# Finite Elements for Higher Order Steel-Concrete Composite Beams
###### Abstract  
This paper presents finite elements for a higher order steel-concrete composite beam model developed for the analysis of bridge decks. The model accounts for the slab-girder partial interaction, the overall shear deformability, and the shear-lag phenomenon in steel and concrete components. The theoretical derivation of the solving balance conditions, in both weak and strong form, is firstly addressed. Then, three different finite elements are proposed, which are characterised by (i) linear interpolating functions, (ii) Hermitian polynomial interpolating functions, and (iii) interpolating functions, respectively, derived from the analytical solution expressed by means of exponential matrices. The performance of the finite elements is analysed in terms of the solution convergence rate for realistic steel-concrete composite beams with different restraints and loading conditions. Finally, the efficiency of the beam model is shown by comparing the results obtained with the proposed finite elements and those achieved with a refined 3D shell finite element model. beam finite element; consistent interpolation; higher order steel-concrete composite beam; interdependent interpolation; shear deformability; shear-lag +
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making it possible to investigate alternative design solutions, especially in the preliminary calculations aimed at sizing and optimizing the steel girders. To provide sufficiently accurate results, theoretical models have to account for several peculiarities that characterize the behaviour of steel-concrete composite bridge decks. For straight beams, the most important issues are (i) the longitudinal deformability of steel connectors, which implies a partial interaction between the concrete slab and steel girder; (ii) the shear deformability of the composite cross section, which significantly affects the deflection for vertical loads, especially in the case of multi-supported continuous beams; and (iii) the in-plane shear deformability of the concrete slab and the flanges of the structural steel girder, which causes a non-uniform distribution of normal stresses within the elements (i.e., shear-lag effects). A wide range of scientific-technical literature covering all features relevant to design and modelling of composite systems is available nowadays. A state-of-the-art review is beyond the scope of this work and only a brief overview of some important papers selected among those published over the last two decades is presented in the following to introduce topics involved in this paper. The composite beam model proposed in this paper is among those aimed at capturing the effects of the shear connection deformability. A pioneering work about this topic was done by [PERSON] [2], who described the partial interaction behaviour of a composite beam by considering two Euler-Bernoulli beams connected with a shear deformable layer. The Newmark model was used to investigate many features connected to particular material properties such as the time-dependent behavior of the concrete and the nonlinear behavior of materials or of the shear connection. Among many other authors, [PERSON] et al. [3] proposed a 1D finite element for beams subjected to bending in which the concrete of the slab is assumed to be an age-dependent viscoelastic material; the solution is achieved in the time domain by means of a step-by-step integration and in the space domain by exploiting the exact closed form of the stiffness matrix. [PERSON] and [PERSON] [4] adopted a closed form solution of the problem of composite beams subjected to transverse load in the framework of the direct stiffness method by adopting algebraic expressions for the time-dependent stress-strain concrete constitutive relationships. [PERSON] et al.