# Finite Elements for Higher Order Steel-Concrete Composite Beams
## 5 Conclusions  
Finite elements for a higher order steel-concrete composite beam model were presented in this paper. The model, which is particularly suitable for the analysis of bridge decks, includes the partial interaction between concrete and steel members and accounts for the overall shear deformability and the shear-lag phenomenon, which strongly characterise the response of both steel and concrete elements.  
Finite elements characterised by different interpolating functions are developed, implementing linear, polynomial, and exponential shape functions; the latter are derived from an analytical solution that exploits exponential matrices. The performance of the presented finite elements is investigated in terms of the solution convergence rate with reference to realistic steel-concrete composite beams with different restraints and loading conditions. The efficiency of the proposed finite elements in providing a reliable prediction of the structural response of composite beams is also addressed through comparison of the results with those achieved with a refined 3D numerical model developed using conventional shell finite elements.  
The following remarks can be drawn:  
* The finite element based on linear shape functions (_GFE_) suffers from locking problems and requires a highly refined discretization to reach an accurate solution of the problem;
* The finite element implementing cubic and quadratic polynomial shape functions (_CIFE_) avoids locking problems and is characterised by a higher converge rate than that based on linear shape functions (_GFE_);
* The finite element with exponential shape functions (_IIFE_) is the most performant and furnishes an almost exact solution, independent of the beam discretization, provided  
Figure 17: (**a**) Longitudinal normal stress on the slab mid plane, (**b**) on the steel girder web, and (**c**) on the steel girder bottom flange for case study S3-_PR_.  
that enough finite elements are adopted to avoid issues in the numerical evaluation of the exponential matrix;
* The _CIFE_ is highly competitive with respect to the _IIFE_, especially in predicting beam displacements and rotations; in some cases, if very accurate solutions are not required, the former may provide results with a lower number of finite elements than that necessary to avoid instabilities in the computation of the exponential shape functions of the _IIFE_;
* In the case of distributed or concentrated loads, the convergence rate relevant to warping intensities of the steel components is much lower than that relevant to the other response parameters; differences in the convergence rate attenuate in the cases of prestressing or concrete shrinkage.  
Provided that a proper discretization of the beam axis is used, depending on the adopted finite element, the beam model is able to capture very well the structural response of composite beams subjected to different loads and restraint conditions obtained from a refined 3D model. In particular, both displacements, stresses, and stress resultants of the 3D model are reproduced by the proposed beam model, which foresees a number of dots about thirty times lower.  
Conceptualization, G.L. and F.G.; methodology, G.L., F.G., and S.C.; software, F.G. and S.C.; validation, G.L. and F.G.; formal analysis, G.L., F.G., and S.C.; data curation, S.C.; writing--original draft preparation, F.G. and S.C.; writing--review and editing, G.L. and F.G.; visualization, F.G. and S.C.; supervision, L.D. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.