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- 1. Elastic Modulus of Concrete Elastic Modulus of Concrete from ACI Code Shear and bulk moduli, G and K.
- 2. For Burj Khalifa (currently the tallest building in the world), the designer specified minimum 43800 MPa for 80 MPa concrete mixes for the vertical elements. In the simplest terms, modulus of elasticity (MOE) measures the stiffness of the material and is a good overall indicator of its strength. It is the ratio of stress to strain. Stress is the deforming force acting per unit area (F/A), and strain is the deformation (change in shape) caused by stress (∆L/L). The stress- strain relationship was first studied by Robert Hook, an English natural philosopher, architect and an expert in many different areas of knowledge. In 1678 he stated that ‘within elastic limit, the stress is directly proportional to strain.'
- 3. Stress α strain i.e., stress/ strain = a constant (this constant is called modulus of elasticity) When stress is applied on a body, strain develops, and the material passes through different stages of deformation as shown in the picture below.
- 4. Elasticity is the property of matter by which the material regains its original shape when the deforming force has been withdrawn. Elastic limit (Yield point) is the amount of stress that a material can undergo before moving from elastic deformation to plastic deformation. In plastic deformation, the material cannot regain its original shape even when the deforming force has been withdrawn. It remains in the deformed shape. Plastic deformation continues up to the breaking point and then it ruptures. This point of stress at which the material breaks, with the sudden release of the stored elastic energy, is termed ultimate tensile strength (UTS). Based on the types of stress (tension, compression or shear) and strain, including the direction, different types of elastic modulus can be identified as detailed below. 1. Young’s Modulus (E) - ratio of linear stress to linear strain, 2. Shear Modulus (G or µ) – ratio of shear stress to shear strain and, 3. Bulk Modulus (K) – ratio of volume stress to volume strain. Young’s modulus enables the calculation of the change in the dimension of concrete members under tensile or compressive loads. For instance, it predicts how much a concrete column can shorten under compression. In other words, elastic modulus tells us how much tension or compression is required to make the material little bit longer or shorter.
- 5. Thomas Young (1773 – 1829) was an English scientist and an expert in many different areas of knowledge. He was much interested in the early experiments and studies of Leonhard Euler (1727) and Giordano Ricatti (1782) on elastic moduli of materials. Young’s modulus (E) = linear stress/ linear strain Linear stress = Force/area = F/A Linear strain = Change in length/ original length = ∆L/ L Therefore, Young’s Modulus (E) = (F/A)/ (∆L/L) = FL/ A∆L The higher the elastic modulus means the concrete can withstand higher stress but the concrete will become brittle and sooner cracks will appear. Low elastic modulus indicates that it will bend and deform very easily. High elastic modulus at early ages (7 days or 14 days) will result in a higher potential for cracking. This is due to high stress produced due to even low strain. Strain can arise from causes other than applied stress like shrinkage. Shrinking and thermal activity can cause very low stress, but due to high elastic modulus, the corresponding stress is high. As the tensile strength of concrete is still low at this early ages, cracks will develop. Hydrated cement paste has lower elastic modulus than the aggregate. Hence, the volumetric content of aggregate is important as far as the elastic modulus of the mix is considered. The elastic modulus of hardened cement paste is around 10 to 30 GPa and that of aggregate is between 45 to 85 GPa. Concrete generally have an elastic modulus varying between 30 to 50 GPa. The factors affecting elastic modulus of concrete are:
- 6. 1- Coarse aggregate properties- like elastic modulus of aggregate, type of aggregate (crushed or natural), petrology and mineralogy, and quantity of aggregate. The higher the volume of aggregate in the mix, the higher the elastic modulus. 2- Mix design- which includes total cementitious content and w/c ratio. Less paste is good for higher elastic modulus. 3- Curing conditions- moist cured specimen showed better results than that of dry cured, due to shrinkage and associated cracks. 4- Loading rate- high loading rate will result in higher compressive strength and higher elastic modulus. 5- Chemical admixture- does not have much influence on elastic modulus. But some type of admixture can produce higher cement dispersion and thus will result in higher compressive strength and elastic modulus. 6- Mineral admixture- as they affect the strength of concrete, they affect the elastic modulus too. The most important factor influencing the elastic modulus of concrete is the aggregate used. It is also affected by the aggregate/cement ratio and age of the concrete. The following table taken from Eurocode- 2, gives the compressive strengths (based on cylinders and cubes), modulus of elasticity and tensile strength for various strength classes of normal weight concrete, which are generally used for design purpose. The above values have been computed by using the following equation provided in Eurocode- 2: Ec = 22000 x [fcy/10] ^0.3 ------ in MPa Where Fcy is the compressive strength of cylinder at 28 days. ACI 318- Structural concrete building code suggests that the elastic modulus (Ec) for concrete shall be calculated by the formula given below: Ec = 33 Wc^1.5 √ fc ----- in psi or Ec = 0.043 Wc^1.5 √fc ----- in MPa
- 7. Where Ec is the elastic modulus, Wc is the weight of concrete (pounds per foot or kg/m3), and fc is the compressive strength of cylinder at 28 days (psi or MPa). These equations are often simplified based on normal density aggregate and normal weight concrete as follows: Ec = 57000 √fc ----- in psi or Ec = 4700 √fc ------ in MPa BS 8110 Structural use of concrete Part 2, under Clause 7.2. Elastic Deformation, suggests the following equation for computing the expected elastic modulus value based on the 28 days’ cube strength results. Ec,28 = Ko + 0.2 fcu,28 Where, Ko is a constant closely related to the elastic modulus of aggregate often taken as 20kN/mm2 for normal weight aggregate and Fcu,28 is the compressive strength of cube at 28 days. Where deflections or deformations are of great importance, tests should be carried out on concrete made with the aggregate to be used in the structure. But without previous data on aggregate or with unknown aggregate, a range of values for Ec based on Ko = 14 kN/mm2 to 26 kN/mm2 shall be considered. The Indian Code of Practice (IS 456) recommends the following equation: Ec = 5000 √fck ACI 363R- Report on High-Strength Concrete and New Zealand Standard NZS 3101-1 give the following equation for elastic modulus of concrete as: Ec = [3320 √fc + 6900] ---- in MPa The Australian Standard AS 3600 recommends the following expression to calculate the value of the modulus of elasticity within an error of ± 20%:
- 8. Ec = ρ^1.5 x [0.024 (fm + 0.12)^0.5] ------- in MPa Where ρ is the density of concrete in kg/m3, and fm is the mean compressive strength in MPa at 28 days. Elastic modulus of concrete is tested by using 150 mm X 300 mm cylinder specimens in accordance with: 1. ASTM C 469- Static modulus of elasticity and Poisson’s ratio of concrete in compression or 2. BS 1881 Part 121- Determination of static modulus of elasticity in compression. Elastic modulus is determined by using compressometer fixed on the cylinder specimen (sometimes extensometer also to compute Poisson’s ratio as shown in the figure above) and loaded at a particular stress level. It can be estimated using the 15 to 85% stress levels in the elastic range. In ASTM the stress level is 40% of the compressive strength of the companion cylinder and in BS 33% of the strength of the companion cylinder. According
- 9. to ASTM test method, the results is reported to the nearest 200 MPa, but to BS test method to the nearest 500 MPa. Other types of strain gauges (compressometer and extensometer) are available. An electrical strain gauge is the most suited method for concrete strain determination, which has to be adhered to the concrete sample, but which requires time and attention from technicians. Ezeagu C.A. and Obasi K.C. (International Journal of Advanced Research) have reported from their studies that concrete made with 20mm maximum nominal size aggregate showed higher elastic modulus than that with 30mm and 60mm. They have calculated the elastic modulus based on different equations and have found different values of elastic moduli. Takafumi Noguchi et all (ACI Structural Journal) have reported that even though, Japanese and American code rules suggest unit weight with an exponent 1.5, their studies have shown that there is a direct proportionality between elastic modulus of concrete and its unit weight power to 2. K. Anbuvelan and Dr. K. Subramanian (International Journal of Engineering and Technology) have reported based on their experimental investigation on elastic properties of concrete containing steel fiber that the IS 456 and EC-2 predict higher modulus of elasticity than BS 8110, ACI 318 and NZS 3101. Based on the results of their study, Walid Baalbaki et all (ACI Materials Journal) concluded that it is unreliable to precisely predict the elastic modulus of high strength concrete from its compressive strength. The following table gives the compressive strength and elastic modulus (trial mix results) of concrete mixes used for the vertical elements of the Burj Khalifa- the tallest tower in the world. The elastic modulus values are very close to the ACI 318 equation.
- 10. Before winding up this article, the author has a question to the readers. It is known to everyone that elastic modulus indicates the stiffness of material. In other words, it represents the tenacity of the material. The tenacity of a material can be of six types as follows: 1- Brittle – The material breaks or powders very easily. 2- Malleable – The material can be pounded into thin sheets, like metal. 3- Ductile – The material can be drawn into wire, like metal. 4- Sectile – The material can be cut smoothly with a knife. 5- Plastic – The material deforms under stress, but it cannot regain its original shape when the force is withdrawn. 6- Elastic – The material deforms under stress but regains its original shape when force is withdrawn. As the elastic modulus increases, the material becomes stiffer and brittle. But compared to steel, concrete is more brittle, even though the elastic modulus of steel is 200 GPa and that of concrete is between 25 to 50 GPa. Why is it so? Diamond has an elastic modulus of 1220 GPa, and is very very brittle