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]