# Finite Elements for Higher Order Steel-Concrete Composite Beams
###### Abstract
Before adopting a model with its relevant solving approach, it is imperative to know not only its theoretical applicability, but also the expected degree of approximation of the solution. The latter depends not only on the accuracy of the analytical model, because of the assumed admissible displacement field, but also on the numerical approach adopted for the solution (e.g., locking problems for finite elements). A further issue is the efficiency of the numerical approach to achieve a desired degree of precision, which reflects on the computation time depending on the refinement of the geometric discretization and the computation capacity. In this paper, three new finite elements for the numerical solution of a theoretical higher order model developed by the authors [30] for the analysis of steel-concrete composite beams (Figure 1) are presented. The model, which accounts for the overall shear deformability and the shear-lag phenomenon on both steel and concrete members of composite beams, is firstly recalled providing a formulation that simplifies the subsequent numerical implementation. Both the kinematics (i.e., the assumed displacement field) and the active stresses field (according to linear elastic laws for the materials) are reported; the balance conditions in both weak and strong forms are provided. Then, three finite elements characterised by different interpolating functions are developed, namely, a general finite element (_GFE_) (a 16 degrees of freedom (dof) element with linear shape functions); a consistent interpolation finite element (_CIFE_) (a 25 dof element with polynomial functions whose degree is chosen to avoid locking problems); and an interdependent interpolation finite element (_IIFE_), adopting shape functions derived from the closed-form solution. Figure 1: Examples of steel–concrete bridge decks: (**a**) continuous twin-girder multi-span viaduct-Serra Cazzola viaduct, AG, Italy; (**b**) continuous box-girder multi-span bridge on the Cesano River, at Mondolfo, PU, Italy. The performance of the proposed finite elements is analysed in terms of the rate of convergence, considering realistic steel-concrete composite beams with different restraints and loading conditions and assuming as a benchmark the solution obtained with the _IIFE_ and a refined discretization. Finally, the overall efficiency of the proposed beam model in capturing the behaviour of a composite bridge deck is shown by comparing the results of the model (implementing all the finite elements) with those achieved with a refined 3D finite element (FE) model developed with SAP2000(r) [32]. The theoretical higher order model for the analysis of composite beams is briefly recalled in Section 2 and different finite elements to solve the problem numerically are developed in Section 3, discussing their effectiveness as a function of the beam discretization. In Section 4, the overall efficiency of the beam model in capturing the behaviour of a composite bridge deck is shown through comparisons with the results from a refined three-dimensional FE model developed with a commercial software.