Elastic properties and modulus of elasticity

Elastic properties

Concrete is a composite of paste, aggregate and interface, and so it is not truly elastic. On loading, the stress-strain curve is nearly straight up to about 30 to 40% of the ultimate stress, then curves as microcracks grow. For design, concrete is treated as elastic at working stress. The elastic properties are the modulus of elasticity $E$, Poisson's ratio $\mu$ (0.15 to 0.20 for concrete) and modulus of rigidity $G = E/[2(1+\mu)]$.

Modulus of elasticity (Young's modulus)

It is the ratio of stress to strain within the elastic range:

$$E = \frac{\sigma}{\varepsilon}$$

• Initial tangent, tangent, secant and chord moduli are used (secant at about 33% of $f_c$ is the static modulus).
• Dynamic modulus from UPV or resonant frequency is higher than static modulus.
• IS 456 (Cl. 6.2.3.1): $E_c = 5000\sqrt{f_{ck}}$ N/mm$^2$. For M20: $E_c = 5000\times4.472 = 22360$ N/mm$^2$.

Factors affecting $E$

• Higher strength (lower W/C) gives a higher $E$.
• Stiffer aggregates (granite) and a higher aggregate content give a higher $E$; light-weight aggregate gives a lower $E$.
• Moist concrete has a higher $E$ than dry concrete; porosity reduces $E$.
• $E$ increases with age. A slower rate of loading gives a lower apparent value because of creep.

$E$ is used to calculate deflection, to find the modular ratio $m = 280/(3\sigma_{cbc})$ (IS 456, Annex B) and to find the stress distribution in composite sections.