5=1.425 Z6=1.4475



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III. MATERIALS AND METHODSMETHODOLOGY

3.1. Materials

The materials used are the combination of cement, water, fine and coarse aggregate, considering the blend ratio and

time lag for a remixed concrete. Ordinary Portland cement conforming to IS 12269-2013[7] has been used for all

mixes .The fine aggregates used as locally available river sand in the present investigation. It should free of impurities

like clay matter ,salt, and organic matter and different properties checked by the various test as per IS 2386-1963

[8].The coarse aggregate of 20mm from crushed basalt rock, conforming to IS 383:1970[9] were used. The aggregates

were free from adherent coating, injurious amounts of disintegrated pieces, alkali, vegetable matter and other

deleterious substances. Care was taken that the aggregates do not contain high concentrations of flaky, elongated

shapes and organic impurities, which might affect the strength, or durability of the concrete. The water, which is used

for concrete mix, should be potable i.e. fresh and free from impurities.

Table 2-Mixture Proportion of Control Points Showing Actual and Pseudo Components

Sr. No. Comp Z1 Z2 Z3 Z4 Z5 Z6 Response X1 X2 X3 X4 X5 X6

1 A1 0.45 1 1.45 1.75 0.33 0.75 C1 0.25 0.25 0.25 0.25 0 0

2 A2 0.55 1 1.45 1.75 0.33 0.75 C2 0.25 0.25 0.25 0 0.25 0

3 A3 0.45 1 1.45 1.95 1 1.5 C3 0.25 0.25 0 0 0.25 0.25

4 A4 0.55 1 1.45 1.95 1 1.5 C4 0.25 0 0 0.25 0.25 0.25

5 A5 0.45 1 1.95 2.55 1.5 2.25 C5 0 0 0.25 0.25 0.25 0.25

6 A6 0.55 1 1.95 2.55 1.5 2.25 C6 0 0.25 0 0.25 0.25 0.25

7 A12 0.9550 0.9717 1.2250 1.2417 1.6167 1.6333 C12 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

8 A13 1.0793 1.1018 1.1913 1.2138 1.4888 1.5113 C13 0.2250 0.2250 0.2250 0.2250 0.1000 0.0000

9 A14 1.1213 1.1438 1.2413 1.2638 1.5638 1.5863 C14 0.2250 0.2250 0.2250 0.2250 0.0000 0.1000

10 A15 0.8018 0.8243 1.0275 1.0500 1.3275 1.3500 C15 0.2250 0.2250 0.2250 0.0000 0.2250 0.1000

11 A16 0.8693 0.8918 1.1400 1.1625 1.4625 1.4850 C16 0.2250 0.2250 0.0000 0.2250 0.2250 0.1000

12 A23 0.9705 0.9930 1.2413 1.2638 1.6763 1.6988 C23 0.2250 0.0000 0.2250 0.2250 0.2250 0.1000

13 A24 1.0943 1.0943 1.3650 1.3650 1.8000 1.8000 C24 0.0000 0.2250 0.2250 0.2250 0.2250 0.1000

14 A25 1.0080 1.0180 1.3725 1.3825 1.9013 1.9113 C25 0.1000 0.0000 0.2250 0.2250 0.2250 0.2250

15 A26 0.9068 0.9168 1.2713 1.2813 1.6875 1.6975 C26 0.1000 0.2250 0.0000 0.2250 0.2250 0.2250

16 A34 0.8393 0.8493 1.1588 1.1688 1.5525 1.5625 C34 0.1000 0.2250 0.2250 0.0000 0.2250 0.2250

17 A35 1.1588 1.1688 1.3725 1.3825 1.7888 1.7988 C35 0.1000 0.2250 0.2250 0.2250 0.0000 0.2250

18 A36 1.0643 1.0743 1.2600 1.2700 1.6200 1.6300 C36 0.1000 0.2250 0.2250 0.2250 0.2250 0.0000

19 A45 1.1588 1.1688 1.3725 1.3825 1.7888 1.7988 C45 0.1000 0.2250 0.2250 0.2250 0.0000 0.2250

20 A46 0.8543 0.8768 1.0900 1.1125 1.4213 1.4438 C46 0.2250 0.2250 0.2250 0.0000 0.1000 0.2250

21 A56 0.7443 0.7668 1.0838 1.1063 1.4250 1.4475 C56 0.2250 0.2250 0.0000 0.1000 0.2250 0.2250

3.2. Method

The specimen for the compressive strength is concrete cube moulds measuring 150mm ×150 mm ×150 mm. The

concrete cubes were cast and cured at 28 days for compressive test. The total of 126 concrete cubes were cast. Sixty

three cubes were cast for experimental test (formulating the model) and another sixty three beams were cast to test the

adequacy of the model (control test). For each of the mixture ratio, three specimens were cast and cured at 28 days.

The three specimen for each mixture were crushed after 28 days of curing and the average compressive strength

recorded [10, 11].

Compressive Strength=Load at failure/Area of Specimen

IV. RESULTS AND DISCUSSION

The results of the compressive strength of the remixed concrete was observed in the laboratory after 28 days curing

under Compression Testing Machine. The laboratory results is shown in Tables 3–5 for the control points.



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Table 3: The Results of Compressive Strength (Response, Yi) Based on a 28-Days Strength

Sr.

No.

Response

symbol

Compressive

Strength

(MPa)

Average

Compressive

Strength

(MPa)

Sr. No. Response

symbol

Compressive

Strength (MPa)

Average

Compressive

Strength

(MPa)

1 Y1 32.4 33 34 Y23 33.2 33

2 Y1 33.2 35 Y23 32.9

3 Y1 33.4 36 Y23 32.9

4 Y2 32.1 31.5 37 Y24 34.21 34

5 Y2 31.8 38 Y24 33.96

6 Y2 30.6 39 Y24 33.83

7 Y3 25.7 25.2 40 Y25 33.9 33.7

8 Y3 24.9 41 Y25 34

9 Y3 25 42 Y25 33.2

10 Y4 24.77 24.4 43 Y26 32.88 33.1

11 Y4 24.2 44 Y26 33.3

12 Y4 24.23 45 Y26 33.12

13 Y5 33.1 32.8 46 Y34 33.05 32.87

14 Y5 32.88 47 Y34 32.79

15 Y5 32.42 48 Y34 32.77

16 Y6 32.76 32.2 49 Y35 32.1 31.5

17 Y6 32.1 50 Y35 31.65

18 Y6 31.74 51 Y35 30.75

19 Y12 28 27.4 52 Y36 34.21 34.11

20 Y12 27.22 53 Y36 34

21 Y12 26.98 54 Y36 34.12

22 Y13 26.54 26.3 55 Y45 34.11 34.21

23 Y13 26.22 56 Y45 34

24 Y13 26.14 57 Y45 34.52

25 Y14 26.89 27 58 Y46 33.53 33.22

26 Y14 27.11 59 Y46 33.12

27 Y14 27 60 Y46 33.01

28 Y15 26.6 26.3 61 Y56 33.96 32.04

29 Y15 26.22 62 Y56 32.11

30 Y15 26.14 63 Y56 30.05

31 Y16 27.1 26.7

32 Y16 26.45

33 Y16 26.55

Table 4: Compressive Strength Values and Their Corresponding Density for the Control Points

Sr.

No.

Response

symbol

Compressive

Strength

(MPa)

Average

Compressive

Strength

(MPa)

Sr. No. Response

symbol

Compressive

Strength

(MPa)

Average

Compressive

Strength

(MPa)

1 C1 32.43 32 34 C23 33.11 33.57

2 C1 31.8 35 C23 32.98

3 C1 31.77 36 C23 34.62

4 C2 28.96 29 37 C24 34.73 33.96



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5 C2 28.93 38 C24 34.75

6 C2 29.11 39 C24 32.4

7 C3 23.94 24 40 C25 32.18 32.73

8 C3 23.95 41 C25 32.17

9 C3 24.11 42 C25 33.85

10 C4 23.02 23.5 43 C26 33.88 33.78

11 C4 23.88 44 C26 34.27

12 C4 23.6 45 C26 33.2

13 C5 31.92 32 46 C34 32.95 33.42

14 C5 31.97 47 C34 32.61

15 C5 32.11 48 C34 34.71

16 C6 31.48 31.5 49 C35 34.91 34.86

17 C6 31.7 50 C35 35.02

18 C6 31.32 51 C35 34.64

19 C12 26.08 26.055 52 C36 34.48 34.45

20 C12 25.98 53 C36 34.38

21 C12 26.1 54 C36 34.48

22 C13 26.29 26.28 55 C45 34.55 34.60

23 C13 26.31 56 C45 34.62

24 C13 26.24 57 C45 34.63

25 C14 26.58 26.3025 58 C46 32.73 32.70

26 C14 25.98 59 C46 32.67

27 C14 26.34 60 C46 32.7

28 C15 25.97 25.92 61 C56 32.6 32.65

29 C15 25.95 62 C56 32.71

30 C15 25.84 63 C56 32.64

31 C16 25.95 25.97

32 C16 25.84

33 C16 26.12

Table 5: Coefficients of the Scheffe’s second degree polynomial for Compressive Strength

α1 α2 α3 α4 α5 α6 α12 α13 α14 α15 α16 α23 α24

32 29 24 23.5 32 31.5 -10.8 -6.88 -5.8 -24.32 -23.12 28.28 54.34

α25 α26 α34 α35 α36 α45 α46 α56

8.92 14.12 38.68 27.44 27.8 36.8 22.48 9.4

Table 6: Experimental Test Result and the Replication Variance

Sr.No. Response

symbol

Replicate Response yi(N/mm2) ∑yi y ∑yi2 (yi-y) ∑(yi-y)2 Si2

1 Y1 1 32.4 99.00 33.0 3267.5 -0.60 0.560 0.280

2 Y1 2 33.2

0.20

0.000

3 Y1 3 33.4

0.40

0.000

4 Y2 1 32.1 94.50 31.5 2978.0 0.60 1.260 0.630

5 Y2 2 31.8

0.30

0.000



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6 Y2 3 30.6

-0.90

0.000

7 Y3 1 25.7 75.60 25.2 1905.5 0.50 0.380 0.190

8 Y3 2 24.9

-0.30

0.000

9 Y3 3 25

-0.20

0.000

10 Y4 1 24.77 73.20 24.40 1786.29 0.37 0.060 0.030

11 Y4 2 24.2

-0.20

0.000

12 Y4 3 24.23

-0.17

0.000

13 Y5 1 33.1 98.40 32.80 3227.76 0.30 0.061 0.030

14 Y5 2 32.88

0.08

0.000

15 Y5 3 32.42

-0.38

0.000

16 Y6 1 32.76 96.60 32.20 3111.06 0.56 0.164 0.082

17 Y6 2 32.1

-0.10

0.000

18 Y6 3 31.74

-0.46

0.000

19 Y12 1 28 86.96 28.99 2252.85 -0.99 8.121 4.061

20 Y12 2 27.22

-1.77

0.000

21 Y12 3 26.98

-2.01

0.000

22 Y13 1 26.54 78.90 26.30 2075.16 0.24 0.085 0.043

23 Y13 2 26.22

-0.08

0.000

24 Y13 3 26.14

-0.16

0.000

25 Y14 1 26.89 81.00 27.00 2187.02 -0.11 0.055 0.027

26 Y14 2 27.11

0.11

0.000

27 Y14 3 27

0.00

0.000

28 Y15 1 26.6 78.96 26.32 2078.35 0.28 0.035 0.017

29 Y15 2 26.22

-0.10

0.000

30 Y15 3 26.14

-0.18

0.000

31 Y16 1 27.1 80.10 26.70 2138.92 0.40 0.001 0.001

32 Y16 2 26.45

-0.25

0.000

33 Y16 3 26.55

-0.15

0.000

34 Y23 1 33.2 99.00 33.00 3267.06 0.20 0.018 0.009

35 Y23 2 32.9

-0.10

0.000

36 Y23 3 32.9

-0.10

0.000

37 Y24 1 34.21 102.00 34.00 3468.07 0.21 0.001 0.001

38 Y24 2 33.96

-0.04

0.000

39 Y24 3 33.83

-0.17

0.000

40 Y25 1 33.9 101.10 33.70 3407.45 0.20 0.009 0.004

41 Y25 2 34

0.30

0.000

42 Y25 3 33.2

-0.50

0.000

43 Y26 1 32.88 99.30 33.10 3286.92 -0.22 0.040 0.020

44 Y26 2 33.3

0.20

0.000



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45 Y26 3 33.12

0.02

0.000

46 Y34 1 33.05 98.61 32.87 3241.36 0.18 0.007 0.004

47 Y34 2 32.79

-0.08

0.000

48 Y34 3 32.77

-0.10

0.000

49 Y35 1 32.1 94.50 31.50 2977.70 0.60 0.002 0.001

50 Y35 2 31.65

0.15

0.000

51 Y35 3 30.75

-0.75

0.000

52 Y36 1 34.21 102.33 34.11 3490.50 0.10 0.004 0.002

53 Y36 2 34

-0.11

0.000

54 Y36 3 34.12

0.01

0.000

55 Y45 1 34.11 102.63 34.21 3511.12 -0.10 0.004 0.002

56 Y45 2 34

-0.21

0.000

57 Y45 3 34.52

0.31

0.000

58 Y46 1 33.53 99.66 33.22 3310.86 0.31 0.002 0.001

59 Y46 2 33.12

-0.10

0.000

60 Y46 3 33.01

-0.21

0.000

61 Y56 1 33.96 96.12 32.04 3087.34 1.92 0.006 0.003

62 Y56 2 32.11

-3055.23

0.000

63 Y56 3 30.05

-1.99

0.000

1 C1 1 32.43 96.00 32.00 3072.28 -0.60 0.560 0.280

2 C1 2 31.80

-0.20

0.000

3 C1 3 31.77

-0.23

0.000

4 C2 1 28.96 87.00 29.00 2523.02 -0.04 0.019 0.009

5 C2 2 28.93

-0.07

0.000

6 C2 3 29.11

0.11

0.000

7 C3 1 23.94 72.00 24.00 1728.02 -0.06 0.018 0.190

8 C3 2 23.95

-0.05

0.000

9 C3 3 24.11

0.11

0.000

10 C4 1 23.02 70.50 23.50 1657.13 -0.48 0.060 0.030

11 C4 2 23.88

0.38

0.000

12 C4 3 23.60

0.10

0.000

13 C5 1 31.92 96.00 32.00 3072.02 -0.08 0.061 0.030

14 C5 2 31.97

-0.03

0.000

15 C5 3 32.11

0.11

0.000

16 C6 1 31.48 94.50 31.50 2976.82 -0.02 0.164 0.082

17 C6 2 31.70

0.20

0.000

18 C6 3 31.32

-0.18

0.000

19 C12 1 26.09 83.39 27.80 2036.60 -1.71 9.091 4.546

20 C12 2 25.98

-1.82

0.000



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21 C12 3 26.10

-1.70

0.000

22 C13 1 26.29 78.84 26.28 2071.92 0.01 0.085 0.043

23 C13 2 26.31

0.03

0.000

24 C13 3 26.24

-0.04

0.000

25 C14 1 26.59 78.91 26.30 2075.65 0.29 0.055 0.027

26 C14 2 25.98

-0.32

0.000

27 C14 3 26.34

0.04

0.000

28 C15 1 25.97 77.76 25.92 2015.55 0.05 0.035 0.017

29 C15 2 25.95

0.03

0.000

30 C15 3 25.84

-0.08

0.000

31 C16 1 25.95 77.91 25.97 2023.36 -0.02 0.001 0.001

32 C16 2 25.84

-0.13

0.000

33 C16 3 26.12

0.15

0.000

34 C23 1 33.11 100.71 33.57 3382.50 -0.46 0.018 0.009

35 C23 2 32.98

-0.59

0.000

36 C23 3 34.62

1.05

0.000

37 C24 1 34.73 101.88 33.96 3463.50 0.77 0.001 0.001

38 C24 2 34.75

0.79

0.000

39 C24 3 32.40

-1.56

0.000

40 C25 1 32.18 98.20 32.73 3216.28 -0.55 0.009 0.004

41 C25 2 32.17

-0.56

0.000

42 C25 3 33.85

1.12

0.000

43 C26 1 33.88 101.35 33.78 3424.53 0.10 0.040 0.020

44 C26 2 34.27

0.49

0.000

45 C26 3 33.20

-0.58

0.000

46 C34 1 32.95 100.27 33.42 3353.90 -0.47 0.007 0.004

47 C34 2 32.61

-0.81

0.000

48 C34 3 34.71

1.29

0.000

49 C35 1 34.91 104.57 34.86 3645.04 0.05 0.002 0.001

50 C35 2 35.02

0.16

0.000

51 C35 3 34.64

-0.22

0.000

52 C36 1 34.48 103.34 34.45 3559.73 0.03 0.004 0.002

53 C36 2 34.38

-0.07

0.000

54 C36 3 34.48

0.03

0.000

55 C45 1 34.55 103.80 34.60 3591.48 -0.05 0.004 0.002

56 C45 2 34.62

0.02

0.000

57 C45 3 34.63

0.03

0.000

58 C46 1 32.73 98.10 32.70 3207.87 0.03 0.002 0.001

59 C46 2 32.67

-0.03

0.000



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60 C46 3 32.70

0.00

0.000

61 C56 1 32.60 97.95 32.65 3198.07 -0.05 0.006 0.003

62 C56 2 32.71

-3165.3

0.000

63 C56 3 32.64

-0.01

0.000

Total 10.74

Sy2 0.268

Sy 0.518

Table 7: t-Test: Paired Two Sample for Means

34.60 29.35

Mean 30.46 29.43

Variance 24.43 26.94

Observations 41.00 41.00

Pearson Correlation 0.62 Hypothesized Mean

Difference 0.00

df 40.00

t Stat 1.49

P(T<=t) one-tail 0.07

t Critical one-tail 1.68

P(T<=t) two-tail 0.15

t Critical two-tail 2.02

Table 8: Experimental Test Results and Scheffe’s Model Test Results.

Sr.No. Symbols

Experimental

test results

(MPa)

Scheffe

model test

results(MPa)

Symbols

Experimental

test

results(MPa)

Scheffe

model test

results(MPa)

1 Y1 33 33 C1 32 33.24

2 Y2 31.5 31.5 C2 29 30.67

3 Y3 25.2 25.2 C3 24 29.51

4 Y4 24.4 24.4 C4 23.5 30.72

5 Y5 32.8 32.8 C5 32 37.91

6 Y6 32.2 32.2 C6 31.5 38.13

7 Y12 28.99 28.99 C12 27.8 34.16

8 Y13 26.3 26.3 C13 26.28 33.66

9 Y14 27 27 C14 26.3 33.44

10 Y15 26.32 26.32 C15 25.92 31.26

11 Y16 26.7 26.7 C16 25.97 32.87

12 Y23 33 33 C23 33.57 32.40

13 Y24 34 34 C24 33.96 39.07

14 Y25 33.7 33.7 C25 32.73 35.05



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15 Y26 33.1 33.1 C26 33.78 35.25

16 Y34 32.87 32.87 C34 33.42 33.82

17 Y35 31.5 31.5 C35 34.86 35.85

18 Y36 34.11 34.11 C36 34.45 36.38

19 Y45 34.21 34.21 C45 34.6 35.85

20 Y46 33.22 33.22 C46 32.7 31.38

21 Y56 32.04 32.04 C56 32.65 31.48

Table 9: t-Test for the Compressive Strength

Sr.No. Symbol Eexpt EScheffe,s Eexpt -EScheffe,s (Eexpt -EScheffe,s)2

1 C1 32 33.24 -1.24 1.54

2 C2 29 30.67 -1.67 2.79

3 C3 24 29.51 -5.51 30.36

4 C4 23.5 30.72 -7.22 52.13

5 C5 32 37.91 -5.91 34.93

6 C6 31.5 38.13 -6.63 43.96

7 C12 27.8 34.16 -6.36 40.45

8 C13 26.28 33.66 -7.38 54.46

9 C14 26.3 33.44 -7.14 50.98

10 C15 25.92 31.26 -5.34 28.52

11 C16 25.97 32.87 -6.90 47.61

12 C23 33.57 32.40 1.17 1.37

13 C24 33.96 39.07 -5.11 26.11

14 C25 32.73 35.05 -2.32 5.38

15 C26 33.78 35.25 -1.47 2.16

16 C34 33.42 33.82 -0.40 0.16

17 C35 34.86 35.85 -0.99 0.98

18 C36 34.45 36.38 -1.93 3.72

19 C45 34.6 35.85 -1.25 1.56

20 C46 32.7 31.38 1.32 1.74

21 C56 32.65 31.48 1.17 1.37

Total -71.11 432.28

tstat =∑(Expt -EModel)/[n*∑(Eexpt - EScheffe,s)2 -∑(Expt-model)2]^0.5 /(n-1)

= -71.11/ [21*432.28--71.112]/ (21-1)]^0.5

= -0.079082359

α = 0.05 and 0.025 for two tail (t-distribution table).

tcritical = 2.086

tstat < tcritical.



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Table 10: Anova: Single Factor

SUMMARY

Groups Count Sum Average Variance

Eexpt 42 1358.26 32.33952 11.37024

Emodel 42 1358.26 32.33952 11.37024

ANOVA

Source of Variation SS df MS F P-value F crit

Between Groups 2.27E-13 1 2.27E-13 2E-14 1 3.957388

Within Groups 932.3596 82 11.37024

Total 932.3596 83

4.1. Regression equation for flexural strength

From Eq. (8), the coefficients of the Scheffe’s second degree polynomial were determined as follows;

Substituting the values of these coefficients into Eq. (7) yields

Y = 32X1 + 29X2 + 24X3 + 23.5X4 + 32X5+ 31.5X6 -10.8X1X2-6.88X1X3 -5.8X1X4-9X1X5 -24.32X1X6-23.12X2X3

+28.28X2X4 +54.34X2X5 +8.92X2X6 +14.12X3X4+38.68X3X5 +27.44X3X6 +36.8X4X5

+22.48X4X6 +9.4X5X6 (10)

Eq. (10) is now a new model for the optimization of the compressive strength of remixed concrete using Scheffe’s

second degree polynomials.

4.2. Replication variance

Mean responses, Y and the variances of replicates Si2 in Table 6 were obtained from equation below

Si2=1/(n-1)

[∑(Yi-Y)2]∑ [𝑌𝑖 − Y]𝑛𝑖=1

2] (11)

Y= Yi /n`

where Yi = responses; Y = mean of the responses for each control point; n = number of parallel observations

at every point; n – 1 = degree of freedom; Si2 = variance at each design point.

For all the design points, number of degrees of freedom [12–13],

Ve=∑ N-2=42-2=40

where N is the number of points.

Sy2= 10.74/40=0.268

where Si2 = variance at each design point

Sy = 0.518

The experimental results and the Scheffe’s model test results were shown in Table 7 below;



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V. VALIDATION AND TEST OF ADEQUACY OF THE MODEL The improved model was analyzed statistically using student’s t-test and ANOVA method; the adequacy of the model

was tested against the experimental results of the control points. The predicted values (Y(predicted)) for the test control

points were obtained by substituting the corresponding values of X1, X2, X3, X4,X5and X6 into the improved model

equation i.e. Eq. (10). These values were compared with the experimental results (Y-observed).

The test for adequacy of the model was done using student’s t-test and ANOVA at 95% confidence level on the

flexural strength at the control points that is, C1, C2, C3, C4, C5,C6, C12, C13, C14, C15,C16, C23, C24, C25,C26, C34, C35

,C36, C45,C46and C56. In this test, two hypotheses were set as follows:

5.1. Null hypothesis

There is no significant difference between the laboratory tests and model predicted strength results.

5.2. Alternative hypothesis

There is a significant difference between the laboratory test and model predicted strength results.

5.3. Student’s t-Test

We do a two-tail test (inequality) and if t Stat > t Critical two tail, we reject the null hypothesis. From the

result presented in Tables 9, t stat = -0.07908 and t critical two tail = 2.086 so t critical > t stat.

Therefore, we accept the null hypothesis.

5.4. Discussion of result

Using Scheffe’s simplex model the values of the compressive strength were obtained. The model gave

minimum compressive strength of 23.02 N/mm2 corresponding to mix ratio of 0.25: 0:0.25:0.25: 0.25 for

cement, CA, FA, W/C ratio, blend ratio and time lag (t) respectively. This further showed that the improved

value of compressive strength was achieved .The maximum strength was found to be 34.91 N/mm2

corresponding to mix ratio of 0.1: 0.225: 0.225: 0: 0.225: 0.225 for cement, CA, FA, W/C ratio, blend ratio

and time lag (t).

The maximum strength value was greater than the minimum value specified by the Indian Standard Code

for the compressive strength of good concrete. Using the model, compressive strength of all points in the

simplex can be derived.

VI. CONCLUSION

Scheffe’s second degree polynomial was used to formulate a model for predicting the compressive strength

of remixed concrete. This model could predict the compressive strength of the remixed concrete cubes if

the mix parameters are known and vice versa. The strengths predicted by the models are in good agreement

with the corresponding experimentally observed results. The optimum attainable compressive strength

predicted by the model at the 28th day within the factor space was 34.91N/mm2 corresponding to mix ratio

of 0.1: 0.225: 0.225: 0: 0.225: 0.225 and the minimum strength was found to be 23.02 N/mm2 corresponding

to mix ratio of 0.25: 0:0.25:0.25: 0.25 for cement, CA, FA, W/C ratio, blend ratio and time lag (t).This

meets the minimum standard requirement stipulated by IS code. With the model, any desired compressive

strength of remixed concrete, given any mix proportions is easily evaluated.



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REFERENCES

[1] N.K.Bairagi and P.P.Jhaveri, (March 1977),” Strength Variation of Composite Mixes by Pure Selfing” Indian Concrete Journal, Vol. 11,Pp 87-89

[2] N.K.Bairagi, A.S.Goyal and P.A.Joshi, (December 1989),"Strength of Composite Mixes Using General Crossing Theory”, the Indian Concrete Journal, Vol.63, Pp.600-605.

[3] N.K.Bairagi,"Selfing and Crossing Concept Applied to the Strength of Blended Concrete Mixes"

[4] H. Scheffé, Experiments with mixtures, J. R. Stat. Soc. B 20 (1958) 344–360.

[5] H. Scheffé, The simplex-centroid design for experiments with mixtures, J. R.

[6] N.N. Osadebe, Generalized Mathematical Modeling of Compressive Strength of Normal Concrete as a Multi-Variant Function of the Properties of its Constituent Components, University of Nigeria Nsukka, 2003.

[7] IS 12269(1989), "Specification for 53 Grade Portland Cement, Indian Standards, New Delhi

[8] IS: 2386 (part I) – 1963 (reaffirmed 2002) Indian standard methods of test for aggregates for concrete part I particle size and shape

[9] IS 383 (1970)," Specification for Coarse and Fine Aggregates from Natural Sources of Concrete", In: Bureau of Indian Standards Reaffirmed 2002, New Delhi, India, p 19

[10] IS 516(1999), " Methods of Tests for Strength of Concrete, (Reaffirmed 1999)", Bureau of Indian Standards, New Delhiand J. Hernandez, " A tutorial on Digital Watermarking ", In IEEE annual Carnahan conference on security technology, Spain, 1999.

[11] IS 456: 2000 Indian standard plain and reinforced concrete - code of practice (fourth revision)

[12] Simon, M. J., Teng, (2003), "Concrete Mixture Optimization Using Statistical Methods", Final Report, FHWA-RD-03-060, Office of Infrastructure Research and Development ,1-168

[13] Simon, M. , Lagergren, E. S., & Snyder, K. A. (1997),"Concrete Mixture Optimization Using Statistical Mixture Design Methods", Proceedings of the PCI/FHWA International Symposium on High Performance Concrete New Orleans, 20-22



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