Embedded beam in Tension

Question

Hello, this regarding the true input of the modulus of elasticity "E" for an embedded beam element, either 2D or 3D. 
 When using this feature for modelling a concrete structure (reinforced concrete piles, grouted nail...), it seems that there are doubts about what "E" to use. For instance, I recall seeing that Bentley recommends for soil nails to use an equivalent elasticity modulus for steel and concrete as follows: Eeq = EsAs + EcAc / As + Ac. While this formula is applicable for an element in compression, when the structure is in tension the difference in the rigidity becomes relatively large and the design becomes way over-conservative, i.e. with larger displacements, which makes it not cost-efficient. 
 The issue becomes more complicated when the element switches its behavior from a phase to another due to loading/unloading of loads. 
 Your expertise is appreciated in this manner.

Vasileios Basas · Answer

Hi Mohamad, The stiffness of concrete is indeed different between tension and compression, but only at high strain, with tensile stiffness being much lower due to cracking. In lower strains or when you model the embedded beam as linear elastic or linear elastic-perfectly plastic, the stiffness of concrete doesn't simply "disappear". It is, in fact, the same in compression and tension. So, until tension is high enough to lead to the development of cracks in concrete, the concrete and its reinforcing steel work together to resist tension, therefore, have an equivalent stiffness from steel and concrete combined. At the location where concrete is cracked, only the reinforcing steel is left in place to resist the tension alone, and therefore the stiffness is much lower. However, in reality, concrete cracks do not occur everywhere but progressively at certain spacings as tension increases. It is always the case that concrete remains attached to the reinforcing steel in between cracks due to bonding. Therefore, when considered on a structural element scale, the effective stiffness of the cracked concrete almost always sits somewhere in between those two states. 
 Suppose you believe that neither the linear elastic nor the linear elastic - perfectly plastic behaviours cover your needs. In that case, you can define an elastoplastic (M-&kappa;) material, which allows you to input a non-linear stiffness according to a defined &Mu;-&kappa; diagram. Alternatively, you can use the concrete model, which allows modelling behaviour in tension as linear elastic until the tensile strength ft followed by a linear strain softening.