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International Journal of Emerging Technology and Innovative Engineering Volume 2, Issue 7, July 2016 (ISSN: 2394 – 6598)
Date of Publication: 07.09.2016
345
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TREND LINE ANALYSIS OF WEIGHT VERSUS
COMPRESSIVE STRENGTH OF CONCRETE
Sagar Tanwani
Graduate Civil Engineering
Quaid-e-Awam University of Engineering,
Science & Technology
Bashir Ahmed Memon
Professor, Department of Civil Engineering
Quaid-e-Awam University of Engineering,
Science &Technology
ABSTRACT
Concrete is most widely used material in construction industry. To ensure its durability during
service life quality control is ecclesial. 28-day strength of concrete particularly compressive
strength is determined to check and ensure its quality. This requires dedicated equipment and
setup. An alternative to it is the numerical analysis. Therefore, in this research paper relationship
between weight and strength of concrete cylinders is developed using trend line analysis
available in Microsoft excel. For the purpose 240 cylinders using 1:2:4 mix with 0.45 water-
cement ratio are prepared and cured for 28-days. After curing weight of the cylinders is
evaluated first followed by determination of compressive and tensile strength. The obtained
results are then analyzed. It is observed that the trend line fit by power function gives results in
good agreement to experimental observations. Individual equations for relationship between
weight and compressive strength and weight and tensile strength are presented. Then both the
equations are combined in one general equation with separate coefficients for compressive and
tensile strength.
Keywords: compressive strength, tensile strength, weight, tend line analysis.
1. INTRODUCTION
Concrete is most widely used material in
construction industry, mainly due to its
flexibility to be shaped in any desirable
shape, less maintenance cost during service
life. The material is very good in
compression however it has some
limitations in tension. This weakness of
concrete is overcome by using
reinforcement which makes it equally good
in tension. To ensure durability of concrete
during service life it is mandatory to check it
strength during construction. Normally 28-
day strength is required for checking which
on contrary is not only time consuming but
also requires dedicated laboratory setup.
Although early age determination of
concrete strength i.e. 1- day, 3-days and 7-
days strength is in practice and there exist
method of computing projected strength for
28-dyas yet actual 28-day strength is
mandatory by codes of practice. Also in case
if the test result does not confirm the design
strength of concrete then whole process need
to be repeated. It will not only time
consuming but also costly. For each failure
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one has repeat the process to confirm the
required design strength of concrete.
Alternatives to laboratory testing are
nondestructive testing and the numerical
determination of the strength.
Nondestructive testing is normally carried
out in the existing structures. Therefore,
numerical methods may be the suitable
alternative for the purpose. The approach
remained the active area of research since
long among the scholars. Several scholars
have made attempts to develop methods for
the purpose. In the following related
literature is summarized.
Rachna et al[1]
conducted research work to
propose the statistical model for predicting
concrete strength using linear regression
analysis. They used water-binder ratio, fine
aggregate-binder ratio, coarse aggregate-
binder ratio and binder content to conduct
their research work. They used fly ash
variation and curing from 28 to 91 days. For
28-day curing they demonstrated that the
quadratic polynomial with co-efficient of
determination greater than or equal to 0.99
gave results in very good agreement with
reference concrete.
Larrossa et al[2]
in their research work
considered compressive strength of concrete
as variable to ensure conformity control of
port construction work in Brazil. Based on
their results the authors showed variations in
conformity control of concrete from
different places and concluded that there is
need of uniformity in the said area.
Liu et al[3]
in their research work used
nondestructive test (surface hardness
rebound ) value, material design parameters
and regression analysis to increase the
accuracy of strength calculations. They used
166 samples of 738-days age in 146 training
examples. They also used 20 test examples
to estimate concrete compressive strength.
Based on the findings the authors used
regression analysis to establish a
mathematical formula. Their study results
indicatethat the correlation coefficient may
reach 0.9622 which indicates that the
proposed method has
referentialvalue.Therefore, can confidently
be used to compute to determineconcrete
strength.
Soutsos et al[4]
in their research conducted
in-situ pull out test following standard
statistical procedures of BS EN 13791:2007
on full scale reinforced concrete building
frame built in laboratory. The procedures
account for the uncertainty of the strength
correlation by requiring either the lower95%
confidence limit curve or a curve shifted
downwards by a margin to be used. Based
on their finding they concluded that
conservative estimates of the in-situ
characteristic strength are observed.
Akthem et al[5]
in their research work used
statistical analysis of compressive strength
of lightweight aggregates concrete of the
Benica Martine Bridge in California. They
used specimen of sizes 4x8 in. and 6x12 in
at site and tested in laboratory at 5-years.
The authors used statistical analysis and
probability theory to compare the results
with 35-days test results provided by
California Department of Transportation
Caltrans. The authors observed that although
the dispersion in data from mean strength
was more for specimen of age of 5-years yet
the strength was more by 3.6% than
specimen of age 35-days. They also
observed that the average compressive
strength of 4x8 in. cylinders was more than
the compressive strength of 6x12 in.
cylinders by 2% at 5 years. The probability
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of compressive strength to fall below the
minimum observed strength was
approximately increased 3 times at 5 years
when compared to those observed at 35
days. The probability of compressive
strength to go above the maximum value
was also increased by 1.23% at 5 years.
From the analysis on the compressive
strength data at 35 days and 5 years, it was
concluded that the probability of falling
below the target specified strength
essentially approaches zero, which means
concrete will never fall below its specified
design strength in the life time of the
structure.
Mahmoud[6]
in his research paper presented
statistical model to predict the compressive
strength of concretecontaining different
matrix mixtures at fixed age or at different
age of 1, 3, 7, 28, 56, 90 and180 days. In
this research the author examined effect of
eight different parameters for the matrix
mixture i.e. time, water, cement, metakaolin,
silica fume, sand, aggregateand
superplasticizer, on the compressive strength
of concrete. Based on the research findings
the author concluded that the predicted
model has highcorrelation to the
experimental results for the concrete
compressive strength.
Baji and Ronagh[7]
in their research work
studied stress-strain block parameters using
statistical analysis. They used database of
experimental results onconcrete equivalent
rectangular stress block parameters. The
authors studied uncertainty involved in the
evaluation of the equivalent
rectangularstress block parameters using
probability-based model errors and
compared with experimental data. Finally,
using Monte Carlo Simulation the authors
studied effect of uncertainty in the concrete
stress block parameters on the ultimate
flexural strengthand curvature. Based on the
results the authors concluded that due to
variations in material and
sectionalproperties, a significantly higher
variability exists in the ultimate curvature of
reinforced concretebeam sections in
comparison to strength and that the ultimate
curvature is sensitive to morerandom
variables comparing to the strength.
Shimizuet al[8]
conducted research to
estimate concrete strength in existing
structures for proper inspection for seismic
resistance and retrofitting of existing
structures in Japan. For the purpose they
used statistical analysis on 10788 cores from
1130 existing buildings. Based on their
findings they concluded that in younger
structures there is less chance of low
strength. Compressive strength showed
changes from floor to floor in a building.
Out of considered buildings significant
number of buildings had low strength.
Jonathan[9]
for his master’s research work
studied the possibility of use of
nanotechnology in the field of cement and
concrete. The author used nanoparticles and
nanofibers & nanotubes in concrete. The
author used Monte Carlo and Bayesian
methods to analyze and compare the results
with those available in literature. Based on
the finding author concluded the use of
statistical prediction to analyze patterns and
trends to set optimal dosage of additives in
concrete.
Kolisho, Hunka and Jung[10] in their
research work performed statistical analysis
of modulus of elasticity and compressive
strength of pre-stresses precast beams. For
the purpose the authors used statistical
analysis between modulus of elasticity and
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compressive strength on 133 test results of
cylinders 150 × 300 mm in size (28-day
curing). Based on their findings the authors
concluded that for this specific case high
skewness is recorded. However, the beta
distribution is better than generally
recommended theoretical distribution for
strength. Moreover, they concluded that
modulus of elasticity is not significantly
affected due to skewness as the mean value
of this parameter is used.
Nowak and Rakoczy[11]
performed research
to develop a statistical model of compressive
strength of lightweight concrete, using new
materialtest data. They used 8000
samplesfrom 8 different sources with
strength ranging from 21 to 50MPa (3000 to
7000 psi). The presented research is focused
on the development of statistical parameters.
For the purpose the authors presented test
data in form of the cumulative distribution
functions plotted on the normal
probabilitypaper. The shape of the CDF is
an indication of the type of distribution, and
sincethe resulting CDF’s are close to straight
lines, they can be considered as normal
random variables. In addition, the
statisticalparameters are determined by
fitting a straight line to the lower tail of the
CDF. The most important parameters arethe
mean value, bias factor and the coefficient
of variation. The authors observed that the
quality of material and workmanshiphas
been improved over the last 30 years and
this is reflected in reduced coefficients of
variation.
Akcay[12]
in his technical note presented
statistical model and results for the
compressive strength of existing buildings in
potential earthquake risk zone in Istanbul.
He used core samples from 244 structures.
The author presented mean value for a total
of 1,892 characteristic strength values equal
to 22.01 N/mm2 with a standard deviation
value of 9.99 N/mm2. Based on the obtained
results the author concluded that
approximately 42% of all the analyzed
structures were below the required quality.
Das and Chatterjee[13]
conducted research to
develop statistical model using Monte Carlo
simulation technique for self-compacting
and normally compacting concrete. The
authors used and tested concrete cubes at 7,
28, 60, 90, 120 and 150 days after normal
water curing. For each case 10 samples were
tested. The authors introduced a new
concept called ―Root Mean Squared
Spacing (RMSS)to estimate adequate
sample size following the principle of
random walk.Theauthors used MATLAB
software for the analysis. The authors
concluded that new proposed approach
based on simulation technique facilitates to
overcome the problem of low sample size
using Monte Carlo and RMSS methods.
They also concluded that comparative
analysis of test results can provide better
meaningful understanding if systematic
approach is adopted through application of
conventional statistical parameters along
with the new proposed procedure.
Zain et al[14]
in their research work used
non-linear regression equation for prediction
of concrete compressive strength at different
ages. Using the model, the authors obtained
the correlation coefficients equal to 0.995
and 0.994 for the prediction of 7 and28-day
compressive strength respectively. Based on
their results the authors concluded that
proposed model has good correlation with
test results.
Adam[15]
in his research work developed
predictive multiple least square regression
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relationship for concrete strength, elastic
modulus, strain at peak stress, ultimate
strain, and stress-strainbehavior, including
the temperature, aggregate type, test type,
and strength at roomtemperature. The
research results reveals that at high
temperatures both modulus of elasticity and
strength are reduces whereas strain at peak
stress and ultimate strain are increased.
Al-Qadi et al[16]
in their research work
developed statistical models by considering
influence of key mixture parameter (cement,
water to powder ratio, fly ash andsuper
plasticizer) on hardened properties affecting
the performance of self-compacting cement
concrete. They used thirty-one mixtures to
derive the numerical models. Based on the
results the authors concluded that the models
were valid for a wide range ofmixture
proportioning.
Cecia et al[17]
in their research work
performed statistical analysis of existing
models for resisting the debonding of FRP
sheets used for strengthening of reinforced
and pre-stressed concrete beams in flexure.
They used 2 different databases for their
work. Based on regression analysis the
authors concluded that there is variation in
different models for calculated loads at
which debonding occurs. They further
concluded that regression analysis helps in
deciding the best model for the purpose.
Yaghi and Hammoud[18] in their research
work developed statistical model based on
modified tolerance based approach for
prediction concrete strength. In their
research article the authors argue that
modified tolerance approach of ACI is based
on normal distribution which has several
draw backs therefore they used reliability
analysis. Also the authors incorporated
lower bound through the use of left
truncated normal distribution. The authors
also compared tolerance factor, modified
tolerance factor and Bartlett and Macgregor
approaches. Based on the results the authors
conclude that the modified tolerance factor
approach givesmore reliable estimates of
equivalent design compressive
strengthespecially for data with high
coefficient of variation.
Novák et al[19]
in their research work used
statistical nonlinear fracture mechanics
analysis to study the size effect of concrete
beams. They used crack band theory based
on fracture energy related in
SBETA/ATENA and introduced statistical
variability of material parameters using
Monte Carlo simulation. They demonstrated
comparison of the results with experimental
data.
Dashrath et al[20]
in their research work used
natural sand, artificial sand, crushed stone
and recycled aggregates to study
compressive and tensile strength of concrete.
Based on the obtained results they
concluded that compressive and tensile
strength reduces with induction of these
aggregates in comparison for reference
concrete. However, they observed that the
reduction in tensile strength is less than
compressive strength. Which shows that
alternative aggregates have promising effect
on strength. However, refined techniques are
required to ensure proper compressive
strength of the resulting material.
From the above discussion it is clear that
scholars remained keen to coin out the
methods based on statistical procedure
which can successfully predict the concrete
strength in different cases. In this research
work an attempt is made to develop
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relationship between weight and
compressive strength of concrete cylinders
of standard size. To achieve the target of the
research 1:2:4 mix with water-cement ratio
of 0.45 is used to prepare 240 cylinders.
Ingredients of concrete mix were ordinary
Portland cement, coarse aggregates and hill
sand. These ingredients were proportioned
by weight to prepare 1:2:4 mix followed by
casing of standard size cylinders in standard
manner. Rodding is used for the compacting
of material in molds. After 28-day water
curing first weight of concrete cylinders is
determined then universal load testing
machine was used to evaluate the
compressive and tensile strength of concrete
cylinders. 120 cylinders are selected for
compressive and tensile strength
respectively. The obtained results are then
analyzed using trend line / regression
analysis to develop equations for both
compressive and tensile strength of concrete.
Several methods are studied and will be
discussed in relevant section. The equations
are then used to reevaluate the concrete
strength. The obtained results show very
good agreement between experimental and
numerical results. It is hoped that the work
presented in this article will be very helpful
for practicing engineers to have good idea of
the concrete strength.
2. MATERIAL AND TESTING
To achieve the target of the research work
300 concrete cylinders are prepared using
1:2:4 mix with 0.45 water cement ratio as
per ACI recommendation of concrete mix
design. Ordinary Portland cement, crushed
stone aggregates of maximum 1 inch size
and hill sand are used as the ingredients of
concrete. Mix proportioning is done using
weight method. Mold are prepared and
cylinders are cast using standard procedures.
As the compaction on site is generally made
by rodding therefore the same is adopted for
compaction of the cylinders. As mentioned
earlier 28-day curing is the standard
requirement of curing for the proper strength
of concrete members therefore the cylinders
are then water cured for 28-days. After
curing time weight of all the cylinders is
recorded. The cylinders are then divided
into two batches and each batch is tested
using universal load testing machine for
compressive and tensile strength
respectively. The obtained results of the
strength are then checked for strength
deviation from mean strength. All the
cylinders having strength deviation of 15%
or more are discarded. Finally 120 cylinders
in each batch are selected for further
analysis.
Table 1: Weight and compressive strength of concrete cylinders
# Weight
Compressive
Strength # Weight
Compressive
Strength # Weight
Compressive
Strength
kg psi Kg psi kg psi
1 13.8 3804.50 41 13.6 3761.85 81 13.6 3597.30
2 13.6 3804.95 42 13.6 3742.98 82 13.8 3852.54
3 13.6 3820.75 43 13.8 3780.62 83 13.8 3854.69
4 13.7 3715.25 44 13.8 3810.56 84 13.6 3675.56
5 13.8 3820.50 45 13.7 3762.61 85 13.6 3684.95
6 13.8 3825.75 46 13.7 3755.98 86 13.7 3763.54
7 13.6 3750.25 47 13.7 3745.87 87 13.7 3761.82
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8 13.6 3820.45 48 13.8 3823.69 88 13.8 3844.98
9 13.6 3821.36 49 13.6 3598.74 89 13.8 3872.66
10 13.8 3754.25 50 13.8 3842.58 90 13.8 3865.74
11 13.7 3678.36 51 13.6 3685.36 91 13.8 3854.65
12 13.8 3845.65 52 13.6 3712.65 92 13.6 3599.87
13 13.7 3655.63 53 13.6 3687.52 93 13.6 3621.54
14 13.7 3687.95 54 13.8 3871.65 94 13.8 3854.96
15 13.7 3750.89 55 13.8 3825.67 95 13.7 3759.68
16 13.6 3726.45 56 13.8 3852.36 96 13.7 3760.51
17 13.6 3765.68 57 13.7 3745.63 97 13.7 3762.43
18 13.7 3755.36 58 13.8 3842.68 98 13.7 3756.82
19 13.8 3838.75 59 13.7 3735.32 99 13.6 3711.97
20 13.8 3780.45 60 13.8 3864.25 100 13.8 3866.98
21 13.7 3756.89 61 13.7 3752.64 101 13.8 3862.78
22 13.7 3779.21 62 13.8 3855.69 102 13.6 3588.96
23 13.7 3563.58 63 13.8 3857.61 103 13.6 3615.87
24 13.7 3598.78 64 13.6 3691.23 104 13.7 3699.87
25 13.6 3524.32 65 13.6 3589.65 105 13.8 3798.99
26 13.6 3598.36 66 13.7 3761.56 106 13.7 3756.93
27 13.6 3496.85 67 13.7 3740.58 107 13.7 3712.85
28 13.8 3452.89 68 13.8 3869.79 108 13.7 3735.68
29 13.8 3476.35 69 13.6 3596.54 109 13.7 3762.59
30 13.7 3682.75 70 13.6 3612.54 110 13.6 3721.54
31 13.6 3679.82 71 13.6 3658.24 111 13.8 3875.12
32 13.8 3742.56 72 13.6 3638.94 112 13.8 3763.21
33 13.8 3752.69 73 13.8 3854.96 113 13.8 3861.23
34 13.6 3685.91 74 13.8 3844.57 114 13.6 3722.65
35 13.7 3812.40 75 13.6 3720.56 115 13.6 3712.36
36 13.7 3761.36 76 13.7 3746.66 116 13.8 3852.65
37 13.8 3785.14 77 13.7 3748.21 117 13.8 3842.86
38 13.6 3764.32 78 13.7 3752.62 118 13.7 3758.71
39 13.8 3719.86 79 13.6 3754.36 119 13.8 3854.36
40 13.6 3754.65 80 13.8 3836.55 120 13.8 3851.26
Table 2: Weight and tensile strength of concrete cylinders
# Weight
Tensile
Strength # Weight
Tensile
Strength # Weight
Tensile
Strength
W1 Psi W1 Psi W1 Psi
1 13.8 422.00 41 13.7 414.00 81 13.7 411.00
2 13.8 424.00 42 13.8 436.00 82 13.8 428.00
3 13.8 435.00 43 13.8 439.00 83 13.8 428.00
4 13.6 388.00 44 13.8 428.00 84 13.8 429.00
5 13.6 392.00 45 13.8 425.00 85 13.6 385.00
6 13.8 432.00 46 13.7 389.00 86 13.6 385.00
7 13.8 438.00 47 13.7 413.00 87 13.7 413.00
8 13.8 439.00 48 13.7 396.00 88 13.7 412.00
9 13.8 436.00 49 13.8 421.00 89 13.7 412.00
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10 13.8 426.00 50 13.8 428.00 90 13.7 398.00
11 13.8 428.00 51 13.8 429.00 91 13.8 436.00
12 13.8 432.00 52 13.8 426.00 92 13.8 435.00
13 13.7 405.00 53 13.8 431.00 93 13.6 389.00
14 13.8 438.00 54 13.8 438.00 94 13.8 439.00
15 13.8 439.00 55 13.8 435.00 95 13.7 399.00
16 13.8 437.00 56 13.7 397.00 96 13.7 414.00
17 13.8 439.00 57 13.7 414.00 97 13.8 439.00
18 13.8 428.00 58 13.6 385.00 98 13.8 436.00
19 13.8 438.00 59 13.7 410.00 99 13.8 438.00
20 13.7 409.00 60 13.6 387.00 100 13.7 415.00
21 13.7 405.00 61 13.6 388.00 101 13.6 390.00
22 13.8 429.00 62 13.8 439.00 102 13.6 388.00
23 13.8 430.00 63 13.8 428.00 103 13.6 382.00
24 13.8 436.00 64 13.8 438.00 104 13.8 432.00
25 13.8 438.00 65 13.8 439.00 105 13.8 430.00
26 13.8 432.00 66 13.8 427.00 106 13.8 439.00
27 13.8 431.00 67 13.8 427.00 107 13.8 436.00
28 13.8 426.00 68 13.8 429.00 108 13.8 436.00
29 13.8 429.00 69 13.8 429.00 109 13.7 412.00
30 13.8 432.00 70 13.8 432.00 110 13.7 410.00
31 13.6 387.00 71 13.6 386.00 111 13.7 413.00
32 13.7 398.00 72 13.6 386.00 112 13.8 439.00
33 13.7 402.00 73 13.7 413.00 113 13.8 439.00
34 13.7 413.00 74 13.8 438.00 114 13.8 435.00
35 13.6 386.00 75 13.6 388.00 115 13.6 388.00
36 13.7 411.00 76 13.8 436.00 116 13.8 437.00
37 13.6 388.00 77 13.8 435.00 117 13.8 435.00
38 13.6 389.00 78 13.8 436.00 118 13.8 435.00
39 13.6 382.00 79 13.8 434.00 119 13.8 439.00
40 13.7 410.00 80 13.7 411.00 120 13.8 436.00
Table 3: Basic strength information of all cylinders
Strength Compressive Tensile
Mean strength 3748.48 419
Standard deviation 92.36286 18.73006
Median value 3756.86 428
Maximum strength 3875.12 439
Minimum strength 3452.89 382
Weight and compressive strength of 120
concrete cylinders is given in table 1,
whereas table 2 shows the results of weight
and tensile strength of 120 cylinders. Basic
information of both compressive and tensile
strength is listed in table 3. From this table it
may be observed that mean compressive
strength is 3748.48 psi and mean tensile
strength is recorded as 419 psi. The mean
tensile strength is 11.17% of the
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compressive strength. In addition, deviation
from mean compressive strength is
computed and is recorded as low as 0.01%
and as high as 7.89% which is less than
15%. Similarly, deviation of tensile strength
from mean value is recorded as minimum
equal to 0.33% and maximum equal to
8.96%. The values are well with in the 15%
permissible limit.
3. RESULTS AND DISCUSSION
3.1 Compressive Strength:Weight and
compressive strength results of the cylinders
are then analyzed using in built function of
trend line analysis. The option offers trend
line analysis with linear, polynomial degree
two, exponential, logarithmic and power
functions. All of these functions are used to
analyze the data and the results are plotted in
figure 1. The obtained equations along with
the value of R2 are listed in table 4. The
value of R2 is an important parameter in
regression analysis which gives percentage
of variation of fitted data. Generally high the
value of the parameter represents higher
accuracy of data fit. For our analysis we
obtained about 35% value for R2 which
shows good agreement of fitted data. All
these functions are then used to reevaluate
the compressive strength as function of
weight of the cylinders. The obtained results
are analyzed for basic parameter. Table 5
gives details of minimum, maximum, mean,
median, and standard deviation of
compressive strength of all function
obtained by trend line analysis and
experimental results. This table also gives
details of correlation coefficient of all trend
line functions with respect to experimental
results. It is observed from table 5 that
although there is standard deviation of 92
psi is in the experimental results but
standard deviation of 55 psi is observed in
calculated data using trend line functions.
Also coefficient of correlation of 0.59 is
recorded for all function. Both standard
deviation and coefficient of correlation
shows that the fitted data is in good
agreement with the experimental results.
Table 4: Trend line fit for compressive data
Function Type Polynomial R2
Exponential y = 347.16e0.1736x
0.3411
Linear y = 651.55x - 5181.1 0.3500
Logarithmic y = 8924.5ln(x) - 19613 0.3498
Polynomial Degree 2 y = 1551.9x2 - 41874x + 286132 0.3560
Power y = 7.4231x2.3777
0.3409
Table 5: Statistical parameters of all function versus experimental data (Compressive Strength)
Parameter Experiment Exponential Linear Polynomial
D2 Power Logarithmic
Min 3452.89 3680.29 3679.98 3685.02 3679.60 3680.57
Max 3875.12 3810.31 3810.29 3814.64 3809.57 3810.85
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Mean 3748.48 3748.38 3748.39 3748.41 3747.74 3749.04
Median 3756.86 3744.73 3745.14 3734.31 3744.26 3745.95
Standard
deviation 92.36286 54.53404 54.64331 55.09461 54.50573 54.62948
Correlation
Coefficient - 0.591979 0.591618 0.596642 0.591829 0.591463
After analysis for basic parameters,
experimental data and compressive strength
obtained from trend line function is plotted
in figure 2 for comparison. Although all the
functions are in good agreement with the
experimental data for compressive strength
but it is observed that the power function
gives lowest deviation from mean strength
with minimum error of 0.01% and
maximum error of 10.33%. These error
values are least compared to other functions.
Therefore, power function obtained from
trend line analysis is taken as the standard
function to represent the relationship
between weight and compressive strength of
concrete cylinders and is listed in equation
(1)
y = 7.4231x2.3777
(1)
where x represents weight and y represent
compressive strength of concrete cylinder.
3.2Tensile Strength:
In similar fashion to compressive strength
data, tensile strength data is analyzed. The
results of trend line analysis for all available
functions is plotted in figure 3. The
functions obtained are listed in table 6. The
obtained functions are then used to
reevaluate the tensile strength. Basic
statistical information of the obtained tensile
strength results is given in table 7.
(a) Linear function (b) Polynomial degree two
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(c) Exponential function (d) Logarithmic function
(e) Power function
Figure 1: Trend line fit for compressive strength data
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Figure 2: Experimental results along with all trend line functions (Compressive Strength)
Table 6: Trend line fit for all tensile strength data
Function Type Polynomial R2
Exponential y = 0.162e0.5718x
0.9251
Linear y = 235.4x - 2815.6 0.9221
Log y = 3226.6ln(x) - 8035.9 0.9219
Polynomial D-2 y = 227.16x2 - 5992.8x + 39873 0.9244
Power y = 5*10-07
x7.8381
0.9249
For the comparison purpose all tensile
strength values versus experimental results
are plotted in figure 4. It may be visualized
from this figure that almost all the functions
give results in good agreement with
experimental values. It is noted that both
standard deviation of tensile strength given
by power function is less compared to other
type of functions. and coefficient of
correlation for power function. Although the
correlation coefficient for power function
.001 remained than that of the exponential
functions but the mean error of power
function is less compared to other functions.
Therefore, power function is taken as the
best fit of the experimental data for tensile
strength and is reproduced as under.
y = 5*10-07
x7.8381
(2)
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(a) Linear Function (b) Polynomial Degree two
(c) Exponential Function (d) Logarithmic Function
(e) Power Function
Figure 3: Trend line fit for tensile strength data
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Figure 4: Experimental results along with all trend line functions (Tensile Strength)
Table 7: Statistical parameters of all function versus experimental data (Tensile Strength)
Parameter Experiment Exponential Linear Log Polynomial
Degree 2 Power
Min 382.00 382.01 387.20 385.75 386.43 383.50
Max 439.00 428.22 434.30 432.86 432.71 429.99
Mean 419.59 414.97 420.96 419.53 419.07 416.68
Median 428.00 428.22 434.30 432.86 432.71 429.99
Standard deviation 18.73006 17.745 17.993 17.983 18.005 17.840
Correlation Coefficient - 0.961 0.960 0.960 0.961 0.961
Combining both equations (1) and (2) in one general equation it may be written as
𝑦 = 𝛼𝑥𝛽 (3)
Where 𝑥 represents weight of cylinders and
𝑦 represent concrete strength with 𝛼 equal to
7.4231 for compressive strength and 5x10-7
for tensile strength. Similarly, 𝛽 equals to
2.3777 for compressive strength and 7.8381
for tensile strength.
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4. CONCLUSION
Quality control of concrete require
determination of its strength particularly
compressive strength of cylinders / cubes
cured for 28-days. This require dedicated
laboratory setup. An alternative of it is the
numerical analysis. This research paper
presents relationship between weigh and
concrete strength (both compressive and
tensile) using trend line analysis. All of the
inbuilt options of trend line analysis of excel
are evaluated for compressive and tensile
strength data of 120 cylinders each. These
cylinders are prepared using 1:2:4 mix with
0.45 water-cement ratio and cured for 28-
days. It is observed that power function
gives results in best agreement to
experimental values. First separate equations
for relationship between weight and
compressive strength and weight and tensile
strength are presented then combined into
one equation by stating coefficients for
compressive and tensile strength evaluation.
It is hoped that the work presented in this
research paper will not only improve the
literature on the topic but also help the
practicing engineers to have good insight of
concrete strength from weight on site.
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