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Effective Elastic Properties

In order to simplify calculations it is possible to determine affective elastic properties of a section. The simplest of these is the area. Consider a section with both concrete and steel with areas AcA_c and AsA_s respectively. The axial stiffnesses are

kc=AcEclks=AsEsl\begin{aligned}k_{c} &= \frac{A_{c}E_{c}}{l}\\ k_{s} &= \frac{A_{s}E_{s}}{l}\end{aligned}

And the total stiffness is then

k=AcEcl+AsEslk = \frac{A_{c}E_{c}}{l} + \frac{A_{s}E_{s}}{l}

To simply calculation we can choose a reference material. So for concrete as a reference material

AeffEcl=AcEcl+AsEsl\frac{A_{eff}E_{c}}{l} = \frac{A_{c}E_{c}}{l} + \frac{A_{s}E_{s}}{l}

or

Aeff=Ac+AsEsEcA_{eff} = A_{c} + A_{s}\frac{E_{s}}{E_{c}}

More generally for a collection of components with a reference section

Aeff=∑i(AiEi)ErefA_{eff} = \frac{\sum_{i}^{}\left( A_{i}E_{i} \right)}{E_{ref}}

For a section made of multiple components the effective centroid is defined as

ceff=∑i{Ai(ci−cref)}Aeffc_{eff} = \frac{\sum_{i}^{}\left\{ {A}_{i}\left( c_{i} - c_{ref} \right) \right\}}{A_{eff}}

As for axial properties effective bending properties can be defined (allowing for the different centroids) as

Ieff=∑i{(Ii+Ai(ci−ceff)2)Ei}ErefI_{eff} = \frac{\sum_{i}^{}\left\{ \left( I_{i} + A_{i}\left( c_{i} - c_{eff} \right)^{2} \right)E_{i} \right\}}{E_{ref}}