# Composite Slabs

Composite slabs are a slab supported on steel decking. These can be modelled as a solid slab with adjustment to the in-plane ($t_p$) and bending ($t_b$) thickness. For a unit width the area of slab is $A$ concrete ($A_c$) and steel ($A_s$) are known as are the second moments of area ($I_c$ and $I_s$) and the $E$ values ($E_c$ and $E_s$).

Referring back to the concrete as the primary material the effective area is

$A_{eff} = A_{c} + \left( \frac{E_{s}}{E_{c}} \right)A_{s}$

And the effective thickness (in-plane) is

$t_{p} = \frac{A_{eff}}{A} = \frac{A_{c} + \left( \frac{E_{s}}{E_{c}} \right)A_{s}}{A}$

Give the centroid of the concrete ($z_c$) and steel decking ($z_s$) the centroid of the composite section is then

$z_{eff} = \frac{A_{c}z_{c} + \left( \frac{E_{s}}{E_{c}} \right)A_{s}z_{s}}{A_{c} + \left( \frac{E_{s}}{E_{c}} \right)A_{s}}$

and the effective second moment of area ($I_{eff}$) is

$I_{eff} = \left\lbrack I_{c} + A_{c}\left( z_{c} - z_{eff} \right)^{2} \right\rbrack + \left( \frac{E_{s}}{E_{c}} \right)\left\lbrack I_{s} + A_{s}\left( z_{s} - z_{eff} \right)^{2} \right\rbrack$

And the effective thickness in bending is

$t_{b} = \sqrt[3]{\frac{I_{eff}}{I}} = \sqrt[3]{\frac{\left\lbrack I_{c} + A_{c}\left( z_{c} - z_{eff} \right)^{2} \right\rbrack + \left( \frac{E_{s}}{E_{c}} \right)\left\lbrack I_{s} + A_{s}\left( z_{s} - z_{eff} \right)^{2} \right\rbrack}{I}}$