RELATIONSHIP BETWEEN COMPRESSIVE STRENGTH ANDMODULUS OF ELASTICITY OF HIGH-STRENGTH CONCRETE

Fuminori Tomosawa and Takafumi NoguchiDept. of Architecture, Fac. of Engineering, Univ. of Tokyo

Modulus of elasticity of concrete is frequently expressed in terms of compressive strength.While many empirical equations for predicting modulus of elasticity have been proposed bymany investigators, few equations are considered to cover the whole data. The reason isconsidered to be that the mechanical properties of concrete are highly dependent on theproperties and proportions of binders and aggregates. This investigation was carried out as apart of the work of the Research Committee on High-strength Concrete of the ArchitecturalInstitute of Japan (AlJ) and National Research and Development Project, called New RCProject, sponsored by the Ministry of Construction. More than 3,000 data, obtained by manyinvestigators using various materials, on the relationship between compressive strengths andmodulus of elasticity were collected and analyzed statistically. The compressive strength ofinvestigated concretes ranged from 20 to 160 MPa. As a result, a practical and universalequation is proposed, which takes into consideration types of coarse aggregates and types ofmineral additions.

INTRODUCTION

Modulus of elasticity of concrete is a key factor for estimating the deformation of buildingsand members, as well as a fundamental factor for determining modular ratio, n, which is usedfor the design of section of members subjected to flexure. Based on the property of modulus ofelasticity of concrete that it is proportional to the square root of compressive strength in therange of normal concrete strength, AlJ specifies the following equation to estimate modulus ofelasticity.

Eq.1 is applied to concrete of a specified design strength 36MPa or less which is defined asnormal strength concrete. A number of experiments have revealed that the modulus of elastic-ity calculated by Eq.1 become higher than the actual values as the compressive strength

increases, as shown in Fig. 1. Accordingly, this study aims to derive a practical and universalequation which is applicable to high-strength concretes with compressive strengths of over36MPa, by regression analysis of numerous results of experiments published in Japan. Theoutline of this study was published in 1990 [1].

REGRESSION ANALYSIS PROGRAM

Before commencing the analysis, it was necessary to create a basic form of the equation formodulus of elasticity. In this study the authors adopted the conventional form, in whichmodulus of elasticity, E, is expressed as a function of compressive strength a and unit weight,

. Since it is self-evident that the concrete with a compressive strength of 0MPa has modulusof elasticity of 0MPa, the basic form of the equation is expressed as Eq.2.

The parameters examined are compressive strength, modulus of elasticity, and unit weight atthe time of compression test, as well as types and mechanical properties of materials forproducing concrete, proportioning, unit weight and air content of fresh concrete, method andtemperature of curing, and age.

ESTIMATION OF UNIT WEIGHT

Of the 3000 pieces of experimental data collected, only one third included measured unitweights of specimens, , at the time of compression tests. In order to express modulus ofelasticity as a function of compressive strength and unit weight in this study, the unit weightsof hardened concrete had to be estimated, where measured unit weights were unavailable, fromthe data on materials used, proportioning, curing conditions, and ages.

EQUATION FOR MODULUS OF ELASTICITY

Evaluation of Exponent b of Compressive Strength, σBAs compressive strength increases, Eq.1 overestimates modulus of elasticity. It is thereforeconsidered appropriate to reduce the value of exponent b of compressive strength, ,to lessthan 1/2, so as to match the estimation with the measured values.

Firstly, the range of possible values of exponent b in Eq.2 was investigated with regard to166sets of data, each set of which had been obtained from identical materials and curing conditions by the same researcher. Fig.2 shows the relationship between the maximumcompressive strengths and the estimated exponents b in the sets of data. Similarly, Fig.3 showsthe relationship between the exponents b and the ranges of compressive strengths in the sets of

γ

γ

σB

data. In Figs.2 and 3, while the estimated values of exponent b vary widely, the values showa tendency to decrease from around 0.5 to around 0.3, as the maximum compressive strengthsincrease and the ranges of compressive strength widen. In other words, whereas modulus ofelasticity of concrete of normal strength has been predictable from the compressive strengthwith exponent b at 0.4 to 0.5, 0.3 to 0.4 are more appropriate as the values of exponent b in ageneral-purpose equation to estimate modulus of elasticity for a wide range of concretes fromnormal to high-strength. Consequently, the authors propose 1/3 as the value of exponent b, inconsideration of the utility of the equation.

Evaluation of Exponent c of Unit Weight, γ

Secondly, by fixing exponent b at 1/3, as mentioned above, exponent c of unit weight, , wasinvestigated. The relationship between the unit weight, , and the value obtained by dividingmodulus of elasticity by compressive strength to the 1/3 power, , is shown in Fig.4 witha regression equation (Eq.3) that was obtained from the data on all aggregates as shown below.

In Fig.4, it can be said that Eq.3 expresses well the effects of unit weight on modulus of elas-ticity, if the concretes are divided into three groups according to unit weight, i.e. concretes withlightweight aggregate, with normal weight aggregate, and with heavy weight aggregate(bauxite, for example). The concretes with normal weight aggregate, however, are scatteredover a rather wide range of 6000 to 12000 of while they gather in a relatively smallunit weight range of 2.3 to 2.5. This is considered to suggest differences in the effects oflithological types of aggregates on modulus of elasticity, which will be discussed later in thispaper. Whereas 1.5 has been used conventionally as the value for exponent c, as indicated inEq. 1, 1.89 was obtained from the regression analysis as the value for exponent c that is ap-plicable for a wide range of concretes from normal to high-strength concretes. In considera-tion of the utility of the equation, the authors propose 2 as the value for exponent c.

Evaluation of Coefficient a

Thirdly, by fixing exponent b and exponent c at 1/3 and 2, respectively, the value for coeffi-cient a was investigated. The relationship between modulus of elasticity, E, and the product ofcompressive strength to the 1/3 power and unit weight to the second power, , is shownin Fig.5, together with regression equation (Eq.4) obtained from the data on all aggregates asshown below.

γγ

E σB1 3⁄⁄

E σB1 3⁄⁄

σB1 3⁄ γ2⋅

The coefficient of determination is as high as 0.769, and the 95% confidence interval ofmodulus of elasticity is within the range of ±8000MPa, as shown in Fig.5. The relationship be-tween modulus of elasticity and therefore be virtually expressed by Eq.4.

Evaluation of Correction Factor k

In the conventional equation for modulus of elasticity, Eq.1, the only difference in the type ofcoarse aggregate taken into account is the difference in the specific gravity, the effect of whichis represented by the unit weight of concrete, . However, use of a wide variety of crushedstone has revealed that the difference in unit weight is not the only factor to account for thedifferences in moduli of elasticity of concretes of the same compressive strength. Lithologicaltype should also be subject to consideration as a parameter of coarse aggregate. Besides it hasalso been pointed out by many researchers that modulus of elasticity cannot be expected toincrease in relation to the increased compressive strength, when the concrete contains amineral addition for high-strength concrete, such as silica fume. This suggests the need toinclude the type of additions as another factor affecting modulus of elasticity. Thus aregression analysis was conducted using the following equation (Eq.5) for each lithologicaltype of coarse aggregate, as well as for each type and level of content of mineral addition, toinvestigate the values of correction factor k.

1. EvaluatIon of Correction Factor for Coarse Aggregate

Fig.6 shows the relationship between the values estimated by Eq.4 and the measured values ofmodulus of elasticity of concretes without additions. According to Fig.6, most of the meas-ured values/the calculated values, i.e. values of in Eq.5, fall in the range of 0.9 to 1.2,indicating that each lithological type of coarse aggregate tends to have an inherent . Thecorrection factor for each coarse aggregate is presented in Table 1. According to Table 1,the effects of coarse aggregate on modulus of elasticity are classified into the following 3groups: the first group, which requires no correction factor, includes river gravel, crushedgraywacke, etc; the second, which requires correction factors of greater than 1, includescrushed limestone and calcined bauxite; and the third, which requires correction factorssmaller than 1, includes crushed quartzitic aggregate, crushed andesite, crushed cobbel stone,crushed basalt, and crushed clayslate. Consequently, the value for each type of coarseaggregate is proposed as shown in Table 2, in consideration of the utility of the equation.

σB1 3⁄ γ2⋅

γ

k1

k1

k1

k1

2. Evaluation of Correction Factor k2 for Additions

Table 3 presents the averages of correction factor obtained for each lithological type ofcoarse aggregate as well as for each type and level of content of addition. When fly ash as suchis used as an addition the value of correction factor is greater than 1, but when additions forincreasing strength of concrete, such as silica fume, ground granulated blast furnace slag, or flyash fume (ultra fine powder produced by condensation of fly ash) are used, the correctionfactor should be smaller than 1. The values of correction factor for additions arcproposed as shown in Table 4, in consideration of the utility of the equation.

Practical Equation for Modulus of Elasticity

Eq.5 was derived as an equation for modulus of elasticity. Meanwhile, conventional equa-tions such as Eq.1 have been convenient in such a way that standard moduli of elasticity can beobtained simply by substituting standard values of compressive strength and unit weight in theequation. In this study as well, the authors propose Eq.6 as the "New RC equation for modulusof elasticity" in consideration of a project known as New RC. The equation is based on6OMPa, a typical compressive strength of high-strength concrete in the project, and takesaccount of a unit weight of 2.4, which leads to the compressive strength of 60MPa.

COMPARISON OF EQUATIONS

Figs.7-10 show the accuracy of estimation by Eq.1 and Eq.6 as well as by ACI 363R andCEB-FIP equations, which are presented mTable 5.

k2

k2

k2 k2

As pointed Out by a number of researchers, the equation by AIJ (Fig.7) tends to overestimatemoduli of elasticity in the range of compressive strength over 40MPa, excepting the caseswhere crushed limestone or calcined bauxite is used as the coarse aggregate. The residuals alsotend to increase in relation to the compressive strength.

The equation by ACI 363R (Fig.8) slightly underestimates moduli of elasticity when crushedlimestone or calcined bauxite is used as the coarse aggregate, regardless of the compressivestrength. In the case of other aggregates, the equation tends to overestimate the moduli, thoughmarginally, as compressive strength increases.

The CEB-FIP equation (Fig.9) leads to clear differences in residuals depending on the litho-logical type of coarse aggregate. When lightweight aggregete is used, the equationoverestimates the moduli, and the value of the residuals tends to decrease as the specificgravity of coarse aggregate increases from crushed quartzitic aggregate to crushed graywacke,crushed limestone, and to calcined bauxite.

The residuals by the New RC equation (Fig.1O), as a whole, fall in the range of ±5000MPa,regardless of the compressive strength levels, although a portion of data displays residuals near+10000MPa. The New RC equation is therefore assumed to be capable of estimating moduli ofelasticity for a wide range of concretes from normal to high-strength.

EVALUATION OF 95% CONFIDENCE INTERVALS

The accuracy of the equation being enhanced by incorporating the correction factors, 95%confidence intervals should be indicated, because the reliability of the estimated values isrequired in structural design and is used when determining materials and proportioning so as toensure safety.

Excluding the case of using fly ash as an addition, only five values of the product of thecorrection factors, and , are possible, i.e. 1.2, 1.14, 1.0, 0.95, and 0.9025.

A regression analysis of Eq.2 was conducted for the combinations of a coarse aggregate and anaddition corresponding to each of the five values of , to obtain 95% confidence intervalsof both estimated and measured moduli of elasticity. The results are shown in Figs.11-15. Thecurves indicating the upper and lower limits of 95% confidence of the expected values for all

are within the range of approximately ±5% of the estimated values, regardless ofcompressive strength and unit weight. The curves indicating those for the observed values arealso within the range of approximately ±20% of the estimated values. Consequently, the 95%confidence limits of the New RC equation (Eq.6) are expressed in a simple form as Eq.7, andthe 95% confidence limits of measured modulus of elasticity can be expressed as Eq.8.

CONCLUSIONS

Multiple regression analyses were conducted using a great deal of data published in Japanregarding the relationship between compressive strength and modulus of elasticity of con-crete, by assuming compressive strength and unit weight as explanatory variables and modu-lus of elasticity as the target variable. As a result, the authors propose the "New RC equation"(Eq.6) as a practical and universal equation for modulus of elasticity. It is applicable to a widerange of concretes from normal to high-strength. The 95% confidence limits of the New RCequation were also examined, and Eq.7 and 8 were proposed as the equations to indicate the95% confidence limits for the expected and observed values, respectively.

ACKNOWLEDGEMENT

The authors are grateful to a lot of researchers for the offers of their valuable data.

REFERENCE

1. Tomosawa, F., Noguchi, T. and Onoyama, K. "Investigation of Fundamental MechanicalProperties of High-strength Concrete", Summaries of Technical Papers of Annual Meeting ofArchitectural Institute of Japan, pp.497-498, October 1990

k1 k2

k1 k⋅ 2

k1 k⋅ 2