690 ACI Structural Journal/September-October 2009 ACI Structural Journal, V. 106, No. 5, September-October 2009. MS No. S-2008-210 received June 26, 2008, and reviewed under Institute publication policies. Copyright © 2009, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the July-August 2010 ACI Structural Journal if the discussion is received by March 1, 2010. ACI STRUCTURAL JOURNAL TECHNICAL PAPER Many empirical equations for predicting the modulus of elasticity as a function of compressive strength can be found in the current literature. They are obtained from experiments performed on a restricted number of concrete specimens subjected to uniaxial compression. Thus, the existing equations cannot cover the entire experimental data. This is due to the fact that mechanical properties of concrete are highly dependent on the types and proportions of binders and aggregates. To introduce a new reliable formula, more than 3000 data sets, obtained by many investigators using various materials, have been collected and analyzed statistically. The compressive strengths of the considered concretes range from 40 to 160 MPa (5.8 to 23.2 ksi). As a result, a practical and universal equation, which also takes into consideration the types of coarse aggregates and mineral admixtures, is proposed. Keywords: analysis; coarse aggregates; compressive strength; high- strength concrete; modulus of elasticity; normal-strength concrete; water- cement ratio. INTRODUCTION To design plain, reinforced, and prestressed concrete structures, the elastic modulus E is a fundamental parameter that needs to be defined. In fact, linear analysis of elements based on the theory of elasticity may be used to satisfy both the requirements of ultimate and serviceability limit states (ULS and SLS, respectively). This is true, for instance, in the case of prestressed concrete structures, which show uncracked cross sections up to the failure. 1 Similarly, linear elastic analysis, carried out through a suitable value of E, also permits the estimation of stresses and deflections, which need to be limited under the serviceability actions in all concrete structures. Theoretical and experimental approaches can be applied to evaluate the elastic modulus of concretes. In the theoretical model, concretes are assumed to be a multi-phase system; thus, the modulus of elasticity is obtained as a function of the elastic behavior of its components. This is possible by modeling the concrete as a two-phase material, involving the aggregates and the hydrated cement paste (refer to Mehta and Monteiro 2 for a review), or three-phase material, if the so-called interface transition zone (ITZ) between the two phases is introduced. 3-5 Nevertheless, according to Aïtcin, 6 theoretical models can appear too complicated for a practical purpose, because the elastic modulus of concrete is a function of several parameters (that is, the elastic moduli of all the phases, the maximum aggregate diameter, and the volume of aggregate). As a consequence, such models can only be used to evaluate the effects produced by the concrete components on the modulus of elasticity. 7 Empirical approaches, based on dynamic or static measurements, 8 are the most widely used by designers. Dynamic tests, which measure the initial tangent modulus, can be adopted when nondestructive diagnostic tests are required. On the contrary, static tests on cylindrical specimens subjected to uniaxial compression are currently used for evaluating E. From these tests, the current building codes propose more or less similar empirical formulas for the estimation of elastic modulus. Because they are directed to designers, the possible equations need to be formulated as functions of the parameters known at the design stage. 9 Thus, for both normal-strength (NSC) and high-strength (HSC) concrete, the Comité Euro-International du Béton and the Fédération Internationale de la Précontrainte (CEB-FIP) Model Code 10 and Eurocode 2 11 link the elastic modulus E to the compressive strength σ B according to (1a) (1b) In Eq. (1a), E and σ B are measured in MPa, whereas in Eq. (1b), E and σ B are measured in ksi. In the case of HSC, in the formula proposed by ACI Committee 363, 12 the elastic modulus of concrete is also function of its unit weight γ E = (3321σ B 0.5 + 6895) · (γ/2300) 1.5 (2a) E = (1265σ B 0.5 + 1000) · (γ/145) 1.5 (2b) In Eq. (2a), E and σ B are measured in MPa, and γ in kg/m 3 , whereas in Eq. (2b), E and σ B are measured in ksi and γ in lb/ft 3 . Similarly, the Architectural Institute of Japan 13 specifies the following equation to estimate the modulus of elasticity of concrete E = 21,000(γ/2300) 1.5 (σ B /20) 1/2 (3a) E = 3046(γ/145) 1.5 (σ B /2.9) 1/2 (3b) In Eq. (3a), E and σ B are measured in MPa and γ in kg/m 3 , whereas in Eq. (3b), E and σ B are measured in ksi and γ in lb/ft 3 . E 22,000 σ B 10 ----- ⎝ ⎠ ⎛ ⎞ 1 3 -- = E 3191 σ B 1.45 --------- ⎝ ⎠ ⎛ ⎞ 1 3 -- = Title no. 106-S64 A Practical Equation for Elastic Modulus of Concrete by Takafumi Noguchi, Fuminori Tomosawa, Kamran M. Nemati, Bernardino M. Chiaia, and Alessandro P. Fantilli