# Finite Elements for Higher Order Steel-Concrete Composite Beams
## 3 Proposed Higher Order Finite Elements
### Convergence Analysis of the Proposed Finite Elements
resultants \(N_{c}\) and \(M_{s}\), trends within the elements are constant and linear for the \(GFE\) and the \(CIFE\), respectively, consistent with the adopted interpolating functions. Overall, convergence for the \(CIFE\) is faster than that relevant to the \(GFE\), especially for displacements for which good results can be obtained with very few \(CIFE\)s. The rate of convergence of each finite element is presented through the COV in Figure 7c as a function of the beam discretization; in this case, results considering up to 150 finite elements are shown and results relevant to the \(IIFE\) are addressed. The COVs confirm the faster convergence of the \(CIFE\) over the \(GFE\), especially for displacements, and show that the slab-grider interface slip \(\Gamma_{z}\) is characterised by the lowest rate of convergence. Furthermore, overall errors lower than 5% can be achieved by using few \(CIFE\) (about 10, excluding \(\Gamma_{z}\), or about 25, including \(\Gamma_{z}\)), while a higher number of \(GFE\)s is needed to get errors of a similar magnitude, due to locking problems (about 60, excluding \(\Gamma_{z}\), or about 125, including \(\Gamma_{z}\)). Figure 7: \(S2\)-_UDL_: Normalized beam response quantities for the (**a**) \(GFE\) and (**b**) \(CIFE\); (**c**) errors with respect to the reference solution. Finally, as already stated, the solution obtained through the _IIFE_ is practically exact for the minimum number of finite elements avoiding computational issues [34]; in this framework, it is interesting to observe that, at least with reference to displacements, a sufficiently accurate solution can be obtained through _CIFE_ by adopting a lower number of elements than that required to avoid computational issues. It is worth stating that the _IIFE_ does not suffer for the usual higher errors in the force evaluation. Figure 8 shows the selected normalised beam response parameters for case study _S3-CL_. Figure 8a refers to the _GFE_, while Figure 8b refers to the _CIFE_. With respect to the previous case, the number of finite elements (equal to 3, 4, 8, 16, and 51) is slightly changed for the need of applying the concentrated load. The previous considerations generally hold. COVs reported in Figure 8c once again demonstrate that the _CIFE_ is overall highly competitive with respect to the _IIFE_. Figure 9, similarly to previous ones, refers to case study _S2-SS_. Discretizations involving 2, 4, 8, 16, and 50 elements are considered. Overall, the previous observation concerning the performance of the _CIFE_ holds even if the _IIFE_ performs better below 50 elements. Figure 8: _S3-CL_: Normalized beam response quantities for the (**a**) _GFE_ and (**b**) _CIFE_; (**c**) errors with respect to the reference solution. Finally, Figure 10 reports the results obtained for case study _S3-PR_. It is worth stating that, in these three last cases, the convergence rate of the _GFE_ and _CIFE_ for forces and interface slippage is sensibly lower than that of the _IIFE_, which provides the exact solution with very few elements. Figure 11 shows the COVs evaluated for all the case studies (derived from the combination of the static schemes with the loading conditions) and for all the kinematic response parameters. Each plot, referring to a specific case study, is further subdivided into three subplots of different colours, relevant to the results obtained considering the _GFE_, the _CIFE_, and the _IIFE_. Overall, it was found that, independent of the adopted finite element, the steel girder warping intensities (\(f_{\mathit{sv}}\) and \(f_{\mathit{sh}}\)) present higher COVs than the remaining parameters; thus, in each plot, a synthetic representation of results is proposed adopting shaded areas to trace envelops of COVs of all response parameters, excluding the steel girder warping intensities, while specific symbols are used for the latter.