# Finite Elements for Higher Order Steel-Concrete Composite Beams
## 4 Numerical Comparison with More Complex Models  
In this section, the overall efficiency of the proposed beam model in capturing the behaviour of a composite bridge deck is shown by comparing the results with those achieved with a refined three-dimensional FE model developed in SAP2000(r)[32]. For this purpose, all the analysis cases previously presented, including loading conditions and static schemes, are investigated. Components of the beam cross section are modelled with four node homogenous thin shell elements that combine membrane and plate-bending behaviour and adopt a four-point numerical integration formulation. The actual position of their relevant mid planes is suitably taken into account and preliminary convergence analyses are performed on the developed FE models to define the suitable mesh dimension. In detail, the concrete slab is schematized with elements of dimensions 0.5 \(\times\) 0.5 m, whereas upper flanges, webs, and bottom flanges of the steel girder are divided into 5, 6, and 12 elements, respectively. The shear connection is modelled with two node frame elements connecting nodes of the concrete slab and the upper steel flanges, located in the relevant mid planes. Two beam  
Figure 11: Trend of the coefficients of variation (COVs) for all the case studies and beam response quantities. elements are adopted, pinned in correspondence of the slab-steel girder interface, having properties suitably calibrated to assure a translational stiffness equal to the one used in the previous applications. A pictorial view of the developed model is reported in Figure 12.  
Figure 13 shows the vertical deflection \(v_{0}\), the slab-girder interface slip \(\Gamma_{z}\), and the longitudinal displacements of the upper (\(w_{s,sup}\)) and bottom flanges (\(w_{s,inf}\)) of the steel girder obtained for case studies _S2-UDL_, _S3-CL_, _S2-SS_, and _S3-PW_. The results obtained considering the different beam elements (_GFE_, _CIFE_, and _IIFE_) are reported with lines of different colours and are compared with those achieved by the refined 3D FE model, represented by black dots. Concerning the proposed beam model, a suitable number of elements is considered, on the basis of the previous applications, to assure the analysis convergence and accuracy. In detail, 200, 100, and 50 elements are considered for the _GFE_, the _CIFE_, and the _IIFE_, respectively. It can be observed that the results from the proposed beam finite elements are  
Figure 12: 3D finite element (FE) model. Figure 13: Comparison between the proposed beam model and the refined 3D FE model for some case studies in terms of vertical displacement \(v_{0}\); interface slip \(\Gamma_{z}\); and the longitudinal displacement of the steel girder bottom and upper flanges, \(w_{s,inf}\) and \(w_{s,sup}\). practically superimposed and all perfectly match the solution achieved with the refined 3D FE model. Figure 14 compares the normal stresses on the concrete slab mid plane, on the steel girder web, and on the steel girder bottom flange obtained from the proposed beam model with those resulting from the refined 3D FE model, for case study _S2-UDL_. As the results from the beam models are almost superimposed, for the sake of simplicity, only those relevant to the _IIFE_ are reported in red. The results from the refined FE model are reported with black lines. Because the response of the slab is symmetric with respect to the longitudinal middle axis, comparisons between the beam and shell finite element models are made by dividing the plot of Figure 14 into two parts, and by presenting the distribution of stresses for half of the slab. The beam model is able to capture very well the slab normal stresses with very few differences in the bridge deck section characterised by the maximum positive bending moment and at the fixed support.