Composite Slabs

Two types of composite element or composite slab can be defined in GSA.

• Layered slabs (e.g. CLT or reinforced concrete)
• Composite steel decking slabs

Layered slabs

Layered slabs are modelled as a stacked set of shell elements with offsets from the mid-surface. The stiffness of each layer is calculated as for a solid shell element with the layer thickness and properties. The strain of the composite shell is assumed to be linear through the thickness of the element so this composite action is included by offsetting the stiffness of each layer from the mid surface $z_i$ through a transformation matrix $T_i$ where

$$T_i= \begin{bmatrix} 1 & & & & z_i &\\ & 1 & & -z_i & &\\ & & 1 & & &\\ & & & 1 & &\\ & & & & 1 &\\ & & & & & 1 \end{bmatrix}$$

with the final stiffness being the sum of the stiffnesses of each layer

$$K = \sum_i{(T_i^T K_i T_i)}$$

Composite steel decking 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

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