i

ENGR. NZEH, REMIGIUS EGWUATU

PG/M.ENG/99/26274

PG/M. Sc/09/51723

STANDARDIZATION OF SANDCRETE – BLOCK’S STRENGTH

THROUGH MATHEMATICAL MODELING AND PRECISION

CIVIL ENGINEERING

A THESIS SUBMITTED TO THE DEPARTMENT OF CIVIL ENGINEERING, FACULTY

OF ENGINEERING, UNIVERSITY OF NIGERIA, NSUKKA

Webmaster

Digitally Signed by Webmaster‟s Name

DN : CN = Webmaster‟s name O= University of Nigeria, Nsukka

OU = Innovation Centre

JUNE, 2008

ii

STANDARDIZATION OF SANDCRETE – BLOCK’S STRENGTH THROUGH MATHEMATICAL

MODELING AND PRECISION

BY

ENGR. NZEH, REMIGIUS EGWUATU

PG/M.ENG/99/26274

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY OF NIGERIA, NSUKKA

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE AWARD OF MASTER OF

ENGINEERING IN CIVIL ENGINEERING (STRUCTURAL

ENGINEERING OPTION)

JUNE 2008

iii

APPROVAL/CERTIFICATION

THIS PROJECT HAS BEEN APPROVED/CERTIFIED FOR THE

AWARD OF THE DEGREE OF MASTER OF ENGINEERING IN CIVIL

ENGINEERING

BY

REV. ENGR. PROF. N. N. OSADEBE

(SUPERVISOR)

EXTERNAL EXAMINER

ENGR. PROF. J. C. AGUNWAMBA

(HEAD OF DEPARTMENT)

DEAN, SCHOOL OF POST GRADUATE STUDIES

iv

TITLE PAGE

STANDARDIZATION OF SANDCRETE – BLOCK’S STRENGTH THROUGH MATHEMATICAL

MODELING AND PRECISION

DEDICATION

v

To my son, Remitregah, a promising young Engineer.

To the contemporary engineers, architects, block manufacturers, builders and home makers.

And

To the catholic diocese of Nsukka

vi

ACKNOWLEDGEMENT

I am grateful to God Almighty who kept me, my supervisor and my other lecturers

alive to the successful end of this project. I thank him also for creating all material resources,

with the inherent factors of safety, which humankind know nothing about but manipulate to

build structures.

I am also grateful to the Catholic Bishop of Nsukka Diocese, His Excellency, Most

Rev. Dr. F. E. O. Okobo who gave me space in his heart and diocese for the practice of civil

engineering. May God bless him more.

I thank my supervisor, Rev. Engr. Prof. N. N. Osadebe for the courage and directions

he gave me even when I was to give up. I thank Engr. Prof. J. C. Agunwamba, the head of

Department of Civil Engineering for his courage and persuasion to complete this project.

I thank my lecturers, Engr. Dr. C. U. Nwoji, Engr. Dr. H. N. Onah, Engr. B. O. Mama

and Dr. F. O. Okafor for their efforts and appreciation of my strengths and my weaknesses.

My thanks also go to the men that helped me in the laboratory work, Mr. Cyril Eze the brick

layer, Silas Ugwuanyi the mason, Ogbozor Thomas the carpenter, Eze Joseph and Asogwa

Joseph the laboratory assistants, my very brother Elias Nzeh, Mr. C. C. Wogu the Chief

Technical Officer, Daniel Onyishi and Patrick Omeje the masons and Ngwu Nwaedego a

helper.

My special thanks go to Engr. T. N. Osadebe, the Director of Works Services

Department, UNN who I was privileged to be closely associated with and who also offered

invaluable time for this project to come through. I greet Mrs. Chinyere Ojobor and Engr.

Charles for typing this manuscript.

My Special gratitude goes to my wife Nzeh Rita I. and my children who helped me in

different ways that money cannot buy.

May God reward you all in Jesus name.

Engr. R. E. Nzeh

UNN

vii

ABSTRACT

The ultimate purpose of this thesis is to find the effects of various mix proportions on the

compressive strength of hollow 225mm x 225mm x 450mm sandcrete block with 40% void

and correlation between it and equivalent cube 150mm x 150mm x 150mm size by

standardization of the strength through mathematical modeling and precision. The

compressive strength tests were done on the topside as cast surfaces of the hollow blocks and

on the side surfaces of the cubes and statistical analyses were used to establish the correlation

between strength values of the hollow blocks and cubes. With the mathematical model so

obtained, one can use any mix proportion to predict the corresponding compressive strength of

the blocks. The strength values showed that the cube has approximately three times the value

of the hollow block strength, and a very good linear correlation between strength of the

hollow blocks and the cubes is Rc = 2.6 + 2.11Rb. The minimum compressive strength was 2.0

N/mm2 and 6.5 N/mm

2 for blocks and cubes respectively; while the maximum compressive

strength was 3.507 N/mm2 and 6.5 N/mm

2 for blocks and cubes respectively. From the

student‟s T distribution, the confidence intervals were found to be as follows

2.3620≤µ≤3.1380 for blocks and 8.6712≤µ≤10.8745 for cubes. A minimum compressive

strength between 1.75N/mm2 to 2.0N/mm

2 is recommended for pure sandcrete hollow blocks

(225mm x 225mm x 450mm) of 40% void.

viii

TABLE OF CONTENT

APPROVAL/CERTIFICATION ii

TITLE PAGE iii

DEDICATION iv

ACKNOWLEDGEMENT v

ABSTRACT vi

TABLE OF CONTENT vii

CHAPTER ONE: INTRODUCTION 1

1.0 DEFINITIONS 1

1.1 CONCRETE BLOCK 1

1.2 CONCRETE 1

1.3 SANDCRETE 1

1.4 BLOCK 1

1.5 SANCRETE BLOCK 1

1.6 HOLLOW LOAD BEARING BLOCK 2

1.7 HOLLOW UNITS 2

1.8 SIZES OF UNITS 3

1.9 WEIGHT OF SANDCRETE BLOCKS 4

1.10 STATEMENT OF PROBLEM 4

1.11 OBJECTIVE OF STUDY 4

1.12 SIGNIFICANCE OF STUDY 5

1.13 LIMITATIONS 5

CHAPTER TWO: LITERATURE REVIEW 6

2.1 STRENGTH VS DENSITY 6

2.2 STRENGTH VS AGGREGATE SIZE 9

2.3 STRENGTH VS W/C RATIO 10

2.4 STRENGTH VS SHELL THICKNESS 11

2.5 STRENGTH VS COMPACTION 11

2.6 STRENGTH VS CURING CONDITION 11

2.7 STRENGTH VS SAND/CEMENT RATIO 13

2.8 CEMENT STABILIZED BLOCKS AS

A CONSTRUCTION MATERIAL 15

2.9 STANDARD SPECIFICATION 16

2.10 GERMAN SPECIFICATIONS 16

ix

2.11 NIGERIAN SPECIFICATION 16

2.12 STRESSES AT THE BASE OF BLOCK WALLS 17

2.13 COMMENT 17

2.14 SURVEY OF BLOCK MOULDING FACTORIES 18

2.15 COMPRESSIVE STRENGHT 18

CHAPTER THREE: SIMPLEX LATTICE METHOD 20

3.1 INTRODUCTION 20

3.2 ADVANTAGES OF LATTICE 21

3.3 SIMPLE LATTICE METHOD 22

3.4 HENRY SCHEFFE‟S SIMPLEX-LATTICE DESIGNS 24

3.5 TESTING THE FIT OF A SECOND-DEGREE POLYNOMIAL 27

3.6 COMPONENT TRANSFORMATION 29

3.7 USE OF THE VALUES IN EXPERIMENT 34

CHAPTER FOUR: EXPERIMENTAL METHODOLOGY 35

4.0 CONSTITUENT MATERIAL 35

4.1 CEMENT QUALITY 35

4.2 SAND, OR FINE AGGREGATE 35

4.3 WATER 35

4.4 MANUFACTURE OF SANDCRETE BLOCKS AND CUBES 35

4.5 FORM AND SIZE 36

4.6 PROPERTIES 36

4.7 DENSITY 36

4.8 STRENGTH OF SANDCRETE BLOCK 37

4.9 SPECIMENS SIZES AND SHAPES

37

4.10 MOULD SIZE AND 40% VOID COMPUTATION 38

4.11 CALCULATIONS FOR MATERIALS VOLUMES & WEIGHT 39

4.12 MIX PROPORTIONS 42

4.13 BATCHING AND MIXING 43

4.14 COMPACTION OF THE SPECIMEN SANDCRETE BLOCKS

AND CUBES 43

4.15 CURING 43

4.16 THE COMPRESSIVE STRENGTH TESTING OF SANDCRETE

BLOCK AND SANDCRETE CUBES ON 28TH

DAY 43

4.17 OBSERVATIONS AND CONCLUSION 44

x

CHAPTER FIVE: RESULTS AND ANALYSIS 54

5.1 RESULTS 54

5.2 PREDICTION OF Yi FOR BLOCKS BY REGRESSION EQUATION 57

5.3 ADEQUACY OF EXPERIMENT 59

5.4 USE OF t-DISTRIBUTION 60

5.5 PREDICTION OF YI FOR CUBES BY REGRESSION EQUATION 63

5.6 CUBE – Y-PREDICTED FOR CUBES 63

5.8 CORRELATION AND REGRESSION ANALYSIS 66

5.9 REGRESSION 67

5.10 THE LINEAR REGRESSION EQUATION 68

5.11 DETERMINATION OF THE CORRELATION BETWEEN

THE STRENGTH OF THE HOLLOW BLOCKS AND THE CUBES 69

CHAPTER SIX: CONCLUSION AND RECOMMENDATIONS 71

6.1 CONCLUSION 71

6.2 RECOMMENDATIONS 74

REFERENCES 75

APPENDIX

xi

CHAPTER ONE

INTRODUCTION

2.0 DEFINITIONS

1.1 CONCRETE BLOCK

This is a two word element, concrete and block, hence Block made from Concrete.

1.2 CONCRETE

Concrete is a mixture of aggregate, cements and water.

Aggregates are made of fine and coarse particles that are sand and stones, pebbles,

chippings etc.

1.3 SANDCRETE

The word sandcrete is a product of only sand, cement and water, without coarse

aggregate or stones. This would be called sand-cement mixtures.

1.4 BLOCK

The word block is a prismatic shape made from any kind of materials such as wood,

metal, glass, etc which could be used for any purpose. In this case this block is moulded or

made, from the concrete or sandcrete mixture, to form any desired shape of the block, hence

the name concrete or sandcrete block.

1.5 SANCRETE BLOCK

This is a masonry unit and is of a mixture of an inert material generally sand only,

Portland cement and water, compressed or compacted into required prismatic shape, used in

construction industries.

In this country, Nigeria, and many other parts of the world, masonry units have been

used in all types of masonry constructions. The use of mud blocks is now fast disappearing

and giving way to the use of masonry units especially sandcrete blocks.

Nowadays there are a lot of sandcrete and concrete block industries all over the place.

So many are using wooden moulds or metal moulds and compaction is done manually with

hand. Although manual compaction reduces cost of machinery, it is found that at times the

block shapes are inconsistent.

Mechanization in production of sandcrete blocks has also set in and the use of

machines moulded and vibrated sandcrete clocks have gained great entry into the society. In

this case, compaction is of greater efficiency and uniformity is achieved. The strength is also

more certain to be greater than in the manual case.

The two basic types of block moulding machines commonly used so far are the egg-

laying machine and the static machine. Rosachometta vibrating machine is popular for

making sandcrete blocks in Nigeria (Sampson et al, 2002).

xii

The ingredient for most sandcrete blocks are fed into a paddle type mixer, and the

mixture fed into a block forming machine where the blocks are moulded under heavy pressure

and vibration. The blocks are then transported into a curing room or location where they are

cured by steam or open air drying after watering for a period of time.

Sandcrete blocks are masonry units made use of in all types of masonry constructions

such as exterior and interior load bearing walls, fire walls, party walls, curtain walls, panel

walls, partition, backings for other masonry, facing materials, fire proofing over structured

steel members, piers, pilasters columns, retaining walls, chimneys, fireplaces, concrete floors,

patio-paving units, curbs and fences (Smith, 1973).

The most common concrete masonry unit is the concrete block, made with both

stone/sand and lightweight aggregates. They are made in five basic types.

1. Hollow load bearing sandcrete blocks

2. Solid load bearing sandcrete blocks

3. Hollow non load bearing sandcrete blocks

4. Sandcrete building tiles

5. Sandcrete bricks

1.6 HOLLOW LOAD BEARING BLOCK

The hollow load bearing block is the one chosen for this project-work, since it is the

most commonly used in the society presently.

Hence, investigation into its constituent materials and strength therein is of paramount

importance.

In this study, a hollow load bearing sandcrete block of 225 x 225 x 450mm nominal

size has been used.

Nominal size of a 200 x 200 x 400mm was given to be of weight approximately (40 to

50 lbs) 18.16 to 22.7 kg when made with heavy aggregates and 25 to 35 lbs when made with

light weight aggregates.

Heavy weight blocks are made from aggregates such as sand, gravel and crushed

stones and air cooled slag

Light weight units are made from cinders, expanded shale, pumice and scoria

A solid sandcrete block is one which is defined by ASIM as having a core area of not

more than 25% of the gross cross sectional area (Smith, 1973).

1.7 HOLLOW UNITS

1. A hollow sandcrete block is one in which the core area exceeds 25% of the cross

sectional area of the block (Smith, 1973).

xiii

2. Hollow units are defined as those having core-void areas greater than 25% of the gross

area. They may be of two-core or three core designs, at the option of the

manufacturer.

3. Sandcrete block is a precast masonry unit and a variety of sandcrete masonry - world

used for building or structural construction.

Generally, the core area of such units will be from 40 to 50 percent of the gross area

for hollow blocks. In this project, 40% of the core area is used as a case study

Sandcrete blocks are made in wide varieties of shapes and sizes to fit the different

construction needs.

Of course, the availability of suitable structural materials is one of the principal

limitations on the accomplishments of a project or an experienced structural engineer. Early

builders depended almost exclusively on wood, stone, brick and concrete or sandcrete and

mud-blocks. The versatility of sandcrete, the wide availability of its component materials, the

unique ease of shaping its forms to meet the strength and functional requirements involved,

together with the exciting potentials of further improvements and developments for practically

all kinds of structural requirements which has been accomplished by careful selection of the

design dimensions, and development of proper cement, selection of proper aggregates and mix

proportions, careful control of mixing, placing and curing techniques, and imaginative

development of construction methods, equipment, and prosecutes, have been used to make

sandcrete a strong good competitor of other materials in construction industry for the future.

The minimum allowable thickness of face shells and interior webs are given in the

table below (Boeck et al, 2000)

Nominal width of units

(mm)

Minimal Face shell

thickness (mm)

Minimal Web. Thickness

(mm)

100 20 20

125 20 25

150 25 25

225 35 25

1.8 SIZES OF UNITS

All sandcrete masonry units are modular in size. The larger units called blocks have

nominal face dimensions of 225mm in height by 450mm in length and nominal thickness of

100mm, 125mm, 150mm and 225mm.

xiv

1.9 WEIGHT OF SANDCRETE BLOCKS

The weight of sandcrete units can be calculated from the equivalent solid thickness of

the block unit, provided that the density of the sandcrete block is known

1.10 STATEMENT OF PROBLEM

In many of the block industries, there re no standard measures taken during processes

of hollow block production. Consequently, the hollow blocks are not cured properly. They

do not have any standard mix proportions for the sand/cement/water ratios and the sands could

be a mixture of gutter or river sands.

The major aim of the producer or proprietor is profit, and as long as he can produce

more blocks per bags of cement and make more money, nothing else matters. There are no

consideration for strength, durability, quality control, mix proportion, curing and methods

used.

1.11 OBJECTIVE OF STUDY

Hence the objective of this research is to subject the production of hollow sandcrete

block to the same conditions upon which blocks industries produce their blocks without

neglecting the above factors.

Thereafter, investigate the effect of various mix proportions on the strength of hollow

sandcrete blocks.

Establish a mathematical model relationship between compressive strength of a full

size sandcrete hollow block of 225 x 225 x 450mm as a function of mix proportion of

its constituent ingredients; water, cement and sand.

To establish the correlation between the compressive strength of sandcrete cube of 150

x 150 x 150mm and its corresponding full size hollow block of 225 x 225 x 450mm of

40% void and of the same ingredient and age.

To provide a guideline for the selection of mix proportion to achieve a target strength.

To eliminate the use of trial and error in the ratios of sandcrete block ingredient or

constituent materials.

Production of quality sandcrete blocks at a minimum cost and yet, profit will not be

neglected.

xv

1.12 SIGNIFICANCE OF STUDY

The eventual gains in this research cannot be overemphasized. There are immense

benefits to those who research in this area. Producers of hollow sandcrete blocks will benefit

immensely from the results of this research work.

Their fears on quality of blocks produced must have been cleared.

They can be sure they achieve the target strength, yet not sacrificing their profit.

Their reputation shall increase and closure by Government for not measuring to

standards required or specification must have been a thing of the past.

Contractors who use these materials for construction would be sure of using standard

hollow blocks produced as such.

The structures built with such blocks shall stand the test of time and the usual cracks

and deterioration in construction industries must be reduced drastically.

The eventual users of these buildings must be set free from unnecessary maintenance

and huge expenses therein, hence very happy with the buildings they live in or use for

office, etc.

Any organization, ministry, proprietor can use the mathematical model to achieve their

required compressive strength for production of such blocks required.

1.13 LIMITATIONS

Most of the sand were hand sieved

Moulding of the sandcrete and cube blocks were compacted by hand or manually.

The wooden mould built could not keep to its shape. There was shrinkage or

expansion effect from the wooden mould.

Measurements of the batches were by weight on a weighing machine.

The block cubes were made in stable metal mould but were hand-rodded while those

of sandcrete blocks were of wood and hand compacted also.

Weighing machine error due to age

Testing machine deficiency due to age

The aggregates were air-dried.

Personal mistakes in mixing, readings of results from the machines

xvi

CHAPTER TWO

LITERATURE REVIEW

Smith (1973) had defined a hollow sandcrete block as one in which the core area

exceeds 25% of the cross-sectional area, and generally the core area of such units would be

from 40 to 50 percent of the gross area1. as given by ASTM p.100 Smith further defines the

compressive strength of blocks as a measure of the block‟s ability to carry loads and withstand

structural stress1.p.104.

He gave also the minimum compressive strength of hollow load-bearing concrete masonry

units as specified by ASTM-eg. 1952 as (800PSi) 5.516N/mm2 for a minimum face shell

thickness of (1¼’’) 32mm for grade „A‟ the strength is (1600PSi) 11.03N/mm

2 and grade B, is

(1000PSi) 6.895N/mm2 but are for blocks with coarse aggregates. Jackson (1984) stated that

in addition to size, compressive strength is the basic requirement of sandcrete blocks except

for non-load bearing blocks with a thickness less than 75mm which are required to comply

with the transverse breaking strength test for handling. The compressive strength of (concrete)

sandcrete blocks is dependent mainly on their mix composition (in particular binder content),

degree of compaction (or aeration) and to a lesser extent on the aggregate type and curing

normally used (Smith, 1973).

2.1 STRENGTH VS DENSITY

In general Smith stated that for a given set of materials, the strength of a concrete

block will increase with its density (Smith, 1973). The range of strength specified in BS6073

part 2 is 2.8 to 35N/mm-2

. Although this did not specify weather for hollow or solid blocks. He

noted also that since all forms of blocks are tested in the same manner, strength being

calculated as

blockofareagrossthe

loadcrushing.

It follows that for a given strength the form of block (solid, cellular, or hollow) will

not affect its load-carrying capacity. This last statement is questionable since even in steel

structures of hollow metals or tubes, the area used is the net area of solid materials and not the

gross area of the tube, circular hollow metal or hollow square metal21

.

NIS – 75 defined hollow sandcrete blocks as one in which one or more large holes or

cavities pass through the block and the solid material is between 50% and 75% of the total

volume of the block calculated from overall dimensions, as stated by Ezeokonkwo(1998). NIS

-75 also defines compressive strength as the failure load during test divided by the gross block

xvii

area, which means that the compressive load bearing capacity is equivalent for both solid and

hollow blocks (Reynolds and Steedman, 1988). This agrees with N. Jackson‟s definition of

compressive strength of block.

But the design codes(BS 8110, 1985) for concrete masonry units, calculated the

compressive strength of hollow blocks on the basis of the net area of the hollow block

(Reynolds and Steedman, 1988). Eze-Uzomaka upheld the above view, that it is the solid area

of the block that actually transmits and sustains the load, hence compressive strength of a

hollow block should be defined with respect to the solid area only (Reynolds and Steedman,

1988). Osili proposed an average 28days compressive strength of 2.07N/mm2 and a minimum

28days strength of 1.72N/mm2 for individual blocks in the absence of any specification for

sandcrete blocks, which Ezeokonkwo (1998) rightly observed, has a limited applications

(Reynolds and Steedman, 1988)).

The Federal Ministry of Works during an Annual Conference in Kano lowered

requirements for sandcrete blocks based on gross area for load bearing walls of 2 or 3 storey

buildings to 2.1N/mm2 for an average of 6blocks and for individual block gave a minimum of

1.75N/mm2. This also has same problem of limitation and does not allow for flexibility in

design as stated by Ezeokonkwo (Reynolds and Steedman, 1988).

Ezeokonkwo (1988) displayed Onyemelukwe‟s over view of the Draft specification for the

NIS – 75 specification for sandcrete block compressive strength, pointing out that standard

organization of Nigeria envisaged specifying two types of Block, type „A’:- being load-

bearing blocks and type ‘B:- being non-load. The load-bearing blocks (type A) are strength-

graded while type „B‟, although not strength graded, is required to possess a minimum

crushing strength of 1.5N/mm2. Below is the table for proposed strength grading.

Table 2.1. Proposed Strength Grading (Reynolds and Steedman, 1988)

Block type and Designation

Minimum Compressive Strength

Average of 10Blocks Lowest Individual Blocks

Type „B‟ N/mm2

N/mm2

1.50 1.00

Type A(2.5) 2.50 1.75

A(3.5) 3.50 2.50

A(5.0) 5.00 3.50

A(7.5) 7.50 5.50

A(10.0) 10.00 7.50

Additional grades advancing at Designated strength grade 75% of designated strength

xviii

increment of 2.5N/mm2

Ezeokonkwo (1988) displayed NIS-75 as follows: NIS – 75, like BS2028 for concrete blocks,

specified three types of blocks.

TYPE A: For load-bearing and non load-bearing external use;

TYPE B: For load-bearing internal use or for load-bearing or non load bearing external

use, if protected by rendering or other effective manner;

TYPE C: For non-load-bearing internal use.

NIS – 75 specified minimum strength requirements for the three types of blocks as shown in

the table below:

Table 2.2, NIS – 75 Minimum Compressive Strength (Reynolds and Steedman, 1988)

Block Classification Average of 8 Blocks Individual Units

N/mm2 N/mm

2

A 7.5 5.5

B 5.0 4.0

C 3.5 2.5

An overview of the two tables shows that the proposed strength grading has a wider

range for design and better than NIS – 75 specifications in tables 2.1 and 2.2. The wider range

enhances freedom in structural design of sandcrete or concrete blocks for walls in building

structures.

Orchard (1973) has concrete blocks divided into three types:-

Type A: For general use including use below the ground level damp-proof course, and

having a density (no allowance being made for cavities) of not less than

1500Kg/m3 (93.61b/ft

3).

Type B: For general use including use below the ground level damp-proof course in

internal walls, and the inner leaf of external cavity walls. They shall have a

density of less than 1500Kg/m3 (93.61b/ft

3).

Type C: For internal non-load-bearing walls and having a density of less than

1500Kg/m3 (93.61b/ft

3).

A minimum compressive strength is specified for Type A and B blocks and a

minimum transverse breaking load for Type C blocks. This specification by Orchard

explained more on types A, B, C, blocks than NIS – 75 and BS2028 by adding locations and

density limits.

xix

Jackson (1984) stated that the compressive strength of concrete blocks tested in

accordance with BS6073: Part 1 is specified as a minimum average value of ten blocks of

which no single value should fall below 80% of the permitted average value.

Ezeokonkwo (1998) stated that NIS – 75 specification for sandcrete blocks when

compared with BS2028 strength grading for concrete blocks of type “A”, showed that the

minimum compressive strength specified for sandcrete blocks of type „A‟ is greater than the

minimum strength-grading for type „A‟ concrete blocks.

Table 2.3 below shows the compressive strength of concrete blocks for types A and B.

Concrete block being a stronger material than sandcrete blocks, have minimum strength grade

of 3.5N/mm2 for type „A‟ blocks, while sandcrete blocks have 7.5N/mm

2 minimum

compressive strength of concrete blocks types A and B.

Table 2.3: Compressive Strength of Concrete Blocks Types A & B

Block type and

Designation

Minimum Compressive Average of ten

Blocks N/mm2

Strength Lowest Individual

Blocks N/mm2

A(3.5) 3.5 2.8

A(7.0) 7.0 Typical structural units 5.6

A(10.5) 10.5 8.4

A(14.0) 14.0 Blocks of these strengths 11.2

A(21.0) 21.0 may not be readily available 16.8

A(28.) 28.0 22.4

A(35) 35.0 28.0

B(2.8) 2.8 Typical structural units 2.25

B(7.0) 7.0 5.6

There is a disadvantage in a wall or partition which has strength greatly in excess of

the needed. Hence a design should be regarded as defective not only when it does not satisfy

the functional standards, in terms of being under-designed, but also when it fails to take full

advantage of all available data to minimize cost, in terms of being over-designed.

2.2 STRENGTH VS AGGREGATE SIZE

Eze-Uzomaka15

in his presentation on the crushing strength of sandcrete blocks in

relation to their production and quality control expressed that;

(1) Sandcrete mixed with coarser sands develops higher crushing strength.

(2) The crushing strength of sandcrete units is very much affected by the total available

surface area per unit weight of sand which has to be coated by the cement paste.

xx

(3) In cases of mixes where the total surface area is too large, the available cement paste

may not be sufficient to coat all the surfaces and thus adequately bind the sand

particles.

(4) A greater quantity of cement is required with fine sand, to obtain the same strength as

with a coarser sand.

(5) Strength is a geometrical function of the total specific surface of the sand particles

used, and they are in the following forms:-

3days water curing;

031

3471

PSSR

6days water curing;

101

6042

PSSR ,

where RS = the ratio of crushing strength and

SP = the specific surface ratio

OBODOH (1999) stated that the above observations by Eze-Uzomaka substantiates the work

done by Tyler in which he found that blocks made from river sand has higher strength than

those made from sea sand for the same mix proportion and curing conditions. Apart from

coarser aggregates improving the blocks strength, it reduces the consumption of cement, water

absorption and permeability and at the same time eliminates the need for renderings and

framings in buildings. The water-cement ratio of a mix used for block production refers to the

water content of the block immediately after moulding and is different from any water it

absorbs later due to curing operation.

2.3 STRENGTH VS W/C RATIO

Tyler, he stated, noticed a trend of increasing strength with decreasing water-cement

ratio from his experiment on blocks. However, studies by Nwoke as well as Eze-Uzomaka

showed that strength increases with water-cement ratios up to a point where it starts

decreasing. It was found that a water-cement ratio of 0.4 to 0.6 was practical. Mixes below 0.4

were found to be unworkable probably under the compacting equipment used, the mixes

lacked sufficient cohesiveness to retain shape after casting. For water-cement ratio above 0.7,

much damage occurs during demoulding, hence discouraged further increase in the water-

cement ratio.

There is an optimum value of w/c ratio most suitable for a particular sand-cement ratio

and curing condition. This has to do with water required for complete hydration of cement as

xxi

well as for better compaction to be achieved due to the lubrication of particles provided by

water. When the water-cement ratio exceeds beyond this optimum value, excess pore water

resists the compaction effort, causing a decrease in strength. The make of mould used in block

compaction has been found to affect the efficiency of compaction.

2.4 STRENGTH VS SHELL THICKNESS

The effects of block geometry were examined by Eze-Uzomaka in six different

geometries of blocks with identical mixes. He stated that from theoretical point of view that

the actual thickness of the solid parts of a hollow block is the most significant, and that the

most obvious way in which this dimension affects strength is that during compaction under a

given pressure, the effect of frictional resistance between the sides of the mould and the

material, which reduces the effectiveness of the compaction, is more pronounced when this

dimension is small. Consequently, the strength of blocks should increase as the dimensions

increase. An average solid thickness was therefore defined as the quotient of the solid bearing

area to the sum of the lengths of the center-lines of the solid parts of the bearing surface. He

also stated that both the strength and average solid thickness of a block type were expressed as

ratios of corresponding values for one of the block types. From his graph on these, he found a

correlation of the ratio of strength to the ratio of average solid thickness and had the

regression curve to be exponential. There was a correlation coefficient of 0.97 and a standard

deviation or error of 0.05 from the curve and was described by the following equation.

tSg

RR 21exp350

where RSg = the ratio of strength in relation to geometry and

Rt = the ratio of thickness.

By virtue of exponential nature, he said, of this relationship, this factor is crucial in the design

of block-moulding machines.

2.5 STRENGTH VS COMPACTION

OBODO (1999) stated that method of compaction during moulding has a marked

effect on the strength of sandcrete blocks. Eze-uzomaka and Ibeh, he said, found that blocks

from factories achieving compaction by hand using wooden rammers had higher strength than

those compacted by mechanical vibration, except when the vibration is carried out with

additional surcharge.

2.6 STRENGTH VS CURING CONDITION

xxii

The curing condition is an important aspect of block production which affects various

properties of sandcrete blocks. Investigations carried out by Thomas, he said, showed that the

effect of poor curing method is to reduce the compressive strength of blocks appreciably and

rapid drying out at an early age is as harmful to sandcrete blocks as it is to other types of

aggregates/cement mixtures. He also observed that even at a temperature of 210C, rapid drying

out will take place if relative humidity is low enough. He suggested that the heat and humidity

during certain months of the year in the tropics would provide good sandcrete block curing

condition, but the condition of the temperature and humidity which are satisfactory for good

curing however, do not exist through out the year, so it would be a bad practice not to control

the drying out of sandcrerte blocks for the first seven days of its life in whatever weather it

happens to be moulded. Thomas carried out tests on air-dried blocks to find out the effects on

the strength of sandcrete blocks, whereby he found that there is some amount of deterioration

in strength due to soaking and that the strength decreases with the age of the blocks. One of

the shortcomings of his investigation was that he failed to show the effect of water-cement

ratio on the compressive strength at different curing conditions. He did not also state the

values of water-cement ratios at which his investigation took place.

Eze-Uzomaka stated that it was clear from his graph that the strength of blocks which

were cured with water sprinkled on them is higher than the strength of blocks cured without

water sprinkling. He went on to show that it was seen that the effect of water sprinkling was

very pronounced at low water-cement ratio, the ratio of strength being about 0.40. This means

that the blocks made from a mix having a water-cement ratio of 0.40 and cured with water

sprinkling is about 250% as strong as an identical block cure without sprinkling. He observed

that as the w/c ratio increases from 0.40 and 0.60, the ratio of strength increases from 0.42 to

0.70 for a mix of sand/cement ratio of 9.0 and 13.0. It was seen also that the effect of

sprinkling is more pronounced for mixes with higher sand/cement ratio. The effect of water

sprinkling is to augment, the water in the mix which is required for hydration, and hence the

effect of water sprinkling becomes more pronounced at low water/cement ratios.

OBODOH (1999) highlights on Nooke‟s reports which stated that sandcrete blocks

cured outside (in the open air) has higher strength than those cured inside the laboratory, while

immersion of specimen in water generally gave highest strength. (This statement goes against

the earlier observation of Thomas who found that air dried blocks have lower strengths.

Nonetheless, it is very important to specify which weather condition for which curing is taken

place to support ones argument). This he explained, however, might be due to the fact that the

immersed specimens were compacted by vibration whereas the others were by hand

compaction. (This explanation also clashed with the observations of Eze-Uzomaka who found

xxiii

that blocks from factories achieving compaction by hand using wooden rammers had higher

strength than those compacted by mechanical vibration). Vibrated blocks with added

surcharge have been found to give higher strength than non-vibrated ones. Yilla (1967)

reported that different curing conditions also affect the moisture absorption capacity and

change in dimension of sandcrete blocks.

2.7 STRENGTH VS SAND/CEMENT RATIO

Eze-Uzomaka observed that for a given sand/cement ratio, a sandcrete mix has an

optimum w/c ratio for strength development, in the range of 0.60 to 0.70. In this region of w/c

ratio a decisive factor is the ease with which the block can be demoulded, strength, he said, is

reduced by increasing the S/C ratio, as would be expected. He further observed that the gain

of strength with age is similar in pattern to that of concrete. About 60% to 70% of the 28-day

strength is developed in 7days while the ratio is 90% at 14days, from date of casting.

Eze-Uzomaka then established what he called a good correlation between strength of

sandcrete blocks with that of sandcrete cubes, in the form;

UC = 1.07 + 2.46 UB

where UC = the strength of cubes and

UB = the strength of blocks

Hence the strength of blocks can be investigated by testing sandcrete cubes which are very

much easier to mould in the laboratories.

OBODOH (1999) relayed Ben George whose view with Thomas and Chinsman that in

immersion of blocks in water might lead to reduction in strength as claimed by Eze-Uzomaka

but that the test was aimed at establishing a standard test since curing of blocks in open air

during dry and wet seasons could produce inconsistent results. Immersion of blocks was

aimed at reproducibility of results. He further expressed Eze-Uzomaka‟s defence of his

recommendation; that the specimens for compression test should neither be bedded nor

immersed in water before they are tested. It was not for the purpose of simulating the natural

conditions, whereby blocks are soaked due to the presence of ground water for blocks used

below the ground level, or the effects of rainstorms that soaking of specimen was included in

the draft specification. He added that for such cases in will be more logical to specify

reduction factor for the strength of blocks based on the results of the test truly simulating such

conditions. What was required in his recommendation was simply to determine the

compressive strengths of blocks after varying, but specified degree of soaking, so that the

strength reduction factors corresponding to different degrees of wetting or soaking can be

tabulated. This would take care of the situation whereby sandcrete blocks are laid in any state,

xxiv

wet or dry on site as well as the situation during construction in the wet season when certain

external block walls may be saturated through exposure to heavy rainfall.

Orchard (1973) states that the compressive strength of a wall depended principally on

the strength of the concrete (sandcrete) blocks and was little affected by the strength of the

mortar except when the blocks were laid to a diagonal pattern when, with weak mortar, failure

might be due to shear along the diagonals. Ken (1995) states that the world is divided as to

whether it is better to assess concrete strength by cube or cylinder specimens. The UK, much

Europe, the former USSR and many ex-British colonies use cubes; the USA, France and

Australia use cylinders. He went further to say that what is more important and concerns us

here is the ratio between cube and the cylinder; that is cube/cylinder ratio. The British

Standard (BS 1881), he said, nominates this ratio as 1.25 for all circumstances but this is not

the author‟s experience, which is that the ratio varies from over 1.35 to less than 1.05 as

strength increases. A formula giving results in accordance with the author‟s experience, but

not claimed to be thoroughly established, is

strengthcylinderstrengthcylinderstrengthCube

19

OR

strengthcubestrengthcubestrengthCylinder

20

where the units of cube and cylinder are the same, MPa or N/mm2.

The table below, Ken (1995) gives an alternative version which has greater official standing.

Table2.4

Table 2.4: Cube/Cylinder Strength Conversion (Orchard, 1973)

Concrete Compressive Strength at 28days mPa (N/mm2)

Grade CLYINDERS

150mm dia x 300mm CUBES

150mm x 150mm x 150mm

C2/2.5 2 2.5

C4/5 4 5

C6/7.5 6 7.5

C8/10 8 10

C10/12.5 10 12.5

C12/15 12 15

C16/20 16 20

C20/25 20 25

C25/30 25 30

xxv

C30/35 30 35

C35/40 35 40

C40/45 40 45

C45/50 45 50

C50/55 50 55

From steel pipe and structural tubing of manual of steel construction, it is clearly

observed that it is the net cross sectional area of the hollow pipe (unfilled) that has been used

and not the gross cross sectional area. In the tables of Dimensions and properties of steel pipe

(unfilled) a 12`` nominal diameter has outside diameter as 12.75

`` and inside diameter as

12.0‟‟and its Area is 14.6m

-2. It can be seen from calculation below that the gross cross

sectional area should be

222

222

1024.808240161930

6761273756

mxmm

mRAG

while the Net cross sectional area is

6375663756

22

22

rRrRrR

rRAN

232

2

10419009400095303310

57914375037512

mm

m

Hence, I also conclude that the area to be used is the net cross sectional area and not the gross

sectional area, of the hollow sandcrete block.

Boeck et-al (2000) in NSE Newsletter, stated factors affecting the strength of

sandcrete hollow block and made some insight study on these factors, such as proportion of

fine and coarse aggregates, cement content and water-cement ratio were dealt in details. They

added that “substantial savings in the cost of cement can be made by adding coarse aggregates

to sand or fine aggregates”. p.25

2.8 CEMENT STABILIZED BLOCKS AS A CONSTRUCTION MATERIAL

Boeck et al (2000) added that, the combination of cement with sand/fine aggregates

under controlled conditions of moisture and density produces a material of distinct physical

and engineering characteristics. These properties depend on five(5) main factors:

1. Nature of sand/aggregates and their particle size

2. Distribution;

3. Cement Content

xxvi

4. Water-Cement Ratio

5. Compactive effort of Block moulding machines

6. Physical conditions such as the method of curing temperature and curing time.

2.9 STANDARD SPECIFICATION

British Standard (B.S) Specification:

The British Standard 2028, 1364, 1968(1) relates to three types of pre-cast solid or hollow

blocks

Type Description Block

Density

Kg/m3

Average

Comp. Strength

N/mm2

MIN.

Compressive

Strength

N/mm2

A Dense aggregate concrete blocks for

general use in buildings.

Not less than

1500

3.5

7.0

and above

2.8

5.6

B Light weight aggregate concrete

blocks for load-bearing walls for

general us in buildings.

Below

1500

2.8

7.0

2.25

5.6

C Light weight aggregate concrete

blocks for internal non-load-bearing

walls, partitions.

Below

1500

Transverse breaking load is

Specifies

L.Bock et-al (2000) defined a hollow block as a block having one or more large holes

or cavities which pass through the block and the solid material is between 50% and 75% of the

total volume of the block, calculated from the overall dimensions.

2.10 GERMAN SPECIFICATIONS

The German Specification, DIN 18153 for Concrete Masonry Units Specifies high

Strengths. The average compressive strength required varies from 2.5N/mm2 to 60.0 N/mm2,

depending upon the class of the block. For hollow blocks similar to the type and size in

Nigeria, German Block class 4 is considered relevant. The average compressive strength

specified is 5.0N/mm2.

2.11 NIGERIAN SPECIFICATION

xxvii

The Federal Ministry of Works and Housing, Nigeria, had specified the same strength

requirements as stipulated in B.S. 2028 for sandcrete hollow block ie 2.8N/mm2 or 400PSi

(average compressive strength in wet state at 28days age). “This strength requirement was

lowered to 2.5N/mm2 or 360 P.S.I, reference” Nigerian Standard NIS:74:1976 U.D.C. 624,

012.8 for burnt clay brick unit.

In the Federal republic of Nigeria, national Building Code35

, First Edition 2006,

section 10.3.144, the strength requirement of sandcrete hollow block is 2.00N/mm2 (300psi)

for an average of 6 blocks and lowest strength of individual block is 1.75N/mm2

(250psi)

“The same strength requirements could be considered applicable for sandcrete hollow

blocks, though the “Proceedings of the Conference on Material Testing, Control and

Research” (1978), Federal Ministry of Works and Housing, Lagos, Nigeria, recommended

2.1N/mm2 or 300PSI for these blocks.

2.12 STRESSES AT THE BASE OF BLOCK WALLS

For walls of single or double storey buildings, the blocks are generally load bearing.

For a building with more than 2-storeys, it is usually a framed structure and the blocks are

non-load-bearing.

The ultimate loads at the base of walls for a typical single and double storey buildings

are given below:

For Single storey building = 1.12N/mm2

For Double storey building = 0.46N/mm2

This shows that the total stresses at the base of a wall for such buildings are very low

compared to the strength of a sandcrete block of 2.5N/mm2 strength. In other words, the safety

factor is very high. The stresses, 2.5N/mm2 will be reached with wall heights of 20.0m, which

are rare in load-bearing blockwork.

2.13 COMMENT:

(1) The above stresses could be said to be relative since the void percentage was not

given.

(2) The block strength was not given.

(3) The block components were not given as to specify whether the block is of only sand

or with coarser aggregates.

(4) The unit weight of the individual block was not given for those stresses at wall base.

If a block size of 225 x 225 x 450mm and of void 40%, this 0.12N/mm2 would be true

for a block strength of 3.3KN/m2 and at a height of 4.91m for a meter strip. For the same

xxviii

block strength of 3.3KN/m2 the stress of 0.46N/m

2 would be true at the base of about 18.818m

for 60% material and 1.0m strip and block size of 225 x 225 x 450mm. Some little

calculations prove this assertion to be true since stress = A

Pf

Area

Load; .

2.14 SURVEY OF BLOCK MOULDING FACTORIES

According to Boeck et al (2000), hollow blocks were collected from 12 major factories

in Abuja limited to Federal Capital Territory only. The survey indicated that the fine aggregate

used by most of the factories is coarse sand. A few factories were using a mixture of river

sand and crushed stone-dust. They were producing 30 to 36 blocks, of size 450 x 225 x

225mm per one bag of cement, weighing 50kg. This indicates that the cement to fine

aggregate ratio generally varies from 1:13 to 1:17.

The test was conducted on the blocks in Dantata and Sawoe laboratories for their

weights, dimensions, volumes of cavities, compressive strength etc. The test results obtained

showed much inconsistency. The blocks weighed between 20.6 to 26.4Kg and compressive

strength between 0.47 to 1.68N/mm2 in dry state and 0.32 to 1.07N/mm

2 in wet state. The

coefficient of variation is quite high. The compressive strength in all cases has been found to

be below the standard requirements.

2.15 COMPRESSIVE STRENGHT

Boeck et-al (2000) also found out in their experimental results that all the three

variables ie, percentage of coarse aggregate, cement content and water of compaction, affect

the strength considerably. The results showed that the compressive strength increases with the

increase in cement content, and the addition of coarse aggregates to the mix. When the cement

content is doubled, the compressive strength is about twice the previous figure. The

compressive strength is maximum when 45% of coarse aggregates are added to the mix. Both

of these components, cement and coarse aggregates increase the cost of the block.

Economically, it is however cheaper to add coarse aggregate in order to increase the strength

than to add more cement.

Another interesting finding is that by carefully controlling the water of compaction,

which does not increase the cost, the strength can easily be increased by 10 – 20%. The

minimum cement content required to manufacture hollow blocks in Nigeria standard (strength

of 2.5N/mm2 in wet state), is stated below:

xxix

(1) 1: 10 Cement Ratio

OR

200Kg Cement/m3

OR

23blocks, per bag of 50Kg, for 450 x 225 x

225mm size.

when no coarse aggregates are added,

and the fine aggregates conform to zone

1 of FMW grading limits.

(2) 1:12 Cement Ratio

OR

170Kg Cement/m3

OR

26blocks per bag of Cement

when 15% coarse aggregates are added

to above fine aggregates.

(3) 1: 14 Cement Ratio

OR

145kg Cement/m3

OR

29blocks per bag of Cement

(4) When fine aggregates do not conform to zone 1 and coarse aggregates cannot be added,

but coarse sand is available; then, 20blocks per bag of cement may only be produced to

have 2.5N/mm2 blocks strength.

The Nigerian National Building Code (2006) has defined sandcrete blocks to mean a

composite material made up of cement, sharp sand and water.

On mix proportion, the Code has specified that mix used for block shall not be richer

than one part by volume of cement to 6 parts of fine aggregate (sand) except that the

proportion of cement to mixed-aggregate may be reduced to 1:2

14 (where the thickness of the

web of the block is one 25mm or less).

On strength Requirements, it states that sandcrete blocks shall possess resistance to

crushing as stated below and the 28 day compressive strength for a load bearing wall of two or

three storey building shall not be less than:

NBC –P317, Art. 19.3.14.4

Average strength of 6 blocks Lowest strength of individual block

2.00N/mm2 (300psi) 1.75N/mm

2 (250psi)

xxx

xxxi

CHAPTER THREE

SIMPLEX LATTICE METHOD

3.1 INTRODUCTION

Mathematically a simplex lattice is a space of constituent variables of x1, x2, x3 -----xn

which obeys these laws; xi ≤ 0; x negative.

1;)1(10

1

i

q

i

ixx (16)

Lattice is an abstract space

Simplex is structurally representation of lines or planes joining assumed positions or points of

the constituent materials of the mixture, and they are equidistant from each other.

A simplex lattice is an abstract space and any point on it is in an abstract space point. At

principal coordinates only a component is 1, others or the rest are zero.

See fig 3.1

Eg.

Fig. 3.1

A simplex is defined as a convex polyhedron that has K+1 vertices produced by K

intersecting hyper-planes in K-dimensional space (Akhnazarova and Kafarov, 1982). In two-

dimensional space, a simplex is a triangle. In three-dimensional space, a simplex is a regular

pyramid having its four vertices produced by intersecting the planes of the respective three

faces. A regular K-simplex is defined as a set of K+1 equidistant vertices (Akhnazarova and

Kafarov, 1982). A two-dimensional regular simplex is an equilateral triangle. A-three-

dimensional regular simplex is a regular tetrahedron. In planning the experiments, regular

simplex designs usually are employed (Akhnazarova and Kafarov, 1982).

A two-dimensional regular simplex is an equilateral triangle.

A1 (1,0,0,0)

A4 (0,0,0,1)

A3 (0,0,1,0)

A2 (0,1,0,0)

xxxii

A three-dimensional regular simplex is a regular tetrahedron. In planning the

experiments, simplex designs usually are employed (Akhnazarova and Kafarov, 1982).

Any given point on the lattice, say point P(x1, x2,-------xn), the sum of the lattice at any

space must be 1. All points also must be positive at principle coordinates.

The above representation is for a 4 component mixture. At more than 4, it is extremely

difficult to visualize. Any 4 component mixture is a good space to study concrete materials

responses such as strength, stability, permeability, impact strength etc. The response function

or surface (model) is given by: 4321

,,, xxxxf

3.2 ADVANTAGES OF LATTICE

(1) It reduces the space by 1

(2) It makes the modeling simple and mathematically simpler.

A simplex is defined pp,213

as a convex polyhedron that has K+1 vertices produced by K

intersecting hyperplanes in K-dimensional space (Akhnazarova and Kafarov, 1982).

According to Akhnazarova and Kafarov (1982), when studying the properties of a q-

component mixture, which are dependent on the component ratio only, the factor space is a

regular (q-1)-simplex, and for the mixture, the relationship holds as

,11

i

q

i

x ------------------[3.1]

0i

xwhere is the component concentration, q is the number of components. For binary

system, (q=2) the simplex of dimension 1 is a straight line segment. And at q=3, the regular

2-simplex is an equilateral triangle with its inferior. Each point in the triangle corresponds to a

certain composition of the ternary system, and conversely, each composition is represented by

one distinct point. The composition may be expressed as molar, weight or volume fraction, or

percentage. Vertices of the triangle represent straight substances while sides represent binary

systems. In the concentration triangle, points lying on a straight line originating from a vertex

correspond to mixtures with a constant ratio of components represented by the other 2(two)

vertices see fig 2.

Fig 2 Fig 3.2

A2 A3 A23

A13 A12

A1

xxxiii

At q = 4,the regular simplex is a tetrahedron where each vertex represents a straight

component, an edge represents a binary system, while a face represents a ternary one. Points

inside the tetrahedron correspond to quaternary system. See Fig 3.

Fig 3.3

So, the component x1 is absent in the face x2, x3 and x4, but as tetrahedron sections parallel to

the face approach vertex x1, component x1 in them grows in concentration.

The representation of property curves of this system in a plane is not possible.

Therefore such a system is to be depicted as sections of a 3-dimensional simplex with planes

perpendicular to one of its axes. The composition of quaternary mixtures belonging to a

section plane is then described by a 2-dimensional simplex, which enables any changes in

system properties to be represented by contour lines. In doing so, only 3 components of the

section are varied.

A transition from one section to another signifies a change in the fourth component

concentration. These make it most appropriate for use in the study of concrete sandcrete

component mixtures which are also 4; namely; cement, sand water and gravel.

3.3 SIMPLE LATTICE METHOD

In designing the experiment to attack mixture problems involving composition-

property diagrams, the property studied is assumed to be a continuous function of certain

arguments and with a sufficient accuracy it can be approximated by a polynomial.

As a rule, the response surfaces in multi-component systems are very intricate (Akhnazarova

and Kafarov, 1982). To describe such surfaces adequately, high degree polynomials are

required and hence a great many experimental trials. A polynomial of degree n in q variables

has

n

nqC coefficients,

x3

x4 x2

x1

A(1,2,3,4)

xxxiv

niiinikjiijk

qkji

jiij

qji

ii

qi

xxxiibxxxbxxbxbby212

111

0,ˆ

------------------[3.2]

The relationship 11

i

q

i

x enables the qth component to be eliminated, and the number of

coefficients reduced ton

nqC

1. But the very character of the problem dictates that all the q

components be introduced into the model. H.Scheffe (Akhnazarova and Kafarov, 1982)

suggested to describe mixture properties by reduced polynomials obtainable from equation 3.2

subject to the normalization condition of Eqn. 3.1 for a sum of independent variables. Such a

reduced second-degree polynomial derived for a ternary system is demonstrated below. The

polynomial has the general form

3

333

2

222

2

1113232311321123322110ˆ xbxbxbxxbxxbxxbxbxbxbby --- [3.3]

and as 1321

xxx ------------------[3.4]

then,

0302010

bxbxbxb ------------------[3.5]

Multiplying Eqn.(3.4) by x1, x2 and x3 in succession gives

32313

2

3

32212

2

2

31211

2

1

xxxxxx

xxxxxx

xxxxxx

------------------[3.6]

Substituting Eqn 3.5 and 3.6 into Eqn. 3.3, and after necessary transformations, this is

obtained

21221112333302222011110ˆ xxbbbxbbbxbbbxbbby

3233222331131113

xxbbbxxbbb ------------------[3.7]

But we denote that

ijiiijij

bbbbbb ;11101

------------------[3.8]

Then, the reduced second-degree polynomial in three variables is arrived at; thus,

322331132112332211ˆ xxxxxxxxxy ------------------[3.9]

Thus, the number of coefficients has reduced from 10 to 6. The reduced second-degree

polynomial in q-variables is:

jiij

qji

ii

qi

xxxy11

ˆ ----------------[3.10]

and includes 122

qqCCq

xxxv

The Reduced Third-degree Polynomial for a Ternary Mixture: has the form

212112322331132112332211ˆ xxxxyxxxxxxxxxy

321123323223313113xxxxxxxyxxxxy ----------------[3.11]

and for a q-component mixture

qkji

kjiijkji

qjii

jiij

qji

jiij

qi

iixxxxxxxyxxxy

111

ˆ ----------------[3.12]

The non-linear part of these polynomials is called the SYNERGISM, if it gives a higher

response as compared with that of the linear part of the equation, and the ANTAGONISM if it

gives a lesser response. So, the term ij in the second-degree polynomial is termed the

quadratic coefficient of binary synergism of the components i and j. In the third-degree

polynomial, the synergism for a ternary system is equal to

jjkjjkkjjkkikiikkiikjijiijjiijxxxxyxxxxxxyxxxxxxyxx

kjiijk

xxx ----------------[3.13]

where the expressions in brackets are binary synergisms in the ternary system; ijk is the cubic

coefficient of the ternary synergism of the components i, j, and k.

3.4 HENRY SCHEFFE’S SIMPLEX-LATTICE DESIGNS

The most common simplex-lattice designs are produced by H.Scheffe. These designs

provide a uniform scatter of points over the (q-1) simplex. The points form a {q-1}-lattice on

the simplex, where q is the number of mixture components, n is the degree of polynomials.

Simplex Lattice Designs are Saturated

For each component, there exist. (n+1) similar levels ,1,2

,1

,0nn

xi

and all

possible combinations are derived with such values of component concentrations. So, for

instance, for the quadratic lattice (q,2) approximating the response surface with second-degree

polynomials (n = 2), the following levels of every factor must be used; 1,2

1,0 and .

For the cubic (n = 3); it is 1,3

2,

3

1,0 and . This goes on and on. Some of {3, n}-lattices are

shown in the figures below in Fig 3.4 a – b(c)

xxxvi

Having written the coordinates of points of the simplex lattice, we obtain the design

matrix. The design matrix is built for the lattices {3, 2}, {3,3} and {3,4} as may be desired.

Subscripts of the mixture property symbols indicate the relative content of each component in

the mixture. For example, the mixture Number 1 (Table 3.1) contains the component x1 alone,

(b) For a third-degree polynomial

x23

x3 x2

x12 x13

x1

(a) For a second-degree polynomial

x1

x3 x2

Fig 3.4

x123

x112 x113

x122

x2 x3

x133

x1

xxxvii

the property of this mixture is denoted by y1 while mixture N.4 includes ½x1 , ½x2 and the

property being designated as y12. The design matrix for the simplex lattice (3,2) is shown in

table 3.1 below:

Table 3.1. Design Matrix for (3,2) Lattice

N X1 x2 x3 yexp.

1 1 0 0 y1

2 0 1 0 y2

3 0 0 1 y3

4 ½ ½ 0 y12

5 ½ 0 ½ y13

6 0 ½ ½ y23

7 31 3

1 31 y123

Coefficients of these polynomials are derived using the design saturation property. To

obtain the coefficients of the polynomials

32233113211232211ˆ xxxxxxxxxy

x we substitute in succession into

the equation, the coordinates of all the six points of the design matrix (Table 3.1).

Then, substituting the coordinates of the first point (x1 = 1, x2 = 0, x3 = 0) gives

11

y ----------------[3.14]

Hence

3322

yandy ----------------[3.15]

And the substitution of the coordinates of the fourth point yields

41

1221

221

121

1221

221

112y ----------------[3.16]

But as i

y1

41

1221

221

112yyy ----------------[3.17]

Thus:

211212

224 yyy ----------------[3.18]

and 211313

224 yyy

322323

224 yyy ----------------[3.19]

The three points defining the coefficient ij lie on one edge. The coefficients of the reduced

second-degree polynomial for q-component mixture

qji

jiij

qi

iixxxy

11

ˆ are determined in a similar manner:

xxxviii

jiijijiiyyyy 224; ----------------[3.20]

3.5 TESTING THE FIT OF A SECOND-DEGREE POLYNOMIAL

After the coefficients of the regression equations have been derived, the statistical

analysis is considered necessary, that is, should be tested for goodness of fit, the equation and

response values predicted by the equation bound into the confidence intervals. In

experimentation following simplex-lattice designs there are no degrees of freedom to test the

equation for adequacy, because the designs are saturated. Thus, to test the adequacy, the

experiments are run at additional, so-called Test Points.

The number of the control points and their coordinates are conditioned by the problem

formulation and experiment nature. Besides, the control points are sought so as to improve the

model in case of inadequacy. The accuracy of response prediction is dissimilar at different

pints of the simplex. The variance of predicted response 2

yS is obtained from the error

accumulation law. To illustrate this by the second-degree of polynomial for the ternary

system, the following assumptions are observed.

1. xi can be observed without errors (Akhnazarova and Kafarof, 1982),

2. the replication variance 2

yS is similar at all the design points, and

3. response values are the averages of ni an nij replicate observations at appropriate

points of the simplex.

Then, the variances of iji

yandy ˆˆ will be

i

y

iy

n

SS

2

2 ----------------[3.21]

and

ij

y

ijy

n

SS

2

2 ----------------[3.22]

In the reduced polynomial

322331132112332211xxxxxxxxxy we replace coefficients by their

expressions in terms of responses

jiijijyi

yyyy 224,

we obtain

31311321211233221122424 xxyyyxxyyyxyxyxyy

xxxix

32212231211323223222222 xxxxxyxxxxxyxxyyy

i

32233113211232313344422 xxyxxyxxyxxxxxy ----------------[3.23]

Using the condition x1 + x2 + x3 = 1 , we obtain by transformation the coefficient at i

y

121222211111321131211

xxxxxxxxxxxxxx ---------------[3.24]

and so on.

Thus,

233213311221333222111444121212 yxxyxxyxxyxxyxxyxxy

----------------[3.25]

Introducing the designation

jiijiiixxaxxa 4,12 ----------------[3.26]

and using Eqns (3.21) and (3.22), gives the expression for the variance 2

yS

qi qji ij

ij

i

i

yy

n

a

n

aSS

1 1

22

22 ----------------[3.27]

If the number of replicate observations at all the points of the design is equal i.e. ni = nij = n,

then all the relations for 2

yS will take the form

nSS

yy

22 ----------------[3.28]

where for the second-degree polynomial

qji

ij

qi

iaa

1

2

1

2 ----------------[3.29]

is the error for predicted values of the response.

The is dependent only on the mixture composition as the Eqns 3.29.

(1) The adequacy is tested at each control points, for which purpose the following statistic

is built.

1222

yyyS

ny

SS

yt ----------------[3.30]

where theory

yyyexp

,

n is the number of parallel observations at every point.

The t statistic has the Student‟s distribution and is compared with the tabulated value

of )(/ vLt at a level of significance , where L is the number of control points and Vc is the

number of degrees of freedom for the replication variance.

xl

The null hypothesis, that the equation is adequate is acceptable if texp < ttable for all the control

points.

(2) The confidence interval for response value is

yyy ----------------[3.31]

yvS

k

t, ----------------[3.32]

where k is the number of polynomial coefficients determined. By Eqn. 3.28

2/1,

n

Sv

k

t y

----------------[3.33]

3.6 COMPONENT TRANSFORMATION

Since x1, x2 and x3 are subjected to3

1

1i

ix , the transformation of 1:6 sandcrete at

say 60% water-cement ratio cannot easily be computed because the x1, x2 and x3 are in pseudo

expressions or component ratios. To achieve this, transformation from actual component of

(0.6: 1:6.5) to pseudo components )1(

3

)1(

2

)1(

1,, xxx for the i

th experimental point, the following

computations are to be done. The arbitrary vertices chosen for the triangle are

6:1:650,5:1:50,56:1:60321

AAA for ACTUAL COMPONENTS. See the

figure below:

From table 3.1, it can be seen that the arbitrary vertices chosen can be depicted as follows: for

the pseudo component

x3 x2

x1

(0, 0, 1) (0, 1, 0)

(1, 0, 0)

Fig 3.6. Triangular Vertices for (3,2) Lattices for (Pseudo Component)

Z2 Z3

Z13

Z1

Z12

(0.6:1:6.5) (0.65:1:8)

(0.5:1:5.5)

Z23

Fig 3.5 Triangular Vertices for (3,2) Lattice. (ACTUAL COMPONENT)

xli

Transformation can be done in 2-ways

(1) Vertical Transformation and

(2) Linear Transformation.

There is a relationship between Pseudo and Actual Components i.e. x and Z.

Eg.

3332321313

3232221212

3132121111

xaxaxaz

xaxaxaz

xaxaxaz

----------------[3.34]

The linear relationship in matrix form is

3

2

1

333231

232221

131211

3

2

1

x

x

x

aaa

aaa

aaa

z

z

z

----------------[3.35]

The matrix of the relation between x and Z is as follows:

If both the pseudo (imaginary) component and the Actual Components are represented in one

triangle, we have

Fig 3.7. Triangular Vertices for (3,2) Lattices for both Pseudo and Actual Components

From fig 3.7 it can be seen that point Z1 has as follows,

5011

a and 6012

a and 65013

a

55

01

31

21

a

a

56

01

32

22

a

a

08

01

33

23

a

a ----------------[3.36]

This can be put in matrice form of eqns 3.33 as

55,1,500,0,1

,1

3

1

2

1

1

1

3

1

2

1

1

11

zzzxxx

ActualzxPseudo

8,1,6501,0,0

,3

3

3

2

3

1

3

3

3

2

3

1

33

zzzxxx

zx

56,1,600,1,0

,2

3

2

2

2

1

21

3

21

2

2

1

22

zzzxxx

zx

x1

x12

x2 x23

x3

x13 (½,½,0) (½,0,½)

(0,½,½)

xlii

100

010

001

85655

111

6506050

xZ ----------------[3.37]

If transformation matrix is denoted by T, then, the relation can be expressed as follows:

Z = Tx such that ----------------[3.38]

085655

111

6506050

21

21

3

2

1

z

z

z

----------------[3.39]

For Point Z1 0,, 21

21

12x

6085655

10111

5500650050

21

21

3

21

21

2

21

21

1

z

z

z

6,1,55012

z

For Point Z13 21

21

13,0,x

756805655

011011

575065006050

21

21

3

21

21

2

21

21

1

z

z

z

For Point Z23, 21

21

3,,0x

257856055

11101

625065060050

21

21

3

21

21

2

21

21

1

z

z

z

2577566

111

62505750550

231312zzz

Note that

Z = T X ----------------[3.38]

1ZT

T

Zx ----------------[3.40]

But

85655

111

6506050

T

xliii

(a) Determinant of TT

85655

111

6506050

T

10165052605150

15556165015581601568150

(b) Finding the Co Factors “C”

1155561

5215581

5115681

13

12

11

A

A

A

00560555650

425065055850

575065056860

23

22

21

A

A

A

100601150

1506501150

0506501160

33

32

31

A

A

A

10150050

05042505750

15251

333231

232221

131211

AAA

AAA

AAA

C

C. Transpose of C = adjT

100501

150425052

050575051

adjACT

1005001

150425052

050575051

10

11T

xliv

015010

5125425

5075515

1T

ZTx1

----------------[3.41]

To Find 6,1,550@32112

zzzx

Since ZTx1

6

1

550

015010

5125425

5075515

x

060115055010

50651125455025

50650175555015

3

2

1

x

x

x

To Find 756,1,5750@32113

zzzx

ZTx1

756

1

5750

015010

5125425

5075515

x

507561150575010

0756511254575025

50756501755575015

3

2

1

x

x

x

57,1,6250@3223

zzzx

257

1

6250

015010

5125425

5075515

x

502571150625010

50257511254625025

0257501755625015

3

2

1

x

x

x

The value for 0,50,5012

x

,, ,, 50,0,5013

x

,, ,, 5050,023

x

at their Z12, Z13 and Z23 points which is correct for the pseudo values.

xlv

Table 3.2: Design Matrix for A(3,2) Lattice

S/N MIX RATIOS

ACTUAL COORDINATES

STRENGTH MIX PROPORTIONS

PSUEDO COORDINATES

Z1 Z2 Z3 N/mm2

x1 x2 x3

1 0.5 1 5.5 y1 1 0 0

2 0.6 1 6.5 y2 0 1 0

3 0.65 1 8.0 y3 0 0 1

4 0.55 1 6.0 y4 ½ ½ 0

5 0.575 1 6.75 y5 ½ 0 ½

6 0.625 1 7.25 y6 0 ½ ½

7 0.583 1 6.667 y7 1/3 1/3 1/3

CONTROL POINTS

8 0.533 1 5.833 y8 2/3 1/3 0

9 0.55 1 6.333 y9 2/3 0 1/3

10 0.617 1 7.0 y10 0 2/3 1/3

11 0.5625 1 6.375 y11 ½ ¼ ¼

12 0.5875 1 6.625 y12 ¼ ½ ¼

3.7 USE OF THE VALUES IN EXPERIMENT

During the laboratory experiment, the actual components were used to measure out the

appropriate proportions of the ingredients, water, cement, and fine aggregate (sand) for

moulding the sandcrete blocks and the cubes.

xlvi

CHAPTER FOUR

EXPERIMENTAL METHODOLOGY

4.0 CONSTITUENT MATERIAL

To be a good structural material, the material should be homogeneous, isotropic, and

elastic. Portland cement concrete or sandcrete is none of these. It is nonetheless, a very

popular construction material (Wilby, 1983). The necessary materials required in the

manufacture of sandcrete blocks are a hydraulic binder, water and fine aggregate (sand).

4.1 CEMENT QUALITY

The hydraulic binder in this project is an Ordinary Portland cement. DANGOTE

cement fleshly delivered from factory was used and complied with the Standard Institute of

Nigeria (NIS) 1974, kept in an air-tight bag.

4.2 SAND, OR FINE AGGREGATE

The sand used was collected from Abanyi River at Orba, Nsukka zonal area, on 11-1-

2001. The sand was spread to dry for about eight months and sieved to remove debris and

gravel particle. It was recombined to achieve the desired grading which was in zone 3, used

for the experiment. The size are, 2.36, 1.18, 0.6, 0.3 and 1.15mm.

4.3 WATER

The water used was pure drinking water which was free from any contamination

4.4 MANUFACTURE OF SANDCRETE BLOCKS AND CUBES:

The production process involved collection of sand which was left to dry. Cement was

added to sand and mixed in dry conditions and the required quantity of water was added into

each batch. Each of these constituent materials were weighed as shown in table 4.4 before

mixing. The floor surface was cleaned, wetted and dried to prevent loss of the water cement

ratio and prevent excess water being added into the mix. The mixed constituent materials were

compacted into a wooden mould, followed immediately by extrusion of the pressed block so

that the mould could be used repeatedly for other batches.

The produced block or cube was self-supporting and able to withstand any movement

and vibration from movement of extrusion exercise very much drier, and higher fine sand

aggregate content and leaner mixes were used than in normal concrete work to be able to

achieve demoulding for immediate continuity.

xlvii

4.5 FORM AND SIZE

There are 3 basic forms of sandcrete blocks; solid, cellular, and hollow, and within

each type a variety of products are available thus providing versatility to block work

construction both in style and function. In this project work, the Hollow Block of 49% void

was used as a case study, which has two formed holes or cavities. Concrete blocks are

commonly referred to as common, facing and special blocks. Common blocks which have an

open texture, are for general use for both load-bearing and non load-bearing walling.

BS 6073: part 1 defines a block as a masonry unit of larger size in al dimensions than

specified for bricks, but no dimension should exceed 650mm nor should the height (in its

normal aspect) exceed either its length or six times its thickness. There are now no specific

limits on formed voidage in relation to overall block volume, except that the external shell

wall thickness should not be less than 15mm or 1.75 times the nominal size of the aggregate

used, whichever is the greater. In this project experiment, 47mm shell wall thickness has been

used to achieve the 40% void.

4.6 PROPERTIES

The properties of sandcrete blocks depend to a varying degree on the type and

proportion of the Constituent materials, the Manufacturing process, and the Mode and

Duration of Curing employed, as well as on the Form and Size of the block itself.

4.7 DENSITY

The density of sandcrete/concrete blocks is largely a function of the aggregate density,

size and grading, degree of compaction or aeration and the block form. The typical range for

dry density is 500 to 2100Kgm-3

with aerated and solid dense aggregate concrete blocks being

on the lighter and heavier end of the scale respectively and light weight and dense aggregate

concrete blocks of cellular and hollow form falling in the middle of the range or sole.

In this experiment the average of

(1) Block: Hollow block of 225 x 225 x 450mm has wt = 25Kg;

Volume = 0.225 x 0.062500 = 0.0140625m3

3/1778

0140625.0

25mKgDensity

(2) Cube 150 x 150 x 150mm wt = 6.8Kg.

Volume = 0.003375m3

3/8152014

0033750

86mKgDensity

xlviii

4.8 STRENGTH OF SANDCRETE BLOCK

In addition to size, compressive strength is the basic requirement of sandcrete blocks,

except for non-load bearing blocks with a thickness less than 75mm which are required to

comply with the transverse breaking strength test for handling. The compressive strength of

concrete blocks is dependent mainly on:

1. Mix proportions or composition in particular binder content.

2. Degree of compaction or aeration and to a less extent

3. On the aggregate type.

4. On curing mode duration normally used.

The strength of sandcrete blocks increases with its density generally. The range specified by

BS6073: part2, is 2.8 to 35N/mm2. But from consideration of cost, the more normal practical

upper limit is about 20N/mm2 and the most commonly used blocks fall within a smaller

strength band of 3.5 to 10N/mm2.

Ezeokonkwo(14)

enumerated other factors affecting those strength of concrete

/sandcrete blocks. They are

i. Sand particle size

ii. Grading and age

iii. Block geometry

iv. Water-cement ratio

v. Sand-cement ratio

vi. Capping

vii. Workmanship

viii. Testing technique and

ix. Rate of loading.

4.9 SPECIMENS SIZES AND SHAPES

The test specimens according to BS1881, parts 1 -5, 1970 part 6 197BSI, must be a

150mm cube except that the maximum aggregate size does not exceed 25mm, then a 100mm

cube may be used. In this experiment, the 150mm cube was used, also the 225mm x 225mm x

450mm for hollow block of 40% void was used. I constructed a wooden mould to suite the

shape and size of the block as herein specified and calculated, for the hollow block mould,

while the metal moulds for cubes were taken from the laboratory stores.

xlix

4.10 MOULD SIZE AND 40% VOID COMPUTATION

DATA

2t + x = 225mm

3t + 2y = 450mm

Cross Sectional Area = 450 x 225mm

40% of C.S.A = (x x y) x 2 2xy = 40% C.S.A

225 x 450 = (2t+x) x (3t + 2y)

Hence, 2t + x = 225 (1)

3t + 2y = 450 (2)

(2t + x)(3t + 2y) = 225 x 450 (3)

225 x 450 = [2t + (225 – 2t)] [ 3t + 2y ]

2

345051225

51225

450450675101250

23225450225

2

tORty

yt

ytmm

yt

40500512254450

..%404050051225222522

tt

ASCttxy

101250 – 675t – 900t + 6t2 = 40500

6t2 – 1575t + 60,750 = 0 = by6

t2 – 262.5t + 10125 = 0

Void Void 22

5m

m

22

5m

m

t

x

t

Fig 4.1

t y t y t

450mm

l

2

101254526252622

t

t = 46.979mm say 47mm

225 – 2t = 225 – 94mm x = 131mm

tt

y 512252

3450

ymm5154570225

4751225

t = 47mm; x = 131mm; y = 154.5mm

BLOCK MOULD SIZE IS AS FOLLOWS

4.11 CALCULATIONS FOR MATERIALS VOLUMES & WEIGHT

From the mix ratios given in table 4.1 below

S1 = water

S2 = cement

S3 = sand

The mix proportions have been computed thus

X1 + X2 + X3 = 1 as exemplified in the one calculations below.

Example 1:

For S/N 1 in the table 4.1

S1 = 0.5; S2 = 1.0 and S3 = 5.5

The mix proportion is calculated thus

0.5 + 1. + 5.5 = 7

071407

501

x

450mm

225

mm

47

t y t y t

t

x

t 47

131

47

154.5 47 47 154.5

Fig 4.2

li

142907

12

x

785707

553

x

1321

xxx

;01785701429007140 OK

Example 2:

From RCDH. Book

(1) A hollow 200mm concrete block weighs from 18.16Kg to 22.7Kg.

Therefore a 225mm would proportionately weigh

.45022522554251

7.22

200

225mmofblockHollowperKg

(2) From plain concrete of weight 2306.8Kg/m3, a cube of 150mm would weigh

Kgm

78571

82306

1

0033750150

3

3

Since each batch would give or manufacture 2 blocks and 2cubes for the experiment, the 2

values will be multiplied by 2

Hence

25.54Kg + 7.79Kg = 33.33Kg

33.33Kg x 2 = 66.66Kg

32% is added to take care of wastes and slump.

66.66 x 1.32 = 87.993Kg say 88Kg

Base on the ratios of constituent materials, their weights were calculated from 88Kg, see table

4.2

Eg from S/N 1. The ratio 0.5 : 1 : 5.5 gives sum as follows for the hollow Block

0.5 + 1 + 5.5 = 7.0

waterofKg2965057127

88

cementofKg5712157127

88

KgTOTAL

sandofKg

88

14695557127

88

The computation above was repeated for all the values as shown for each batch of the 12 with

the constituent materials ranging as show and as given. The blocks and cubes were moulded

lii

and demoulded immediately their shapes were obtained to stand on themselves. There are

altogether 24 – 225 x 225 x 150 Hollow blocks and 24 – 150mm cubes, all moulded within

3days.

Serial Numbers 1 – 3 were moulded on 1st day

,, ,, 4 – 9 ,, ,, ,, 2nd day

,, ,, 10 – 12 ,, ,, ,, 3rd day

Table 4.1

Table 4.2

Table 4.1: Mix Ratios/Proportions

MIX RATIOS RESPONSE MIX PROPORTION

S/N S1 S2 S3 Y(N/mm2) X1 X2 X3

Water Cement Sand

1 0.5 1 5.5 0.0714 0.1429 0.7857

2 0.6 1 6.5 0.0741 0.1234 0.8025

3 0.65 1 8.0 0.0674 0.1036 0.8290

4 0.55 1 6.0 0.0728 0.1325 0.7947

5 0.575 1 6.75 0.0691 0.1201 0.8108

6 0.625 1 7.25 0.0704 0.1127 0.8169

7 0.583 1 6.667 0.0707 0.1212 0.8081

CONTROL

8 0.533 1 5.833 0.0724 0.1357 0.7919

9 0.55 1 6.333 0.0698 0.1268 0.8034

10 0.617 1 7.0 0.0716 0.1161 0.8123

11 0.5625 1 6.375 0.0709 0.1260 0.8031

12 0.5875 1 6.625 0.0715 0.1218 0.8067

liii

Table 4.2: Mix Ratios and Weights of Component Mix for the Experiment

S/N Ratios Yi Sand (Kg) Cement (Kg). Water (Kg)

1 0.5 : 1 : 5.5 y1 69.1 12.6 6.3

2 0.6 : 1 : 6.5 y2 70.6 10.9 6.5

3 0.65 : 1 : 8.0 y3 73.0 9.1 5.9

4 0.55 : 1 : 6 y12 70 11.6 6.4

5 0.575 : 1 : 6.75 y23 71.3 10.6 6.1

6 0.625 : 1 : 7.25 y23 72 9.8 6.2

7 0.583 : 1 : 6.667 71.11 10.67 6.22

8 0.533 : 1 : 5.833 C1 69.68 11.95 6.37

9 0.55 : 1 : 6.333 C2 70.7 11.16 6.14

10 0.617 : 1 : 7 C3 71.49 10.21 6.30

11 0.5625 : 1 : 6.375 C12 70.677 11.087 6.236

12 0.5875 : 1 : 6.625 C13 70.990 10.715 6.295

4.12 MIX PROPORTIONS

Yilla (1967) revealed that the average value of water cement ratio in Nigeria block

industries is between 0.6 to 0.7 while the sand-cement ratio is 8.0. The values used in the work

are as follows: 0.5, 0.6, 0.65, 0.55, 0.575, 0.625, 0.583, 0.533, 0.55, 0.617, 0.5625, 0.5875 as

the water-cement ratios, (w/c ratios). The values for the sand-cement ratios (S/C ratios) are as

follows: 5.5, 6.5, 8.0, 6.0, 6.75, 7.25, 6.667, 5.833, 6.333, 7.0, 6.375, 6.625. The last 7 values

in each case were to serve as control points as depicted in the tables 4.1 and 4.2. A total of 12

batches were produced and each batch had 2specimen blocks and 2specimen cubes.

liv

4.13 BATCHING AND MIXING

The sand was sieved to eliminate the unwanted materials such as silt dust dry leaves

and coarse aggregates that were retained inBSNo.7 sieve or bigger than 2.36mm. This practice

is not done or observed in the block industries in the cities or towns. Instead they use All-In

aggregate or Pit-Run aggregate which in turn reduce the strength of the hollow block.

4.14 COMPACTION OF THE SPECIMEN SANDCRETE BLOCKS AND CUBES

Two test specimens each of blocks and cubes were made from each of the 12 different

batches and each result was the average of the two similar specimen from same batch. In the

moulding of the specimens, compaction was hand. 35 strokes, of the compaction rod, were

used for the 150 x 150 x 150 cubes, before demoulding and replacement. This was done for

the 12batches. Compaction of the blocks was by hand also using a wooden rod also.

After each demoulding, the moulds were cleaned and oiled to get set for the next

batch. Due to the frame work of the wooden-mould, there were variations in the number of

strokes. One major constraints in compaction by hand is the variability in human hand

compaction.

4.15 CURING

After casting, compaction, the specimens were demoulded as soon as they could sand

on their shape. The specimens were cured under laboratory condition. In other words, they

were left inside the laboratory space. These specimens were wetted every morning for 7days

and then left to harden until the 28th day for the compressive strength test. The specimens

were moulded on the 5th, 6th, and 7th days of August 2002 respectively. They were also

tested on the 28day of their moulds/manufacture.

4.16 THE COMPRESSIVE STRENGTH TESTING OF SANDCRETE BLOCK

AND SANDCRETE CUBES ON 28TH

DAY

The test used was the compressive test on the specimens. Utmost care was observed

and the conditions were rigidly controlled so that comparative results were obtained

irrespective of the machine that was used for the test to have a meaningful result. the machine

used was the DENISON Hardened Concrete Testing Machine in the Civil Engineering

Workshop/in the Concrete Technology Laboratory, U.N.N. Although the principal factors

which may cause variations of the test results are (Orchard, 1973);

lv

(1) Eccentricity of loading due to a misalignment of various parts of the testing

machine.

(2) Tilting of the platens due to a lack of lateral rigidity of the structure of the testing

machine.

(3) The inability to maintain a uniform rate of loading right up to the point at which

the cube or block fails.

(4) A lack of the planeness of the platens.

(5) A variable amount of friction in the spherical seating of a self-aligning to the

platen.

(6) As the specimen block or cube nears the failure point in the compression test, its

rate of yield increased considerably and this means that the movement of the

platens of the testing machine must be speeded up in order to maintain a constant

rate of application of load. The B.S. 1881 specifies that the load shall be applied

continuously without shock at a rate of 4MPa per minute until the cube or block

fails. It is extremely difficult to ensure this specification right up to the failure

point with any machine.

(7) The human errors in operating the machines and loading and also reading the

results are great factor militating on the accuracy of the results, most of these

known factors were controlled to obtain a meaningful and acceptable results.

4.17 OBSERVATIONS AND CONCLUSION

The specimens were subjected to the crushing machine to obtain the compressive

strength. The readings were in tons and converted to N/mm2 by use of the formular below:

AreaSectionalCrossNet

LoadMaximum

mm

N 10009649

2

The results are as tabulated below for the 12 batches in table 4.4. The weights of the

specimen blocks and cubes were also recorded as shown in the same table 4.4.

4.18 CROSS SECTIONAL AREA OF THE SPECIMEN HOLLOW BLOCK

The cross sectional area of the specimen block is as follows:-

(1) The theoretical calculation as taken from section 4.10

22

4047925202391315154 mmmmAREAVOID

2

101250450225

.....

mm

ASCVOIDASCTOTALAREASOLID

lvi

2

6077140479101250.. mmASCNet

(2) The Practical Moulded Specimen C.S.A

Four numbers of the specimen hollow blocks were practically measured as shown

below and the average of the cross sectional areas was taken.

(1) (2)

Block Gross Section at Area Void Area Net Area

mm2 Mm

2

mm2

BLOCK 1 104420 41580 62840

,, 2 103966 41950 62016

,, 3 103737 42400 61337

,, 4 105782 41976 63806

417905 167906 249999

AreaVoidAreaGross

AreaVoid%17840401780

417905

167906

4

1

4

1 OK.

Average C.S.A = 22

006250075624994

249999mmmm

CONVERSION OF TONS TO N/mm2

AreaSectionalCrossNet

Load

mm

N 10009649

2

13

2

13

2

23

0

454

158

13

1 161

22

9

454

158

(1) (2)

13

2

160

22

9

453

160

13

3

13

2

13

2

454

155 163

(3) (4) Fig 4.3

lvii

FOR BLOCK = 22

1594062500

9964

mm

NL

mm

LOAD

FOR CUBE 22

22500150 mmmm

22442840

22500

9964

mm

NL

L

mm

N

Table 4.4: CRUSHING STRENGTH TEST-RESULT

Block wt

(Kg)

Average

(Kg)

Crushing

Load

(Tons)

N/mm2

CUBE

wt

(Kg).

Average Crushing

Load

(Ton)

N/mm2

1

a

25.8

26

22

3.5072

6.9

6.95

22.5

9.964 b 26.2 7.0

2

a

24.8

25

2.232

6.8

6.8

18

6.95 b 25.2 6.8

6.8

18

7.45

3

a

25.2

25.1

17

2.00

6.8

b 25.0 6.8

4

a

25.5

25.8

19

3.216

7.1

6.85

20.5

9.10 b 26.1 6.6

5

a

28.8

24.8

14.5

2.578

6.7

6.55

13

7.45 b 23.8 6.4

6

a

25.8

7

lviii

b 24.7 25.3 15 2.195 6.5 6.75 21 6.5

7

a

25.4

25.4

21.5

2.750

7

6.8

27

8.23 b 25.3 6.6

CONTROL

POINT

8

a

26.2

26

20.5

3.394

7.1

6.95

28.4

10.5 b 25.8 6.8

9

a

25.4

25.7

21

2.391

6.8

6.85

21.5

9.521 b 25.9 6.9

10

a

25.6

25.8

23

2.393

6.7

6.6

20.5

11.15 b 26 6.5

11

a

27.1

26.65

22

2.865

6.7

6.4

19.5

9.05 b 26.2 6.1

12

a

26.6

26.8

22.5

2.709

6.7

6.8

23

10.185 b 27.0 6.9

Table 4.5. COMPRESSIVE STRENGTH OF SPECIMEN HOLLOW SANDCRETE

BLOCK WITH 40% VOID

Dimension Cross

Section

Block

wt.

Crushing Strength

S/N Length Breat

h

Height

Surface

Area mm2

Kg TON N/mm2

Average

Strength

1 450 225 225 62500 26 22 3.5073

lix

2 ,, ,, ,, ,, 25 14 2.232

3 ,, ,, ,, ,, 25.1 12.5 2.00

4 ,, ,, ,, ,, 25.8 20.17 3.216

5 ,, ,, ,, ,, 24.8 16.1 2.578

6 ,, ,, ,, ,, 25.3 13.77 2.195

7 ,, ,, ,, ,, 25.4 17.25 2.75

8 ,, ,, ,, ,, 26.0 21.29 3.394

9 ,, ,, ,, ,, 25.70 17.79 2.391

10 ,, ,, ,, ,, 25.75 15.00 2.393

11 ,, ,, ,, ,, 26.65 17.97 2.865

12 ,, ,, ,, ,, 26.8 16.99 2.709

Table 4.6. COMPRESSIVE STRENGTH OF THE 150 x 150 x 150 mm2 SPECIMEN

CONCRETE CUBES

S/N Size Surface

Area mm1

Weight of

Cube

Crushing load Compressive

strength

TON N/mm2

1 150 x 150 x

150

22500 6.95 22.5 9.964

2 ,, ,, 6.80 15.7 6.952

3 ,, ,, 6.80 16.82 7.45

4 ,, ,, 6.85 20.55 9.10

5 ,, ,, 6.55 16.82 7.45

6 ,, ,, 6.75 14.68 6.50

7 ,, ,, 6.80 18.58 8.23

8 ,, ,, 6.95 23.71 10.500

9 ,, ,, 6.85 21.50 9.5212

10 ,, ,, 6.60 25.2 11.18

11 ,, ,, 6.40 20.44 9.05

12 ,, ,, 6.80 23.00 10.1854

442840

/2

mmNLLoadEquivalent

CubeL

AreaSectionalCrossNet

Load442840

10009649

lx

HOLLOW BLOCK STRENGTH VS S/C RATIO

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3 3.5 4

Compressive Strength of Hoolow Blocks (N/mm2)

S/C

Rati

o

S/C Ration

lxi

Cube Strength vs S/C Ratio

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12

Compressive Strength of Hoolow Blocks (N/mm2)

S/C

Rati

o

S/C Ratio

Block Strenght W/C Ratio

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5 2 2.5 3 3.5 4

Compressive Strength of Hoolow Blocks (N/mm2)

W/C

Rati

o

W/C Ratio

lxii

Cube Strength W/C Ratio

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12

Compressive Strength of Hoolow Blocks (N/mm2)

W/C

Rati

o

W/C Ratio

Cube Strength Hollow Block Strength

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12

CUBE COMPRESSIVE STRENGTH (N/mm2)

BL

OC

K C

OM

PR

ES

SIV

E S

TR

EN

GT

H (

N/m

m2)

Hollow Block Strength

lxiii

Table 4.7. DESIGN MATRIX FOR (3,2) SIMPLE LATTICE FOR BLOCK

S/N

Coord

inat

e poin

t

PESUDO

CORDINATES

RESPONSE (N/mm2)

BLOCK

CUBE

X1 X2 X3 Z1 Z2 Z3

1 A1 1 0 0 Y1 = 3.5072 9.964 0.5 1 5.5

2 A2 0 1 0 Y2 = 2.232 6.95 0.6 1 6.5

3 A3 0 0 1 Y3 = 2.000 7.45 0.65 1 8.0

4 A12 ½ 0 ½ Y12 = 3.2160 9.10 0.55 1 6.0

5 A13 ½ 0 ½ Y13 = 2.5780 7.45 0.575 1 6.75

lxiv

6 A23 0 ½ ½ Y23 = 2.195 6.5 0.625 1 7.25

7 A123 1/3 1/3 1/3 Y123 = 2.750 8.23 0.583 1 6.667

8 C1 2/3 1/3 0 C1 = 3.394 10.5 0.533 1 5.833

9 C2 2/3 0 1/3 C2 = 2.391 9.5212 0.55 1 6.333

10 C3 0 2/3 1/3 C3 = 2.393 11.15 0.617 1 7.0

11 C4 ½ ¼ ¼ C4 = 2.865 9.05 0.5625 1 6.375

12 C5 ¼ ½ ¼ C5 = 2.709 10.1854 0.5875 1 6.625

2313123211952563221539951986250733 XXXXXXy

Black

3,2,1, iyii

, -------,n

jiijij

yyy 224

211212

224 yyy

3232331313

24,24 yyyy

lxv

CHAPTER FIVE

RESULTS AND ANALYSIS

5.1: RESULTS

The adequacy of the results can be tested and analyzed using the “STUDENT‟S” t

DISTRIBUTION29

since this experiment is of a small sample size.

With a small sample size, the confidence interval for the expected value is constructed by

having recourse to STUDENT‟S t DISTRIBUTION, or simply the t distribution

(Akhnazarova and Kafarov, 1982). The Student‟s t distribution holds for a random variable or

test statistic.

/

xt

S n [5.1]

Its probability density function has the form

1

2 2

1

1 2( ) 1

2

vv

tt

v vv

- t + [5.2]

Where (v) = gama function

V = number of degree of freedom of the sample

If S2 and x are derived from the sample,

Then v = n – 1

It is seen that the Student‟s distribution depends on the number of degrees of freedom v, with

which the sample variance has been determined.

Hence the t-distribution is employed. When xi from table 5.1 is substituted into the equation

5.3 or 3.2b

aij = xi(2xi – 1)

aij = 4xi xj [5.3]

The theoretical positions of the response are obtained which are comparable with

experimental results in table 5.1

lxvi

Table 5.1: Control Point Values for (3,2) Lattice

Yexperimental Ypredicted Yexp – Ypred. X1 X2 X3

7 2.75 2.691 0.059 1/3 1/3 1/3

8 3.394 3.390 0.004 2/3 1/3 0

9 2.391 2.849 -0.458 2/3 0 1/3

10 2.393 2.225 0.168 0 2/3 1/3

11 2.865 2.917 -0.052 1/2 1/4 1/4

12 2.709 2.662 0.047 1/4 1/2 1/4

Therefore using control point No 7 as an example

Xi = 1/3 ; Xj = 1/3

ai = xi(2 xi – 1)

aij =4xixj

(1) at a12 ; xi = 1/3 , xj = 1/3 ; 13

12

3

1i

a

13

2

3

1i

a

3

1

3

1i

a

=-1/9 = -0.1111

And aij = 4xixj

3/13/14ij

a = 4/9 = 0.4444

(2) at a13 ; xi = 1/3 , xj = 1/3

ai = 0.1111

aij = 0.4444

(3) at a23 ; xi = 1/3 , xj = 1/3

ai = 0.1111

aij = 0.4444

From equation 5.4 which states that

lxvii

v

xx

n

xx

S

n

i i

n

i i 1

2

1

2

2

1

S2 = 0.137 37.0137.0S

694.01137.0

5059.0

12

y

cal

S

nyt

7541.01749.0

1319.0cal

t

7541.0cal

t << ttab = 2.57

The same procedure is used for C7 to C12 and the tables show the results as in table 5.2

ttab is shown in the appendix table of Student‟s t distribution.

Table 5.2: t – Statistic for Control Points, Compressive Strength Based on Scheffe’s (3,2)

Polynomial

N CN i j ai aij ai2

aij2

ai2 –

aij2

Yexp Ypred ΔY= (Yexp

- Ypred)

tcal ttab

1/3 1/3 -0.1111 0.4444 0.0123 0.1975 0.2098 2.75 2.691 0.0590

7 C7 1/3 1/3 0.1111 0.4444 0.0123 0.1975 0.2098 2.75 0.0590 0.7541 2.57

1/3 1/3 0.1111 0.4444 0.0123 0.1975 0.2098 2.75 0.0590

Σ 0.0369 0.5925 0.6294

2/3 1/3 0.2222 0.8889 0.0494 0.7901 0.8395

8 C8 2/3 0 0.2222 0.0000 0.0494 0.0000 0.0494 3.394 3.390 0.004 0.0474

1/3 0 -0.1111 0.0000 0.0123 0.0000 0.0123

Σ 0.1111 0.7901 0.9012

2/3 0 0.2222 0.0000 0.0494 0.0000 0.0494

9 C9 2/3 1/3 0.2222 0.4444 0.0494 0.1975 0.2469 2.391 2.849 -0.485 -6.5647

0 1/3 0.0000 0.0000 0.0000 0.0000 0.0000

Σ 0.0988 0.1975 0.2963

0 1/3 0.0000 0.0000 0.0000 0.0000 0.0000

10 C10 0 1/3 0.0000 0.0000 0.0000 0.0000 0.0000 2.393 2.225 0.168 2.025

2/3 1/3 0.2222 0.8889 0.0494 0.7901 0.8395

lxviii

Σ 0.0494 0.7901 0.8395

1/2 1/4 0.0000 0.5000 0.0000 0.2500 0.2500

11 C11 1/2 1/4 0.0000 0.5000 0.0000 0.2500 0.2500 2.865 2.917 -0.052 -0.6515

1/4 1/4 -0.375 0.2500 0.1406 0.0625 0.2031

Σ 0.1406 0.5625 0.7031

1 2 -0.375 0.5000 0.1406 0.2500 0.3906

12 C12 1 3 -0.375 0.2500 0.1406 0.0625 0.2031 2.709 2.662 0.047 0.5660

2 3 0.0000 0.5000 0.0000 0.2500 0.2500

Σ 0.2812 0.5625 0.8437

5.2 PREDICTION OF Yi FOR BLOCKS BY Regression Equation

In the absence of blunders and systematic errors, the Expected value of a random

variable coincides with the true results of observations experiments .

Therefore, the estimate of the MEAN is important in processing observation data or

experimental data(29)

. From Equation 3.18 to 3.20 represented below:

322331132112332211xxxxxxxxxy

i ------------------[5.9]

I = yi ; i = 1, 2, 3

3,2,1,224 jiyyyiiijij

----------------[5.10]

From Table 4.7, the values of Block specimen for yi is as follows

002

2322

50723

3

2

1

y

y

y

1952

5782

2163

23

13

12

y

y

y

385612322250723221634

224211212

yyy

702402250723257824

224311313

yyy

3160222322219524

224322323

yyy

3160;70240;38561231312

bbb

lxix

5.2.1 Ypredicted FOR BLOCK

3123322331122112332211xxbxxbxxbxxbxbxbxby

Eqn. 3.9

3160

70240

38561

1952

5782

2163

002

2322

50723

23

13

12

23

13

12

3

2

1

b

b

b

y

y

y

y

y

y

2221213213160702403856102232250723 xxxxxxxxxy

i ------[5.11]

3

1

3

13160

3

1

3

170240

3

1

3

138561

3

12

3

13232

3

150723

7y

6912035007801540667074401691

393307907440

3382702403

1

3

238561

3

12232

3

250723

8y

84921560667033823

1

3

270240

32

3

250723

9y

225207020667048813

1

3

23160

3

12

3

22322

10y

917201980087801732050558075361

4

1

4

13160

4

1

2

1702100

4

1

2

13861

4

12

4

12322

2

150723

11y

66220395004390173050116187704

1

2

1

2

13160

2

1

4

170240

2

1

4

138501

4

12

2

12322

4

150723

12y

lxx

Table 5.3: Statistic for Control Points, Compressive Strength Based on Scheffe’s (3,2)

Polynomial

YEXPER YPREDICTED Yexp - Ypred 2

exp prYY

predic

pred

Y

YYexp

2

7 2.75 2.691 0.059 0.003481 0.0012935

8 3.394 3.390 0.004 0.000016 0.000004720

9 2.391 2.849 -0.458 0.209764 0.0736272

10 2.393 2.225 0.168 0.028224 0.0126849

11 2.865 2.917 -0.052 0.002704 0.00092698

12 2.709 2.662 0.047 0.002209 0.008290

0.0885

08850

2

exp

pred

pred

Y

YY

Hence Equation 5.11 is the mathematical model for the sandcrete Hollow Blocks ie.

3231213213160702403856102232250723 xxxxxxxxxY

B

5.3 ADEQUACY OF EXPERIMENT

Since the number of the control points and their coordinates are conditioned by the

problem formulation and experiment nature, the Adequacy of the Experiments are checked

and run at these additional points. The points are No. 7 to 12 as show in the table above, Table

5.3

Taking values from table 4.7 and from 7 – 12 we have for Block Specimen values of Yi as

follows:

lxxi

Table 5.4. Hollow Block Values for Control Points

x x xx 2

xx

7 2.75 2.75 0.00 0.00

8 3.394 ,, 0.644 0.41474

9 2.391 ,, -0.359 0.1288

10 2.393 ,, -0.357 0.12745

11 2.865 ,, 0.115 0.01323

12 2.709 ,, -0.041 0.00168

x = 16.502 = 0.68598

7526

50216x

n

xx

n

i 1 1

On the basis of 6 replicate control points i.e. Nos 7 to 12 of the table, the Average x

2

1752

6

50216

mm

N

n

xx

n

i i

The Error mean square or Replication Variance S2 is in Eqn. 5.4

v

xx

n

xxS

n

i

n

i 1

2

1

2

2

1

13705

683980

16

6839802S

3701370S

5.4 USE OF t-DISTRIBUTION

The Student‟s t Distribution is also used to ascertain the Confidence Interval for the

expected value due to the small sample size. Introducing t-distribution which is symmetrical

about t = 0, in view of this symmetry, resort is often made to the designation t , v, where v is

lxxii

the number of degree of freedom and is the probability that t lies outside the interval

2/1

2

, tt

The student‟s distribution depends only on the number of degrees of freedom, v, with whch

the sample variance has been determined.

Hence, introducing the formulation

2/12/1t

n

Sxt

n

Sx ----------------[5.12]

whereby on rearrangement, leads to the following one-sided confidence estimates of the mean

(1) the Upper Estimate:

1

tn

Sx ----------------[5.13]

(2) the Lower Estimate:

1

tn

Sx ----------------[5.14]

On the basis of 6 replicate determination the average is 752x

370S

Number of degree of freedom is 5

1 - = 0.9 and at V = 5 for = 0.05 gives t0. 95 = 2.57.

The Upper and Lower state limits are

(1) Upper State Limit

1

tn

Sx

1383

3880572

5721510752

5726

570752

(2) Lower State Limit

3622

3880752

5721510752

lxxiii

Table 5.5: 611

Cubes (150 x 150 x 150mm) Mix Ratios, Compressive Strength and Mix

Proportions

MIX RATIO

STRENGHT

(Yi)

MIX PROPROTION

S/N Z1 Z2 Z3 N/mm2

X1 X2 X3

1 0.5 1 5.5 Y1 = 9.964 1 0 0

2 0.6 1 6.5 Y2 = 6.95 0 1 0

3 0.65 1 8.0 Y3 = 7.45 0 0 1

4 0.55 1 6.0 Y12 = 9.10 ½ ½ 0

5 0.575 1 6.75 Y13 = 7.45 ½ 0 ½

6 0.583 1 7.25 Y23 = 6.50 0 ½ ½

7 0.583 1 6.667 8.23 1/3 1/3 1/3

8 0.533 1 5.833 10.5 2/3 1/3 0

9 0.55 1 6.333 9.5212 2/3 0 1/3

10 0.617 1 7.0 11.15 0 2/3 1/3

11 0.5625 1 6.375 9.05 ½ ¼ ¼

12 0.5875 1 6.625 10.1854 ¼ ½ ¼

323

3212

3211

085655

0

650600

xxxZ

xxxZ

xxxZ

lxxiv

5.5 Prediction of Yi for Cubes by Regression Equation

ij for Cubes 611

(150 x 150 x 150)

1. b1 = y1 = 9.964 4. y12 = 9.10

2. b2 = y2 = 6.95 5. y13 = 7.45

3. b3 = y3 = 7.43 6. y23 = 6.50

322331132112332211xxxxxxxxxy

C

3,2,1; iyii

3,2,1,224 jiyyyjiijij

57229562964921094

224;21212

yyybfori

02854572964924574

224;311313

yyyb

90245729562564

224;322323

yyyb

902;0285;5722231312

5.6 CUBE – Y-Predicted For Cubes

322331132112332211xxbxxbxxbxbxbxbY

C

902

0285

5722

23

13

12

b

b

b

32

121321

902

3028557224579569649

xx

xxxxxxxYi

-----------[5.15]

52627

3222055870285804833231672321333

1

3

1902

3

1

3

150283

3

1

3

15722

3

1457

3

1956

3

19649

7

7

Y

Yfor

5319

0057160031672642763

1

3

25722

3

1956

3

29649

8

8

Y

Yfor

008781173148332642763

1457

3

29649

9Yfor

lxxv

4722.66444048332633343

1

3

292

3

1457

3

2956

10Yfor

09378181306285032150862517375198244

1

4

192

4

1

2

10255

4

1

2

15722

4

1457

4

1956

2

19649

11Yfor

4752736250314303215086251475349124

1

2

192

4

1

4

10285

2

1

4

15722

4

1457

2

1956

4

19649

12Yfor

Table 5.6: Cube Values for Control Points

S/N Y-Experimental Y-Predicted Yexp – Ypredicted 2

exp predYY

pred

pred

Y

YY2

exp

7 8.23 7.5262 0.7038 0.4953 0.06581

8 10.50 9.531 0.969 0.9390 0.09852

9 9.5212 8.0087 1.5125 2.287 0.28565

10 11.150 6.4722 4.6778 2.8818 3.37830

11 9.050 8.0937 0.9563 0.9145 0.1130

12 10.1854 7.4732 2.7122 7.3560 0.98432

= 4.9256

Therefore,

92564

2

exp

pred

pred

y

yy

Hence, Eqn. 5.15 is the mathematical model for the cubes ie.

323121321902028557224579569649 xxxxxxxxxY

C

lxxvi

Table 5.7. ADEQUACY OF THE EXPERIMENT (CUBES)

S/N i

x x xx 2

xx

7 8.23 9.7728 -1.5428 2.3802

8 10.50 ,, 0.7272 0.5288

9 9.5212 ,, -0.2516 0.0633

10 11.15 ,, 1.3772 1.8967

11 9.05 ,, -0.7228 0.5224

12 10.1854 ,, 0.4126 0.1702

= 58.6366 = 5.5616

Using Equation 5.11,

772896

6366581

n

xx

n

i i

Using Eqn. 5.4

1123215

561651

2

2

v

xxS

n

i

051112321

1123212

S

S

5.7 Using the ‘t’ Distribution and Employing eqn. 5.13 and 5.14

Upper Limit 1

* tn

Sx

Lower Limit 1

tn

Sx

Then, from t-distribution table, at v = 5, = 0.05 give t 0.95 = 2.57.

(1) Upper Limit

lxxvii

8745101016177289

57242870772895726

05177289

(2) Lower Limit

671285724287077289

5.8 CORRELATION AND REGRESSION ANALYSIS

Random variables are usually related in such a way that a change in one enables or

entails a change in the distribution of the other. This is known as stochastic relationship. The

change brought about in a random variable Y by a change in another random variable X will

generally contain two components, stochastic (connected to the dependence of Y on X) and

random. If the stochastic component is absent, the two variables Y and X are independent of

each other. If the random component is absent, Y and X are connected by a functional

relationship. If both components are, the relationship between them defines the Strength

(closeness) of Association. If two random variables are independent, the variance of their sum

is equal to the sum of their variances;

yVarxVaryxVar ----------------[5.16]

The relationship between X and Y can be deduced from the inequality

0yx

yxE --------------- [5.17]

Eq. 4.17 is called the Cross Covariance Function or the Covariance, of the random variables X

and Y, abbreviated Cov(XY) or Covarxy.

The dimensionless quantity

yx

yxyxE

----------------[5.18]

is called the Correlation Coefficient.

If the two variables fluctuate in absolute independency, the value of r will be zero. It may also

have the value zero for some dependent variables which are then referred to as uncorrelated.

For normally distributed random variables, a Correlation Coefficient of Zero indicates that

there is no relationship between the variables. The Correlation Coefficient will remain

unchanged, if we add any non-random terms to X and Y, or if we multiply X and Y by a

positive number. If we multiply one of the two variables by –1 while leaving the other

variable unchanged, the correlation coefficient will likewise be multiplied by –1.

lxxviii

The Correlation Coefficient characterizes a linear, rather than just any relationship.

This linear probabilistic relationship between random variables, consists in that an increase in

one causes the other to show the tendency to increase (or decrease) linearly. As already noted,

the correlation coefficient gives a measure of the degree of relationship between two

variables. If the random variables X and Y are connected by a rigorously linear functional

relationship such that,

xbby10

----------------[5.19]

then 1xy

r , the sign being the same as that of the constant b1

In the general case, when X and Y are connected by an arbitrary stochastic

relationship, the limits to the value of r are +1 and –1, that is,

11xy

r ----------------[5.20]

At rxy >0, the correlation is positive;

At rxy <0, the correlation is negative.

The correlation coefficient is equally responsive to a high degree of randomness and to a

marked nonlinearity in the relationship. The correlation coefficient may remain less than

unity, although the relationship between X and Y is a rigorously functional one.

Qualitatively, the existence OR absence of association between two random variables

can be ascertained from the appearance of the Correlation Field as can be observed in figure

5.1

Fig. 5.1 Correlation Field for the Hollow Block and Cube Strengths for the Same w/c ratio

5.9 REGRESSION

The relationship between random variables is completely defined by the conditional

distribution function. For a system of two random variables, the conditional distribution

function f (x, y) is a function of two variables, etc. In practice, however, it is difficult to use

the conditional distribution functions; instead, recourse is had to conditional means, c, and

conditional variances, 2

C. The dependence of

2

C on the parameter x is called the

SCEDASTIC RELATION, but it is used relatively seldom. Practically, if the variances are

homogeneous, they are averaged. The dependence of the conditional mean C on x is called

the REGRESSION of C on x.

In processing experimental data, one usually finds an approximate regression equation

and evaluates the magnitude and probability of the uncertainty. Thus, the problem reduces to

lxxix

developing an approximate regression equation from a given sample of size n and to

evaluating the accompanying error. This problem is solved by the methods of Regression And

Correlation Analysis.

5.10 THE LINEAR REGRESSION EQUATION

In this linear regression equation, the method of least squares is used to estimate the

coefficients of a linear equation of regression.

xbby10

----------------[5.29]

from a sample of size n.

The method of least squares determines the best straight line entirely by calculation, using the

sets of recorded results such that the sum of the squares of the distances to a given straight line

from the given set of points is a minimum (Boeck et al, 2000)

The set of normal equations for the above case will be n

i 1 (Akhnazarova and

Kafarov, 1982).

010 iixbby

010 iiii

xxbbxy

OR

ii

yxbnb10

iiii

yxxbxb2

10 ----------------[5.22]

The coefficients b0 and b1 can readily be found, using the expressions;

22

2

0

ii

iiiii

xxn

yxxxyb ----------------[5.23]

and

21

xx

yyxxb

i

ii

22

ii

iiii

xxn

yxyxn ----------------[5.24]

It is simpler, however to find b0, once b1 is known, from first line of Eqns. [5.22]

xbyb10

----------------[5.25]

where y and x are the sample means of y and x.

lxxx

Among other things, Eq.[5.25] suggests the existence of a correlation between b0 and b1. To

evaluate the strength of correlation, Eq. [5.21], we should find the sample coefficient of

correlation *;

yx

ii

SSn

yyxx

1

* ----------------[5.26]

where Sx and Sy are the sample standard deviations.

From Eqs. [5.24] and [5.26], we have

22

22

1

1*

ii

ii

y

x

yyn

xxnb

S

Sb ----------------[5.27]

5.11 DETERMINATION OF THE CORRELATION BETWEEN THE STRENGTH

OF THE HOLLOW BLOCKS AND THE CUBES

Using the specimens values from table 4.7 for the first sample size of n = 6, we have as

tabulated below the experimental data. Values in N/mm2

Hollow Block Strength: x = 3.507, 2.232, 2.0, 3.216, 2.578, 2.195.

Cube-Strength y = 9.964, 6.952, 7.45, 9.10, 7.45, 6.5;

we find the coefficient of the regression equation of the form xbby10

Table 5.8: Raw Experimental Data for Blocks and Cubes

Exp.

No.

Block xi Cube yi 2

ix ii

yx 2

iy ii

yx 2

iiyx

1 3.507 9.964 12.299 34.944 99.281 13.471 181.468

2 2.232 6.952 4.982 15.517 48.330 9.184 84.346

3 2.000 7.450 4.000 14.900 55.503 9.450 89.303

4 3.216 9.100 10.342 29.266 82.810 12.316 151.684

5 2.578 7.450 6.646 19.206 55.503 10.028 100.561

6 2.195 6.500 4.818 14.268 42.250 8.695 75.603

i

x

iy

2

ix

iiyx

2

iy

iiyx

2

iiyx

15.728 47.416 43.087 128.101 383.677 63.144 682.965

A check on the tabulated results could be made using this equation

222

2iiiiii

yyxxyx ----------------[5.28]

lxxxi

Therefore

966682677383101128208743965682 , acceptable.

This is an indication that the calculations have been carried out correctly.

Using Eq.(5.24), bi can be found to be

0492

15211

84722

72815087436

41647728151011286

1

21

b

b

The coefficient b0 can be found using the expression in Eq. (5.22)

n

xbyb

ii 1

0

6

20515

6

72815049241647

53420

b

Using Eqn. (5.27), we find the sample coefficient of correlation to be

22

22

1

*

ii

ii

yyn

xxnb

2

2

416476773836

728150874360492

78553

152110492

9330

93304553500492

*

As can be seen, the correlation coefficient * is nearly unity, so the dependence of y on x in the

range of values in question is practically linear and has the form

xbby10

xy 04925342ˆ

A check on the first specimen result would give

YC = 2.434 + 2.049 x 3.5072 = 9.72N/mm2 against 9.954N/mm

2.

Little adjustment would yield a better result such as

YC = 2.55 +2.0

Therefore the resultant correlation between the Hollow 225 x 225 x 450 block and 150 x 150 x

150mm cube is

lxxxii

BCRR 112602

within the limit of this experiment.

Where RC = Compressive strength of the cube

RB = Compressive strength of the Hollow block

lxxxiii

CHAPTER SIX

CONCLUSION AND RECOMMENDATIONS

6.1 CONCLUSION

From observations in this experimental work and research in this project, conclusions

are drawn as written below.

Sandcrete block is a block moulded into variety of shapes from a mixture of sand,

cement, and water, exclusive of coarse aggregates (stones).

Much studies and research work had gone towards establishing the reactions of

sandcrete blocks to loads of different magnitudes in order to make use of this material for

construction at minimum acceptable cost and establish minimum collapse of structures.

Sandcrete blocks (Boeck et al, 2000). John Hancock Callender: Time Saver Standards

manufactured by many proprietors of blocks industries have the same constituent

materials such as sand, cement and water but due to none observation of specification and

standards set for the mix ratios or proportions of these materials there is a resultant low

compressive strength than required by NIS, NCP, NBC, IS or BS specifications.

This experimental work has proven that using the same materials as natural as they are,

such as the fine aggregates (sand), cement and water with obedience to standard rules and

specifications, these minimum strength specified are obtainable.

The low strength achieved by local proprietors in the block industries is due to the

following factors. The neglect to factors influencing the strength such as

1. Improper mix ratio or proportions

2. Lack of adequate curing and its duration

3. Inadequate compaction

4. Desire to maximize profit

5. Ignorance and inexperience of the manufacturer

6. Poor workmanship

The consequences of the low strength block are that blocks of these low strength are

sold to the public who use them to their own losses and detriment.

There are cracks in the buildings and structures due to low strength to withstand the

loads.

The structures in many cases would collapse or crush occupants.

Surveys of sandcrete block making factories in various parts of the country show that

the average compressive strength as 1.21N/mm2 and 0.73N/mm

2 in dry and wet states

respectively.

lxxxiv

These values are far below the requirements of NIS-0.74 which stipulates a minimum

strength of 2.5N/mm2 when in wet state

Federal Ministry of Works and Housing, Nigeria had specified 2.8N/mm2 for sandcrete

hollow block as in BS 2028 but later lowered to 2.5N/mm2

Federal Ministry of Works and Housing Lagos Nigeria (1978 Research) recommends

2.1N/mm2 or 300psi for these blocks.

The Nigerian National Building Code 200635

recommended a minimum crushing

strength of 2.0N/mm2 and 1.75N/mm

2 for average of blocks and lowest of individual

block for two or three storey building.

The simplex design method by Henry Scheffe was used and proved a very good result

since it made use of 4 component mixture. It is a good space to study sandcrete

materials responses such as strength, stability, permeability, impact strength etc.

The constituent materials for sandcrete blocks as used in this experimental work are

sand, cement and water. Block size was 225 x 225 x 460mm and cube size was 150 x

150 x 150mm

Sand, cement and water were weighed according to given mix ratios in Table 4.2. The

blocks were weighed and loads converted from Kg to Newton with this formular

AreationalcrossNet

xxcrushingloadMax

mm

N

sec

1000964.9)(

2

The average cross sectional area of 4 blocks was 625000mm2 while that of cube was

225000mm2 ie 0.0625m

2 for block and 0.0225m

2 for cube. The weight of blocks averaged

26kg and the weight of cubes averaged 6.8kg.

This indicates a cube/block ratio of 1:3.82. The maximum crushing strength of block

was 3.5027N/mm2 while that of cube was 9.964N/mm

2. This indicates a ratio of block/cube

of 1:2.845.

It can be seen that the block weighed about three to four times that of cube but the

crushing strength of the cube was about three times that of the block.

The densities of both block and cube averaged 1845kg/m3 and 2015kg/m

3 respectively.

There is a very good linear correlation from this experimental work between the

strength of hollow sandcrete blocks and sandcrete cubes which is as follows

Rc = 2.6 + 2.11 RB

Where RC = compressive strength of sandcrete cube

RB = compressive strength of sandcrete block

a. and 2.11 are correlation constants

lxxxv

The correlation coefficient r is nearly unity (0.933) which shows a very good

dependence of Y on x in the range of values used.

In checking the adequacy of the experiment, a confidence coefficient of

1-x = 0.9 was assumed. The confidence interval for the population standard deviation

was computed as 0.062 ≤ 2 ≤ 0.601

0.249 ≤ 2 ≤ 0.775

Using student t-distribution, the confidence interval was found to be as follows

Block Cube

1. Upper State limit = 3.138 10.8745

2. Lower State limit = 2.362 8.6712

Observing the compressive strength, those of cube values are approximately three

times of hollow block strength.

From the experimental work and research carried out it was observed that the

compressive strength of sandcrete block is affected considerably by

1. the percentage of coarse aggregates

2. the cement content

3. the water of compaction

The compressive strength increases with increase in cement content and addition of

coarse aggregates to the mixture. When the cement content is doubled, the strength increases

to about twice the previous figure.

The compressive strength is maximum at 45% addition of coarse aggregates to the

mix. With further increase, the strength drops down due to increase in voids.

By careful control of water of compaction which does not add to the cost, the strength

can be increased by about 10 to 20%. This study agrees with Ezeokonkwo‟s and

Ezeuzomaka‟s, that the thickness of the solid parts of a hollow block affects the strength of the

block. The thicker the webs and the outer shells, the higher the block density and

consequently increase in the compressive strength.

It is observed that this experimental work obtained the strengths that are higher than

the recommendations of some of the bodies such as NIS, BS and especially that of the NBC.

lxxxvi

6.2 RECOMMENDATIONS

I recommend that

1. there is an urgent need for the enforcement to produce proper and quality sandcrete

blocks for the construction industries to avoid cracks, collapses and slumping of

buildings and other structures.

2. Good sand should be used instead of pit sand for block making since pit sand is

found to be uneconomical. It takes much cement content to obtain the required

strength.

3. Coarse aggregates should be used to increase strength of hollow sandcrete block

since it is cheaper than cement

4. Since 98% or more of blocks used are manufactured by block industries without

sieving, it is better to establish strength of these blocks manufactured from natural

sand that is not sieved.

5. It is strongly recommended that workshops and seminars be organized by Local

Governments, State Governments, Educational Institutions and Churches to

sensitize the block manufacturers and create good awareness and educate them n

the grave dangers of low strength blocks.

6. The athor recommended that the minimum compressive strength of 225 x 225 x

450mm2 hollow sandcrete block be accepted with the ranges of 1.75N/mm

2 to

2.0N/mm2

lxxxvii

REFERENCES

Adams, E. C.(1976) Science In Building For Craft Students & Technicians. 3 Materials. 4th

Impression Jan 1976. Hutchinson, London.

Akhnazarova, S. and Kafarov, V. (1982) Experiment, Optimization In Chemistry & Chemical

Engineering, 1982. Mir Publisher, Moscow.

Anthony, K, D. (2005). Engineering Properties of Ghanaian Sandcrete Blocks.

Boeck, L., Chaudhuri, K. P. R. and Aggarwal, H. R. (2000). Sandcrete Blocks for Building:

A Detailed Study on Mix Compositions, Strength and Cost. NSE Journal Jan – Mar

2000 pp 24 - 33

British Standard (1985) Structural Use of Concrete. BS 8110: Part 1: 1985.

Callender, J. H. – Editor-In-Chief, Time Saver Standards for Architectural Design Data 5th

Edition, 1974. Mc Graw-Hill Book Company, USA.

Ching, F. D. K. (1976) Building Construction Illustrated, Van Nostrand Reinhold Coy. N. Y.

USA.

Chudley, R (1997). Building Construction Handbook Review to meet 1995 Building

Regulation 2nd

Edition, Laxton, Oxford, Boston

Day, K. W. (1995) Concrete Mix Design Quality Control & Specification. 1st Edition. E &

FN Spon, London.

Enwelu, C (2002): Optimization of Compressive Strength of Lateritic building Blocks.

Ezeokonkwo, J. C. (1998). Uniaxial Compressive Strength of Sandcrete Hollow Blocks and

Its Dependence on Geometry. Civil Engineering Dept, UNN, September, 1998.

Eze-Uzomaka, O. J.: The Crushing Strength of Sandcrete Blocks in relation to Their

Production and Quality Control. Op. Cit.

Federal Republic of Nigeria: National Building Code. 1st Edition 2006. Lexis Nexis,

Butterworths.

Jacknson, N (1984) Civil Engineering Materials, Macmillian ELBS, London. Edited – 3rd

Edition.

Kaminetzky, D (1991). Design and Construction Failures, McGraw-Hill USA.

Manual of Steel Construction. 7th

Edition. AISC Inc. NY 1973. AISC. Inc. N. Y. 1973.

Mays, G (1992). Durability of Concrete Structures, Investigations, Repairs, Protection.

Edited 1992. E & FN Spoon, London.

Neville, A. M. (1997). Properties of Concrete. 4th

Edition 1997. Longman ELBS, England.

Neville, A. M. and Brooks, J. J (1991). Concrete Technology, Longman Scientific Technical

N. Y. USA

lxxxviii

Norris, C. H and Wilbur, J. B. Elementary Structural Analysis: 2nd

Edition. Publishers –

McGraw Hill.

O‟Brien, J. P. E. Construction Inspection Handbook P. 1974. Van Nostrand Reinhold Coy. N.

Y. USA.

Obam, S (2003). Mathematical Models for Optimization of Mechanical Properties of Concrete

Made From Rice Husk Ash.

Obodoh, D. A. Optimization of Components Mix in Sandcrete Blocks using Fine Aggregates

from Different Sources. Aug. 1999. UNN.

Orchard, D. F (1973). Concrete Technology vol. 2, 3rd

Edition 1973. Applied Science

Publisher Ltd., London.

Orchard, D. F. (1973) Properties of Materials, Concrete Technology vol. 1, 3rd

Edition.

Applied Science Publisher Ltd., London.

Osadebe, N. N.: Notebook. Dept of Civil Engineering, University of Nigeria, Nsukka.

Oyenuga, V. O (2001). Simplified Reinforced Concrete Design (A Consultant/Computer

Based Approach). 2nd

Edition, 2001. ASBROS Ltd., Lagos, Nigeria.

Parker, H. and Ambrose, J. Simplified Design of Concrete Structures. 6th

Edition. John

Wiley & Sons, Inc. N. Y. USA.

Parker, H. (1975). Simplified Engineering for Architects & Builders – 5th

Edition, John

Wiley & Sons, N. Y.

Reynolds, C. and Steedman, J. C. (1998). Reinforced Concrete Designer‟s handbook, 1988.

Spoon Press, London

Samson, D. Elinwa, A. U. and Ejeh, S. P. (2002). Quality Assessment of Hollow Sandcrete

Blocks, Nigerian Journal of Engineering Research and Development vol. 1 No. 3.

July-Sept. 2002.

Samuel, F. E. (1994). Reinforced Concrete Technology

Smith, R. C (1973). Materials of Construction, 1973. Mc Graw-Hill Book Company, USA.

Spence, W. P (1972) Architecture: Design: Engineering Drawing, 2nd

Ed. McKnight &

McKnight Publishers Coy.

Steel Designer‟s Manual. 6th

Ed. Edited by Buick & Graham W. Owens. Blackwell Science,

Oxford, UK.

Stroud, K. A. (1990) Engineering Mathematics. 3rd

Edition. Macmillan, London

Wilby, C. B. (1983). Structural Concrete, Butterwoth, London, UK.

Yilla, I. S (1967). The Effect of Mix proportion and Curing Condition on Shrinkage of

Sandcrete Blocks R. I. L. E. M Bulletin, New Series No. 34, march, pp 87-90.

lxxxix

APPENDIX

COMPUTER PROGRAM

REM

REM

LET COUNT = 0

70 GOSUB 100

80 END

90 REM

100 REM

110 LET YMAX = 0

120 PRINT

130 PRINT “OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING

FINE”

132 PRINT “AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A

DESIRED STRENG

170 PRINT

185 INPUT “ENTER DESIRED STRENGTH =”, YIN

190 GOSUB 400

200 FOR X1 = 0 TO 1 STEP .01

210 FOR X2 = 0 TO 1 – X1 STEP .01

220 X3 = 1 – X1 – X2

240 LET YOUT = 3.5073 * X1 + 2.985 * X2 + 1.995 * X3 + .2754 * X1 * X2 - .7526

250 GOSUB 500

260 IF (ABS(YIN – YOUT) <= .001) THEN 270 ELSE 290

270 LET COUNT = COUNT + 1

280 GOSUB 600

290

291 NEXT X2

292 NEXT X1

295 PRINT

300 IF (COUNT > 0) THEN GOTO 310 ELSE GOTO 340

310 PRINT “THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL

IS”; YMAX;

320 SLEEP 2

330 GOTO 360

340 PRINT “SORRY, DESIRED STRENGTH OUT OF RANGE OF MODEL”

350 SLEEP 2

360 RETURN

400 REM

410 PRINT

415 PRINT “PLEASE …CALCULATION IN PROCESS”

418 PRINT

420 PRINT “COUNT X1 X2 X3 Y Z1 Z2

X3”

430 PRINT

440 RETURN

500 REM

510 IF YMAX < YOUT THEN YMAX = YOUT ELSE YMAX = YMAX

xc

520 RETURN

600 REM

610 LET Z1 = .5 * X1 + .6 * X2 + .65 * X3

620 LET Z2 = X1 + X2 + X3

630 LET Z3 = 5.5 * X1 + 6.5 * X2 + 8 * X3

650 PRINT TAB(1); COUNT; USING “# # # #.# # #”; X1 X2; X3; YOUT; Z1; Z2; Z3

660 RETURN

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES, CORRESPONDING TO A DESIRED

STRENGHT

ENTER DESIRED STRENGTH = 2.75

PLEASE…CALCULATION IN PROCESS

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 0.010 0.870 0.120 2.750 0.605 1.000 6.670

2 0.080 0.770 0.150 2.750 0.600 1.000 6.645

3 0.150 0.570 0.180 2.750 0.594 1.000 6.620

4 0.220 0.570 0.210 2.751 0.589 1.000 6.595

5 0.290 0.470 0.240 2.751 0.583 1.000 6.570

6 0.380 0.340 0.280 2.749 0.576 1.000 6.540

7 0.450 0.240 0.310 2.750 0.571 1.000 6.515

8 0.520 0.140 0.340 2.751 0.565 1.000 6.490

9 0.540 0.110 0.350 2.749 0.564 1.000 6.485

10 0.610 0.010 0.380 2.750 0.558 1.000 6.460

TH MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS 3.507298

N/SQ.MM

Press any key to continue

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

ENTER DESIRED STRENGTH = 3.507298

PLEASE…CALCULATION IN PROCESS

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 1.000 0.000 0.000 3.507 0.500 1.000 5.500

THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS

3.507298 N/SQ.MM

Press any key to continue

xci

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

ENTER DESIRED STRENGTH = 3.507298

PLEASE…CALCULATION IN PROCESS

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 1.000 0.000 0.000 3.507 0.500 1.000 5.500

THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS

3.507298 N/SQ.MM

Press any key to continue

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

ENTER DESIRED STRENGTH = 3.507298

PLEASE…CALCULATION IN PROCESS

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 1.000 0.000 0.000 3.507 0.500 1.000 5.500

THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS

3.507298 N/SQ.MM

Press any key to continue

EXECUTION OF PROGRAMME

Below are the results of computer program to execute the models, both Block model

and Cube model.

A user-friendly manner is prompted for an input of the desired strength, and will then

proceed to print out all possible combinations of the pseudo and Actual components that will

match that strength configuration within a tolerance of 0.001N/mm2. When there are no

matching combinations, the program will also inform the user of this. However, it will print

out the optimum strength predictable by each model.

xcii

VARIOUS COMBINATIONS USING THE MATHS MODEL FOR

225 x 225 x 450mm HOLLOW SANDCRETE BLOCK OF 40% VOID

3231213213160702405856102232250723 xxxxxxxxxY

B

VARIOUS COMBINATIONS USING THE MATHS MODEL FOR

150 x 150 x 150mm CUBE

323121321902028557224579569649 xxxxxxxxxY

C

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 0.020 0.150 0.830 7.001 0.639 1.000 7.725

2 0.030 0.920 0.050 7.000 0.600 1.000 6.545

3 0.190 0.210 0.600 6.999 0.611 1.000 7.210

4 0.190 0.270 0.540 7.000 0.608 1.000 7.120

THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS

9.963995 N/SQ.MM

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 0.230 0.740 0.030 7.999 0.579 1.000 6.315

2 0.270 0.650 0.080 8.001 0.577 1.000 6.350

3 0.480 0.250 0.270 8.000 0.566 1.000 6.425

4 0.540 0.160 0.300 8.001 0.561 1.000 6.410

THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS

9.963995 N/SQ.MM

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

COUNT X1 X2 X3 Y Z1 Z2 Z3

1 0.610 0.310 0.080 9.000 0.543 1.000 6.010

2 0.730 0.150 0.120 9.001 0.533 1.000 5.950

THE MAXIMUM VALUE OF STRENGTH PREDICTABLE BY THIS MODEL IS

9.963995 N/SQ.MM

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

[8]

[7]

[9]

xciii

COUNT X1 X2 X3 Y Z1 Z2 Z3

SORRY, DESIRED STRENGTH OUT OF RANGE OF MODEL

OPTIMISATION OF COMPONENTS MIX IN SANDCRETE BLOCKS USING FINE

AGGREGATES FROM DIFFERENT SOURCES CORRESPONDING TO A DESIRED

STRENGTH

[[

COUNT X1 X2 X3 Y Z1 Z2 Z3

SORRY, DESIRED STRENGTH OUT OF RANGE OF MODEL

[10]

[11]