**Mathematical Regression Model for the Prediction of **

**Concrete Strength **

### M. F. M. Zain

1### , Suhad M. Abd

1### , K. Sopian

2### , M. Jamil

1### , Che-Ani A.I

11

_{Faculty of Engineering and Built Environment, }

2_{Solar Energy Research Institute, }

### Universiti Kebangsaan Malaysia,

### 43600 UKM Bangi, Selangor Darul Ehsan,

### Malaysia

*Abstract: - *In this study a new mathematical models were proposed and developed using non-linear

regression equation for the prediction of concrete compressive strength at different ages. The variables used in the prediction models were from the knowledge of the mix itself, i.e. mix proportion elements. According to the analysis the models provide good estimation of compressive strength and yielded good correlations with the data used in this study. The correlation coefficients were 0.995 and 0.994 for the prediction of 7 and 28 days compressive strength respectively. Moreover, the proposed models proved to be significant tool in prediction compressive strength of different concretes in spite of variations in the results.

**Key-Words: - mathematical model, statistical analysis, compressive strength, strength prediction, concrete**

**1 Introduction **

Concrete is such a construction material that is widely used in the world. The advantages of concrete are low cost, availability of construction, workability, durability and convenient compressive strength that make it popular near engineers and builders. However, these advantages seriously depend on the correct mix, placing and curing [1]. In 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 [2].

Neither waiting 28 days from such a test would serve the rapidity of construction, nor neglecting it, would serve the quality control process on concrete in large construction sites. Therefore, rapid and reliable prediction for the strength of concrete would be of great significance [3]. For example, it provide a chance to do the necessary adjustment on the mix proportion used to avoid situation where concrete does not reach the required design strength or by avoiding concrete that is unnecessarily strong, and also, for more economic use of raw materials and fewer construction failures, hence reducing construction cost .

Prediction of concrete strength, therefore, has been an active area of research and a Considerable number of studies have been carried out. Many

attempts have been made to obtain a suitable mathematical model which is capable of predicting strength of concrete at various ages with acceptable (high) accuracy [4].

**2 Statistical Analysis for Strength **

**Prediction **

The strengthening of concrete is a complex process involving many external factors. A number of improved prediction techniques have been proposed by including empirical or computational modeling, statistical techniques and artificial intelligence approaches. Many attempts have been made for modeling this process through the used of computational techniques such as finite element analysis. While, a number of research efforts have concentrated on using multivariable regression models to improve the accuracy of predictions. Statistical models have the attraction that once fitted they can be used to perform predictions much more quickly than other modeling techniques, and are correspondingly simpler to implement in software.

S. Popovics [5], augments Abrams model, a widely accepted equation relating the water cement ratio w/c of concrete to its strength with additional variables such as slump, and uses least square regression to determine equation coefficients. Using this approach improved strength prediction and insights into concrete compositions were achieved. M. Nagesha et al. [2], used multivariable

regression techniques on concrete composition data to predict 28 days compressive strength with reasonable accuracy, proposing a formula readily applicable for on-site use.

Apart of its speed, statistical modeling has the advantage over other techniques that it is mathematically rigorous and can be used to define confidence interval for the predictions. This is especially true when comparing statistical modeling with artificial intelligence techniques. Statistical analysis can also provide insight into the key factors influencing 28 days compressive strength through correlation analysis. For these reasons statistical analysis was chosen to be technique for strength prediction of this paper [2].

**3 Experimental Program **

Physical properties of the materials used in this study are shown in Table (1). Locally produced ordinary Portland cement (OPC) was used. It has a specific gravity of 3.1 and specific surface of 3500 m2/kg. Fineness modulus was 2.82 for fine aggregate. 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) to study the effect of the ordinary mix proportion on the compressive strength of concrete.

Since the aim of this study is studying 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 test was performed and evaluated in accordance to BS 1881: Part 116:1983. Specimens were immersed in water until the day of testing at 3, 7, 28 days. Table (3) show the compressive strength test results. Table 4 shows the results of compressive strength test at the age of 3, 7 and 28 days

### Table 1: Physical Properties of Materials

### Table 2: Chemical composition of OPC

Table 3: Details For Mix Proportions

Table 4:** **Compressive Strength for the

Concrete at 3, 7 and 28 Days

**Mix no. ** **Compressive strength **

**(MPa) **
**Density **
**(Kg/m3) **
**3 **
**days **
**7 **
**days **
**28 **
**days **
1 2333.25 17.9 24.5 34
2 2323.2 17.4 22.5 32.5
3 2310 16.3 21.6 32.5
4 2300 16.1 21.5 32.3
5 2293.6 15 21.1 30.5
6 2275.5 14.6 20.4 30.3
7 2268 14.1 20.3 29.2
8 2244 14.1 20 28.9
9 2234 13.9 18.5 27.7
10 2203 13.7 17.6 25.9
11 2176 13.2 17.3 24.5
12 2148 12.3 14.6 23.8
**Mix **
**No ** **Water **
**Kg/m3**
**Cem **
**Kg/m3**
**Sand **
**Kg/m3**
**Agg **
**Kg/m3**
**w/c ** **Density **
**Kg/m3**
1 _{180 400 600 1200 }0.45 _{2333.2 }
2 _{195 390 588 1170 }0.5 _{2323.2 }
3 209 380 570 1140 0.55 2310
4 _{222 370 555 1110 }0.6 _{2300 }
5 _{234 360 540 1080 }0.65 _{2293.6 }
6 _{245 350 525 1050 }0.7 _{2275.5 }
7 146 325 650 1300 0.45 2268
8 _{160 320 640 1280 }0.5 _{2244 }
9 _{173 315 630 1260 }0.55 _{2234 }
10 _{186 310 620 1240 }0.6 _{2203 }
11 198 305 610 1220 0.65 2176
12 _{210 300 600 1200 }0.7 _{2148 }
13 _{233 517 517 1034 }0.45 _{2430 }
14 _{252 504 504 1008 }0.5 _{2421 }
15 270 491 491 982 0.55 2378
16 _{287 479 479 958 0.6 } _{2374 }
17 _{304 468 468 936 }0.65 _{2356 }
18 _{320 457 457 914 0.7 } _{2352 }
**Materials Properties **
Cement (C)
Ordinary Portland
Cement (OPC)
Specific Gravity: 3.1
Specific surface (by Blain) : 3500

cm2_{\g}
Fine Aggregate
(FA):
Sand (S)
Specific Gravity: 2.60
Fineness Modulus: 2.34
Coarse Aggregate
(CA):
Crushed Stone
Specific Gravity: 2.7
Maximum particle size: 20 mm

**Oxide ** **(%) **

Silicon dioxide (SiO2) 22.1

Aluminum Trioxide (Al2O3) 5.96

Ferric oxide (Fe2O3) 3.04

Calcium oxide (CaO) 61.5

Magnesium oxide (MgO) 2.5

Sodium oxide (Na2O) 0.16

Loss on ignition (L.O.I) 1.50

Insoluble residue (I.R) 1.10

13 2430 26.1 31 44 14 2421 23 29.9 39.4 15 2378 21.4 28.3 37.5 16 2374 19.6 26.7 36.1 17 2356 19.5 25.8 35.2 18 2352 18.4 25.7 34.6

**4 Modelling the Prediction of **

**Compressive Strength of Concrete **

The most popular regression equation used in
prediction of compressive strength prediction is:
*c*

*w*

*b*

*b*

*f*

### =

_{0}

### +

_{1}

### /

...**Eq.1**where:

ƒ: compressive strength of concrete w/c: water/cement ratio

b0,b1: coefficients

The previous equation is the linear regression equation .The origin of this equation is Abram’s Law [5] which relate compressive strength of concrete to the w/c ratio of the mix and according to this law, increasing w/c ratio will definitely lead to decrease in concrete strength. The original formula for Abram is:

*c*
*w*

*B*

*A*

*f*

### =

_{/}...

**Eq.2**where:

ƒ: compressive strength of concrete A, B: empirical constants

Lyse [6] made a formula similar to Abram but he
relate compressive strength to cement /water ratio
and not water /cement ratio. According to Lyse
strength of concrete increase linearly with increasing
c/w ratio .the general form of this popular model
was:
*w*
*c*
*B*
*A*
*f* = + / ... ...**Eq.3**
Where:

ƒ: compressive strength of concrete c/w: cement /water ratio

A, B: empirical constants

The quantities of cement,fine aggregate and coarse aggreagate were not included in the model and not accounted for the prediction of concrete strength.So, for various concrete mixes were their w/c ratio is constant ,the strength will be the same and this is not true.Therefore, efforts should be concenerate on models taken into account the influence of mix constituents on the concrete

strength in order to have more reliable and accurate results for the prediction of concrete strength.

So, Eq. 1 which reffered to Abrams Law was extended to include other variables in the form of multiple linear regression equation and used widely to predict the compressive strength of various types of concrete as below:

*c*

*w*

*b*

*b*

*f*

### =

_{0}

### +

_{1}

### /

...**Eq.1**

linear least square regression (reffered to Abram)

*C*

*FA*

*b*

*CA*

*b*

*c*

*w*

*b*

*b*

*f*

### =

_{0}

### +

_{1}

### /

### +

_{2}

### +

_{3}

### +

...**Eq.4**

multiple linear regession Where:

ƒ: compressive strength of concrete w/c: water/cement ratio

C: quantity of cement in the mix

CA: quantity of coarse aggregate in the mix FA: quantity of fine aggregate in the mix

According to Eq. 4 all the variables related to
the compressive strength in a linear fashion, but this
is not always true because the variables involved in
a concrete mix and affecting its compressive
strength are interrelated with each other and the
additive action is not always true. Here, it appears
that there is a need to another type of mathematical
model can reliably predicts strength of concrete
with acceptable high accuracy. So, if we took the
general form of the multiple linear regressions as
below;
*m*
*m*

*X*

*a*

*X*

*a*

*X*

*a*

*X*

*a*

*a*

*Y*

### =

_{0}

### +

_{1}

_{1}

### +

_{2}

_{2}

### +

_{3}

_{3}

### +

### ....

...….... multiple linear regression (**Eq.4**)

For situations where the multiple dependency is curvilinear (non-linear) the logarithmic transformation can be applied to this type of regression [7]:

)
log(
...
)
log(
)
log(
)
log(
)
log(
)
log(*y* = *a*0 +*a*1 *X*1 +*a*2 *X*2 +*a*3 *X*3 + *am* *Xm*

### ...

**Eq.5**

### This equation could be transform back to a

### form that predicts the dependent variable (Y)

### by taking the antilogarithimto yeild an

### equation of the type:

*m*
*a*
*m*
*a*
*a*
*a*

*X*

*X*

*X*

*X*

*a*

*Y*

1### .

2### .

3### ...

3 2 1 0### =

### ..

**Eq.6**

engineering, variables are often dependent on several independent variables, this functional dependency is best characterized by the equation mentioned earlier, and is said to give results that are more realistic too. In this study, the multivariable power equation was found to be very suitable for prediction strength of concrete (as a dependent variable). Factors affecting this strength were the elements of the concrete mix itself.

**5 Results And Discussion **

It is very important to analyze the effect of mix constituents on strength of concrete. Mix design is a specific combination of raw materials that are used in a particular concrete to reach a given target strength. So the significant factor in 28 days compressive strength is the concrete composition. Concrete theory suggested that water to cement ratio (w/c) of concrete is a primary factor influencing the strengthening process, both the final strength and the rate of hardening are affected [2]. Also, it is well known that decreasing water content increases strength for the concrete. This explanation is well represented in Figure (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. Conventionally, strength is related to density and the denser the concrete the higher the strength as shown in figure (2). Furthermore, strength of concrete is highly affected by cement content and amount of fine and coarse aggregate used in the mix as well as any other aditional material added to the mix in order to improve specific property for the concrete like fly ash, silica fume and slag or admixture like superplasticizer.

Fig. 1 Relationship Between 28 Days Compressive Strength and (w/c) Ratio

Fig. 2 Relationship Between 28 Days Compressive Strength and Density

Table (4) shows the relatioship between the compressive strength at the age of 7 and 28 days with the variables provided from the experimental work and are going to be used in the proposed model. This relationship is represented by the correlation coefficient between each variable and each strength.From this table, it can be seen that some variables have significant correlation with the predicted strength at the specified age.The highest correlations were density followed by the cement content in the mix.

Table 5. Correlations between 7&28 Days compressive Strength and Variables Used in the

Proposed Model
**Variable **
**7 Days **
**Compressive **
**Strength**
**28 Days **
**Compressive **
**Strength**
Water/cement(w/c) 0.379 0.41
Water (W) 0.580 0.538
Cement (C) 0.970 0.95
Sand (FA) 0.723 0.680
Aggregate (CA) 0.723 0.683
Density (ρ) 0.98 0.986

After analyzing the influence of mix constituent on the strength at the age of 7 and 28 days, the proposed model was used to predict compressive strength at the specified ages comprises all the variables mentioned earlier.The final form of the proposed strength prediction model for both ages was:

6
5
0
7 . . . /
ƒ _{=} _{a}* _{C}a*1

_{W}*a*2

*3*

_{FA}a

_{CA}*a*4

_{ρ}

*a*

_{w}*………..*

_{c}a**Eq. 7**6 5 0 28 . . . / ƒ

*a*1

*a*2

*a*3

*a*4

*a*

*a*

*c*

*w*

*CA*

*FA*

*W*

*C*

*a*ρ =

### ...

**Eq.8**

### Table (6) gives the regression coefficients of the

### prediction model above, for the prediction of 7

### and 28 days compressive strength respectively,

### as well as the value of coeffient of correlation

### C.C (R).

### Figure (3) and Figure (4) show the relationship

### between the observed(actual) value of the

### compressive strength obtained from the

### experimental work, and the predicted values

### obtained using the proposed model for 7 and 28

### days respectively. It is so obvious that almostly

### 99% of the data lacated on the line of equality

### which means that the actual and the predicted

### values for the concrete compressive strength are

### identical with each other.This is quite true

### because the correlation coefficients were 0.995

### for 7 days prediction and 0.994 for 28 days

### prediction.Moreover almostly 99.047% of the

### variance was explained for the 7 days prediction

### while 98.8% of the variance was explained for

### the 28 days predictions.

Table 6: Regression coefficients for the 7 & 28 days compressive strength prediction models

Coefficient 7 days prediction model 28 days prediction model A0 0.2335 0.34262 A1 -4.8139 -28.7310 A2 4.0703 28.0856 A3 -4.1368 -28.3023 A4 -3.9896 -1.9259 A5 2.5945 0.72819 A6 1.4920 1.61814 C.C 0.995 0.994 Variance explaine 99.047% 98.8%

Fig. 3 Relationship Between the Observed and Predicted Values for 7 Days Compressive Strength

Fig. 4 Relationship Between the Observed and Predicted Values for 28 Days Compressive Strength

**6 Comparison with Other Data**

To test the proposed model obtained from this study ,it was decided apply the model using data from other sources or data from other researchers .This comparison is very important to check the validity of the proposed model for the prediction of 7 and 28 days compressive strength of concrete for any set of data. These data were imported from literature belong to Jee Namyong et al. [6]. Table (7) shows full details of the data imported and used to check the proposed model. The data comprises on 59 different kind of mixture with specified compressive strength of 18-27 MPa, w/c ratio of 0.39-0.62, maximum aggregate size of 25 mm and slump of 12-15 cm.

The reason to choose this set of data is, the large number of concrete mixes (which mean large number of sample) and these mixes were from different plants of ready mix concrete ,and this is also a good prove that the proposed model could valid even for ready mix concrete . Another reason that these data were from Korea, so, this is also a good prove that the model could work for any type and any place inspite of variation of data. Variation in concrete strength of the test specimens depends on how well the materials, concrete manufacture and testing is controlled. Especially construction practices may cause variation in strength of in-situ concrete due to inadequate mixing, poor compaction, delay and improper curing [6].

The variables used in the model were these available from the data .The correlation coefficient for the prediction of 28 days compressive strength was 0.7579 and 0.7267 for 7 days prediction, these results consider to be good results concerning the variations in the data. Moreover, there are some relationships in previously published studies that can predict the 28 days compressive strength from

So, if we use the concept of early age strength to predict later age strength in this case, i.e, the strength at 7 days (ƒ7) will be one of the variables

used in the model. The coefficient of correlation in this case will improve significantly from 0.7579 to 0.866 which prove the importance of this concept. Table 7. Data for 7 & 28 days compressive strength of the proposed models

**Weight of unit volume (kg/m3 _{) }**

**Compressiv**
**e strength **
**(MPa) **
**w/c **
**% **
**S/a **
**% ** **W ** **G ** **S ** **g **
**Age**
**nt ** **7 D ** **28 D **
60.21 51.2 174 289 933 900 0.86 15.5 21.2
59.74 52.1 184 308 927 860 0.91 16.3 24.2
60.6 52.1 183 302 926 860 0.91 16.3 23
57.48 52.7 173 301 961 863 0.9 21.5 26.2
60.32 50.9 190 315 904 862 0.47 18.6 24
61.49 51.2 190 309 911 859 0.46 17.4 22.5
59.55 46.1 184 309 821 975 0.92 15.8 22.6
50 48.5 164 328 886 942 1.64 23.2 34.7
47.83 45.1 176 368 805 988 0.77 19.1 26.9
49.44 48.8 178 360 858 914 1.08 23.3 30.7
52.35 47.9 178 340 839 931 0.51 22.6 28.8
44.47 44.9 165 371 810 1000 1.85 20.7 27.6
44.69 47.6 164 367 847 940 1.84 18.9 28.5
48.56 49.6 169 348 882 902 1.74 24.1 31.8
48.92 49.4 181 370 866 887 1.11 23 30.5
50 49.5 171 342 885 913 1.71 23 31.6
49.73 49.9 181 364 865 874 1.09 21.6 31.7
44.75 48.8 179 400 835 894 2.8 22 30.2
45.34 46.1 180 397 790 939 2.78 22.4 30.2
46.56 43.9 183 393 759 981 1.18 20.4 29.7
50 46 175 350 804 955 1.05 16 29.8
47.04 44.7 183 389 778 962 1.17 19.8 26.7
47.3 44.7 184 389 778 962 1.17 18.1 25.3
48.04 47.1 184 383 810 924 1.15 20.1 27.8
48.41 47.4 183 378 807 902 1.13 22 29.1
48.41 47.4 183 378 818 924 1.13 22.8 29.2
47.79 47.6 184 385 812 922 1.16 21.2 27.6
45.69 48.3 175 383 846 905 0.77 22.8 28.9
46.76 50.3 173 370 889 878 1.11 21.3 27.3
46.74 50.5 179 383 879 862 1.15 21.2 27.6
44.21 49.0 168 380 868 903 1.9 21.1 29.1
47.77 46.1 182 381 812 956 0.8 20 26.2
45.14 47.2 172 381 815 940 1.91 21.5 28.3
48.41 47.4 183 378 807 923 1.13 20.7 27.5
45.89 43.5 184 401 754 978 2.01 23.7 31.3
48.56 45.8 185 381 800 946 1.91 21.9 28.7
45.69 48.3 175 383 863 888 1.92 20.6 28.2
47.78 45.1 172 360 794 965 1.08 21 27.7
45.87 44 189 412 730 947 1.65 25 30.9
45.99 43.9 189 411 732 951 1.44 20.7 30.7
42.76 46.3 180 421 784 927 0.85 22.7 30
40.89 42.5 184 450 713 977 1.35 22.6 30.7
40.62 42.0 184 453 704 983 1.59 23.6 31
41.97 47.5 183 436 804 889 0.87 22.8 29.8
44.1 48.3 187 424 812 882 1.27 22.4 30.2
43.57 46.8 183 420 785 920 1.26 22.8 30.7
44.31 46.6 183 413 780 921 1.24 23.9 30.1
44.31 46.6 183 413 791 923 1.24 23.6 30.9
40.98 46.4 168 410 811 936 2.05 22.6 33.3
42.45 47.7 180 424 813 891 1.27 22.9 30.8
44.31 46.6 183 413 783 956 0.87 20.8 29.3
48.4 47.6 196 405 786 879 0.61 22.4 30.7
45.17 48.5 187 414 818 883 1.24 22.8 29.5
44.31 46.6 183 413 780 900 1.24 25.9 34.3
44.31 46.6 183 413 783 914 1.24 24.5 32.4
39.37 45.2 176 447 760 970 1.34 24.7 29.4
43.75 44.7 182 416 771 954 2.08 23.4 30.8
41.47 45.5 175 422 782 937 0.84 23.7 31.5
43.85 44.6 171 390 768 969 1.17 24.9 30.6

**7**

**Conclusions**

From this study ,a mathemaical reggretion model was developed.

i) The importance of the influence of mix constituents on the strength of concrete was approved

ii) Previouse models that deal with the prediction of concrete compressive strength lack of including other variables affecting strength gaining in concrete.

iii) A mathematical models for the prediction of concrete compessive strength at the ages of 7 and 28 were proposed and developed (using non-linear regressions) from the knowledge of the mix constituents, i.e, the variables used are the mix proportions elements.

The prediction models developed in this study are:

5 6
0
7 . . . /
ƒ = _{a}* _{C}a*1

_{W}*a*2

*3*

_{FA}a*4 ρ*

_{CA}a*a*

_{w}*5 6 0 28 . . . / ƒ*

_{c}a*a*1

*a*2

*a*3

*a*4

*a*

*a*

*c*

*w*

*CA*

*FA*

*W*

*C*

*a*ρ =

iv) These models prove to be used with any set of data inspite of variations in test results of the concrete in question.

v) The concept of using early age strength to predict strngth at later ages proved to be valid and could be used sucussfuly.

*References: *

[1] N. Hamid-zadeh, A. Jamali, N. Nariman-zadeh, H., A Polynomial Model For Concrete Compressive Strength Prediction Using GMDH-type Neural Networks and Genetic Algorithem [2]Darren Williams, Concrete Strength Prediction

From Early-Age Data-Technical Paper, Honour Project, Technical Paper, University of Adelaide [3]Suhad M.A., Mathematical model for the

prediction of cement compressive strength at the ages of 7&28 days within 24 hours, MSc Thesis, Al-Mustansiriya University, college of engineering, civil engineering department, 2001.

[4] Kheder G.F.,Al-Gabban A.M. & Suhad M.A.
Mathematical model for the prediction of cement
compressive strength at the ages of 7&28 days
within 24 hour *materials and structures* 2003.

36: 693-701.

[5] Sandor popovics, Analysis of Concrete Strength Versus Water-Cement Ratio Relationship, ACI Material Journal,Vol.87, No.5, September-October 1990, Pp.517-529

[6] Jee Namyong, Yoon Sangchun, Cho Hongbum, Prediction of Compressive Strength of In-Situ Concrete Based on Mixture Proportion, Journal of Asian Architecture and Building Engineering, 16, may 2004.

[7]Steven, C. Chapra, Raymond & P. Canale. Numerical Methods for engineers with personal computer applications, 1989.