# Finite Elements for Higher Order Steel-Concrete Composite Beams
## 3 Proposed Higher Order Finite Elements
### Convergence Analysis of the Proposed Finite Elements  
In this section, the results of applications carried out with the proposed beam finite elements are presented and discussed to compare their rate of convergence. Analyses are performed with reference to a steel-concrete composite bridge deck with a constant box cross section, having the geometry and mechanical properties reported in Figure 5, which are representative of a concrete grade C25/30 and structural steel (European grade S). The three static schemes reported in Figure 6b are considered; _S1_ is representative of single span simply supported bridge decks, while _S2_ and _S3_ are representative of edge and inner spans of continuous bridges, respectively. Furthermore, four loading conditions are analysed (Figure 6a); these include a uniformly distributed load (_UDL_), a concentrated load (_CL_), the concrete slab shrinkage (_SS_), and the prestressing of wires in the concrete slab (_PW_). Hereafter, analysis cases will be labelled combining the static scheme with the loading condition (e.g., _S1-UDL_ refers to simply supported beam with uniformly distributed load). Figure 4: (**a**) Nodal generalized displacements of the consistent interpolation finite element (_CIFE_); (**b**) interpolation of longitudinal displacement, warping intensities, and rotations; and (**c**) interpolation of vertical displacement. Figure 5: Cross section adopted in the applications. The rate of convergence of models implementing the presented finite elements is analyzed by progressively increasing the number of finite elements adopted to discretize the bridge deck and by comparing the beam response with the reference one obtained considering the _IIFE_ and a high number of finite elements (200 elements). For the sake of brevity, only some of the response parameters are monitored; the selected set includes displacements (vertical displacement \(v_{0}\) of the composite beam, longitudinal displacement \(w_{s,inf}\) of the steel girder bottom flange, warping intensity \(f_{c}\)), stress resultants (axial force \(N_{c}\) on the concrete slab, bending moment \(M_{s}\) on the steel girder), and the slab-girder interface slip \(\Gamma_{z}\). Furthermore, in order to discuss the rate of convergence of the solutions, the following coefficient of variation (COV) is calculated for each response parameter by varying the beam discretization:  
\[\sigma_{p}^{*}=\frac{1}{\mu_{|p|,ex}}\sqrt{\frac{1}{n}\sum_{i=1}^{n}\lvert p_{i }-p_{i,ex}\rvert^{2}} \tag{54}\]  
where \(p\) denotes the generic response parameter, \(n\) is the number of nodes of the discretization, \(p_{i}\) is the value of the \(p\)-th response parameter at the \(i\)-th node, \(p_{i,ex}\) is the relevant value of the reference solution (exact), and \(\mu_{|p|,ex}\) is the arithmetic mean of the absolute values of the reference solution (i.e., the arithmetic mean of the absolute \(p_{i,ex}\) values). The above coefficient is representative of the overall error within the beam, normalised with respect to a quantity proportional to the area of the graph of the generic response parameter. Figure 7 plots the considered response parameters along the non-dimensional beam length z/L for case study _S2-UDL_. The results are normalised with respect to the maximum absolute value of the relevant quantity of the reference solution. In detail, Figure 7a refers to the _GFE_, while Figure 7b refers to the _CIFE_; for both cases, the results obtained by discretizing the beam length into 2, 4, 8, 16, and 50 elements are reported. The solution obtained by implementing the _IIFE_ is not shown as it is almost exact, independently of the beam discretization, provided that enough elements are used to avoid computational issues in the numerical evaluation of the exponential matrix [34]. With reference to stress  
Figure 6: (**a**) Load conditions and (**b**) static schemes considered in the applications. UDL, uniformly distributed load; CL, concentrated load; SS, concrete slab shrinkage; PW, prestressing of wires in the concrete slab.