# Finite Elements for Higher Order Steel-Concrete Composite Beams
###### Abstract
[14] developed a finite element for shear-deformable two-layer beams by exploiting exact stiffness matrices derived by analytical solutions of the problem. [PERSON] and [PERSON] [21] proposed a steel-concrete composite beam model with partial interaction in which a Timoshenko element was used for the steel component; by means of a wide parametric analysis, they determined conditions requiring the modelling of the shear deformation of the steel components. An analytical solution of two-layer beam was derived by [PERSON] et al. [22] considering the interlayer slip and the shear deformation of elements, demonstrating their importance for high shear connection stiffness, for beams with small length/height ratios, and for beams with high normal/shear moduli ratios. As known, for composite systems characterized by elements with a high width/thickness ratio, the assumption of preservation of plane cross section, on which the [PERSON] and [PERSON] theories are based, is no longer valid, and warping of cross section was introduced by many researchers to capture nonuniform stress distributions within the elements. [PERSON] et al. [23] proposed a theoretical model for steel concrete composite bridge decks by including shear lag in the reinforced concrete slab and partial interaction between steel and concrete components; the same authors [24] demonstrated how the shear lag phenomenon is important over the whole length of the beam, mainly when vertical loads are applied, whereas shrinkage and other kinds of actions like support settlements and prestressing produce effects only in limited regions [25]. [PERSON] et al. [26] proposed a finite element analysis to evaluate long-term concrete slab effective width by also considering the concrete cracking in hogging regions (i.e., with negative bending moments). [PERSON] et al. demonstrated the versatility of a specific finite element derived by the theoretical model in analyzing real structural systems like cabled stayed bridges [27] or arch bridges [28]. The most recent and advanced models, in terms of enhanced kinematics, are those developed in the framework of the generalized beam theory and those denoted as higher order models. [PERSON] and [PERSON] [13] presented a two-layer composite beam model capable of capturing cross-sectional deformations and partial interaction in both the longitudinal and transverse directions by introducing a set of deformation modes derived by modal analysis of planar frames representing the cross sections. Similarly, [PERSON] and [PERSON] [29] presented a model able to capture the effects of cross-sectional deformation for a wide range of analyses including vibration and lateral-torsional-distortional buckling of composite cross sections typical for bridge applications. [PERSON] et al. [30] presented a higher order model suitable for the analysis of composite bridge decks accounting for the overall shear deformability, warping of the slab cross section and of the steel beam, and partial shear interaction. [PERSON] et al. [31] proposed a finite element beam model that includes shear warping and considers longitudinal and vertical partial interaction between the components. As also mentioned in the above discussion, solving methods can be roughly divided into two large classes: (i) methods providing the \"exact\" solution, such as closed-form expressions or analytical formulations such exponential matrix method; and (ii) methods providing approximated solutions, such as the finite difference method and the finite element method. Despite their name, \"exact\" solutions are affected by approximations due to precision of calculators and often by numerical issues deriving from the adopted algorithms; furthermore, analytical solutions, especially the closed-form ones, are only available for specific cases (e.g., specific geometric, loading, and restraint conditions). On the other hand, numerical solutions, although approximated, offer a wider versatility, being able to account for generic geometry as well as generic restraint systems and loading conditions.