INTRODUCTION
Concrete is a construction material that is widely used throughout the world.
The advantages of concrete include its low cost, availability of construction,
workability, durability and convenient compressive strength, which make it popular
among engineers and builders (Abd et al., 2008).
However, these advantages are strongly dependent on correct mixing, placing
and curing. In the construction industry, strength is a primary criterion in
selecting a concrete for a particular application. Concrete used for construction
gains strength over a long period of time after pouring. The characteristic
strength of concrete is defined as the compressive strength of a sample that
has been aged for 28 days (Hamidzadeh et al., 2006).
Waiting 28 days to complete such a test would not suit the construction industry’s
need for speed, while neglecting the test would not satisfy the quality control
process of concrete at large construction sites. Therefore, rapid and reliable
prediction of the strength of concrete would be of great significance (Kheder
et al., 2003). The ability to rely on model predictions would enable
mix proportion adjustments in advance of placing and curing; this predictionbased
technique would enable the avoidance of situations in which the concrete is
too weak or unnecessarily strong and would facilitate a more economic use of
raw materials and fewer construction failures, hence reducing construction costs.
Prediction of compressive strength for cement and concrete, therefore, has
been an active area of research. A considerable number of studies have been
carried out in this area (Zain and Suhad, 2009; Kheder
et al., 2003; Zain et al., 2002; Tsivilis
and Parissakis, 1995; Zelic et al., 2004;
Akkurt et al., 2004; Hwang
et al., 2004; Sang et al., 2003).
Many attempts have been made to obtain a suitable mathematical model that is
capable of predicting the strength of concrete at various ages with acceptably
high accuracy (Popovics, 1990; Jee
et al., 2004; Steven et al., 2002; Mehta
and Monteiro, 2006). This study proposes a mathematical model to predict
concrete strength using concrete characteristics from its mix proportion elements.
STATISTICAL ANALYSIS FOR STRENGTH PREDICTION
The strengthening of concrete is a complex process involving many external factors. Multiple attempts have been made to model this process. A number of improved prediction techniques have been proposed by including empirical or computational modelling, statistical techniques and artificial intelligence approaches. Some models have used computational techniques such as finite element analysis, whereas others have used multivariable regression models to improve prediction accuracy. Statistical models are attractive in that, once fitted, they generate predictions much more quickly than other modelling techniques and are correspondingly simpler to implement in software.
There are many factors affecting concrete strength. Water to cement ratio (w/c)
consider to be one of the most important factor in this respect. Many research
works attempt to utilise this factor in models relating strength of concrete
with factors affecting it. Popovics (1990) augmented
Abram’s model, a widely accepted equation relating the water/cement ratio
(w/c) of concrete to its strength, with additional variables such as slump;
he used least squares regression to determine equation coefficients. This approach
improved strength prediction and provided insights into concrete compositions.
Apart from its speed, statistical modelling has two advantages over other techniques: it is mathematically rigorous and it can be used to define confidence intervals for the predictions. These advantages are particularly apparent when comparing statistical modelling with artificial intelligence techniques. Statistical analysis can also provide insight into the key factors influencing 28day compressive strength through correlation analysis. For these reasons, statistical analysis was chosen as the technique for strength prediction in this study.
EXPERIMENTAL PROGRAM
This study was conducted in National University of MalaysiaUKM between 2007
and 2010. The physical properties of the materials used are shown in Table
1. Locally produced Ordinary Portland Cement (OPC) was used; its chemical
composition is shown in Table 2. The fineness modulus was
2.82 for the fine aggregate.
Table 1: 
Physical properties of materials 

Table 2: 
Chemical Composition of OPC 

Table 3: 
Details for mix proportions 

The coarse aggregate was 20 mm maximum size crushed stone; its specific gravity
was 2.7. No admixtures or additives were used in this study; only the ordinary
constituents of concrete (cement, sand, gravel, water) were used to study the
effect of an ordinary mix proportion on the compressive strength of concrete.
Since the aim of this study was to determine the effect of mix proportions on
the compressive strength of concrete, different mixes were used. The details
of all mix proportions are shown in Table 3.
Compressive strength tests were performed and evaluated in accordance with BS 1881: Part 116:1983. Specimens were immersed in water until the day of testing.
MODELLING THE PREDICTION OF CONCRETE COMPRESSIVE STRENGTH
The most popular regression equation used in the prediction of compressive strength is:
where, f is the compressive strength of concrete, w/c is the water/cement ratio
and b_{0} and b_{1} are coefficients. Equation
1 is the linear regression equation for Abram’s Law (Popovics,
1990), which relates the compressive strength of concrete to the w/c ratio
of the mix. According to this law, an increase in the w/c ratio leads to a decrease
in concrete strength. The original formula for Abram’s Law is:
where, f is the compressive strength of concrete and A and B are empirical constants.
Lyse (Jee et al., 2004) established a formula
similar to that of Abram, but he related the compressive strength to the cement/water
(c/w) ratio rather than the water/cement ratio. According to Lyse, the strength
of concrete increases linearly with increasing c/w ratio. The general form of
this popular model is:
where, f is the compressive strength of concrete, c/w is the cement/water ratio and A and B are empirical constants.
The quantities of cement, fine aggregate and coarse aggregate were not included in the model and did not influence the predicted concrete strength. Thus, any concrete mix with the same w/c ratio is predicted to have an equivalent strength, but this is not empirically true. Therefore, models should take into account the influence of mix constituents on the concrete strength to yield more reliable and accurate predictions of concrete strength.
Consequently, Eq. 1 was extended to include other variables in the form of multiple linear regression equations. The resulting Eq. 4 has been used widely to predict the compressive strengths of various types of concrete:
Equation 1 is linear least squares regression and Eq. 4 multiple linear regression:
where, f is the compressive strength of concrete, w/c is the water/cement ratio, C is the quantity of cement in the mix, CA is the quantity of coarse aggregate in the mix and FA is the quantity of fine aggregate in the mix.
According to Eq. 4, all of the variables relate to the compressive strength in a linear fashion; however, this is not always true, because the variables involved in a concrete mix and affecting its compressive strength are interrelated to each other and the additive action does not always increase strength. Thus, there is a need for another type of mathematical model that can reliably predict strengths of concrete with high accuracy. Consider the general form of the multiple linear regression as below:
multiple linear regression
For situations in which the multiple dependencies are curvilinear (nonlinear),
logarithmic transformation can be applied to this type of regression (Steven
et al., 2002):
This equation can be transformed back to a form that predicts the dependent variable (Y) by taking the antilogarithm to yield:
This equation is called the multivariable power equation (Steven
et al., 2002). In engineering, when variables are dependent on several
independent variables, this functional dependency is best characterised by Eq.
7, which gives more realistic results. In this study, the Multivariable
Power Equation was found to be very suitable for predicting the strength of
concrete. The factors affecting this strength were the elements of the concrete
mix itself.
RESULTS AND DISCUSSION
It is very important to analyse the effects of mix constituents on the strength of concrete. The mix design is the specific combination of raw materials that are used in a particular concrete to achieve a given target strength. In 28day compressive strength, the significant factor was the concrete composition. Concrete theory suggests that the watertocement ratio (w/c) of concrete would be the primary factor influencing the strengthening process, affecting both the final strength and the rate of hardening. Additionally, decreasing water content has been shown to increase the strength of concrete.
Furthermore, strength is related to density; the denser the concrete, the higher
the strength. The w/cstrength relationship in concrete can be easily explained:
an increase in the w/c ratio increases the porosity of the cement paste at a
given degree of hydration and this increase in porosity weakens the whole matrix
and leads to a decrease in the concrete strength. This relationship would be
clearer in highstrength concrete where very low w/c ratios are used to achieve
a large increase in compressive strength. This increase would be attributed
mainly to a significant improvement in the strength of the interfacial transition
zone at very low w/c ratios (Mehta and Monteiro, 2006).
This explanation is well represented in Fig. 1 which shows
the relationship between the 28 days compressive strength and the water to cement
ratio (w/c) for the concrete used in this study.

Fig. 3: 
The combined effect of both density and watertocement ratio
on 28day compressive strength 
Furthermore, it is well known that strength is related to density and the denser
the concrete the higher the strength as shown in Fig. 2.
The combined effect of both the w/c ratio and the density is well represented
in Fig. 3 and 4, which show a 3D relationship
between the 28day compressive strength and 7day compressive strength with
both density and w/c ratio. From this relationship, the highest compressive
strength of concrete occurred for the area of maximum density and minimum w/c
ratio. This outcome makes logical sense given that a decreased w/c ratio leads
to increased strength and lowered permeability for an overall increase in density
(Popovics, 1992). Figure 1 clearly shows
this trend for the 28day compressive strength.

Fig. 5: 
The combined effect of both cement content and watertocement
ratio on 28day compressive strength 
The density of concrete is higher at 28 days due to the continued hydration
process and reactions between cement constituents.
Furthermore, the strength of concrete is highly affected by the cement content,
as shown in Fig. 5. The maximum strength obtained in this
study resulted from using a high cement content with a minimum w/c ratio. On
the other hand, increased cement content led to increased density and consequently
increased the strength of concrete at both 7 and 28 days, as shown in Fig.
6 and 7.
The amount of fine and coarse aggregate used in the mix, as well as any other
additional materials added to the mix, would also significantly affect the strength
development of the concrete, since these materials are added to the mix to improve
a specific property of the concrete.
Additives can include fly ash, silica fume and slag or an admixture like superplasticiser.
Strength is also affected by the quality of the constituent materials and the
mixing and curing methods (Zain et al., 2002).
The compressive strength of concrete also depends on the W/B ratio. The lower
the W/B ratio, the higher the strength of the concrete, because strength of
concrete is mainly determined by the strength of the paste and the strength
of its bond to the aggregate and w/c ratio plays a great role in this stage
(Neville, 1996).
Table 4: 
Correlations between 7 and 28day compressive strength and
variables used in the proposed model 

Table 5: 
Regression coefficients for the 7 and 28day compressive
strength prediction models 

Table 4 shows the relationship between the compressive strength
at 7 and 28 days and the variables collected from the experimental work. These
table entries were used in the proposed model. The relationship between 7 and
28 days and the variables in Table 4 is represented by the
correlation coefficient between each variable and each strength. Table
5 reveals that some variables correlate significantly with the predicted
strength at the specified age. The highest correlation was for density, followed
by the cement content in the mix.
After analysing the influence of mix constituents on the compressive strength at 7 and 28 days, the proposed model was used to predict the compressive strength at the specified ages while incorporating all of the variables mentioned previously. The final form of the proposed strength prediction model for both ages was:
The regression coefficients of the prediction model above, for the predictions of 7 and 28day compressive strength, respectively, are given in Table 5. The value of the coefficient of correlation (CC) is also given.
Figure 8 and 9, show the relationship between
the actual and predicted values of the compressive strength at 7 and 28 days,
respectively.
Almost 99% of the data are located on the line of equality, which means that
the actual and predicted values for the concrete compressive strength were identical.
The correlation coefficients provide further evidence of this agreement: 0.99538
for the 7day prediction and 0.99537 for the 28day prediction. Moreover, the
7day prediction explained almost 99.047% of the variance while the 28day prediction
explained 98.8% of the variance.
COMPARISON WITH OTHER DATA
To validate the proposed model presented in this study, the model was tested
using data from other researchers (Jee et al., 2004).
Table 6 shows the full details of the data imported and used
to verify the proposed model. The data consist of 59 different kinds of mixtures
with specified compressive strengths of 1827 MPa, w/c ratios of 0.390.62,
maximum aggregate sizes of 25 mm and slumps of 1215 cm.
This data set was selected for its large number of concrete mixes (large number
of samples), which came from different readymix concrete plants.
Table 7: 
Observed (actual), predicted values of the 28 days compressive
strength and difference between both of them 

Success of the model with this data set would provide good proof that the proposed model was valid even for readymix concrete.
Variations in the concrete strength of the test specimens depend on how well
the materials and concrete manufacturing and testing processes are controlled.

Fig. 10: 
Relationship Between the Observed and Predicted Values for
28 Days Compressive Strength (Jee et al., 2004) 
Construction practices may cause considerable variations in the strength of
insitu concrete due to inadequate mixing, poor compaction, delay and improper
curing (Jee et al., 2004). The variables used
to validate the model were those available from the data set. The correlation
coefficient was 0.7579 for the prediction of the 28day compressive strength
and 0.7267 for the 7day prediction; these were considered to be good results
given the variations in the data.
Some relationships presented in previously published studies can predict the
28day compressive strength from 7day values (Hamidzadeh
et al., 2006), or even from earlier values (Kheder
et al., 2003). If we use the 7day strength to predict the 28day
strength, the coefficient of correlation in this data set improves significantly,
from 0.7579 to 0.866, which proves the importance of this concept (using early
strength to predict strength at later ages). Using this model, the observed
(actual) and the predicted values of 28 days compressive strength with the difference
between them are given in Table 7 and the relationship is
plotted in Fig. 10.
CONCLUSIONS
In this study, mathematical regression models for the prediction of concrete
compressive strength at 7 and 28 days were proposed and developed using nonlinear
regression. The models, which relied on knowledge of the mix constituents, are
as follows:
The importance of the influence of mix constituents on the strength of concrete was demonstrated. Previously models proposed by other researchers for predicting concrete compressive strength do not incorporate variables from the mix proportions elements that are affecting strength of concrete. The models developed in this research were proven effective with another set of data, despite variations in test results of the concrete in question. The concept of using earlyage strength to predict strength at later ages proved to be valid and could be used successfully.