# Finite Elements for Higher Order Steel-Concrete Composite Beams
###### Abstract
[5] investigated the effectiveness of various integration techniques in the space domain (finite elements, finite differences, direct stiffness, and exact solution), focusing on the discretization needed to get converge and to limit errors below established thresholds; even in this case, beams subjected to transverse loads are considered by adopting the age-adjusted effective modulus method to take into account the concrete creep. The Newmark model was demonstrated to also be very versatile in analyzing the effects of construction phases of bridges; [PERSON] et al. [6] proposed a model capable of accounting for creep and shrinkage of the early age concrete, as well as for the complex change of the static scheme due to the fractionated slab pouring by means of a step-by-step time integration procedure. [PERSON] et al. [7] exploited the Newmark model in combination with specific time-dependent concrete effective modulus to investigate the effectiveness of various casting techniques in order to avoid early cracking of concrete slabs in composite bridges. Another field deeply investigated by means of the Newmark model is the nonlinear behavior of composite beams. [PERSON] and [PERSON] [8] proposed a direct stiffness approach in an iterative procedure based on the secant stiffness method; the model was validated against several experimental cases, focusing on discretization issues. Among others, [PERSON] and [PERSON] [9] also investigated the combined effects of mechanical nonlinearity and time-dependent behavior of concrete. [PERSON] and [PERSON] [10] proposed a mixed finite element formulation for the nonlinear analysis of composite beams to improve the solution for problems usually characterized by bad convergence of usual finite element approaches. Typical kinematic assumptions of the Newmark beam model were progressively relaxed, improving the analysis capabilities and the accuracy of new models. The first improvements regard the connection system, for which the multidirectional behaviour was introduced into 1D beam models; among others, [PERSON] et al. [11] and [PERSON] et al. [12] developed finite elements for two-layered composite beams capable of accounting for the deformability of the slab-girder shear connection in the longitudinal and transverse-vertical directions. Partial interaction in both the longitudinal and transverse-lateral directions was instead introduced by [PERSON] and [PERSON] [13] in a very sophisticated model capable of capturing the behavior of elements such as bridge composite box girders, in which cross-section deformation occurs. Discrete connections were also considered by [PERSON] et al. [14] by considering friction arising at the interface of the two components in the contact areas. Zero-thickness interface elements were developed by [PERSON] and [PERSON] [15] to account for partial composite actions in the simulation of steel-concrete composite elements, layered beams, or other structural systems in which components are connected with the possibility of interlayer slip. One of the features that makes the Newmark model sometimes inaccurate is neglecting the shear deformability of elements [16]. The solution of partial interaction composite beams based on the Timoshenko beam theory [17] was discussed by [PERSON] and [PERSON] in [18], pointing out when the evaluation of the flexural-slip-shear deformation is affected by shear deformability of components. The static behavior of composite beams with refined shear deformation theories is presented by [PERSON] and [PERSON] [19], accounting for all coupling coming from the material anisotropy of layers composing the beam. [PERSON] and [PERSON] [20] proposed finite element models for the nonlinear analysis of steel concrete composite beams, in which Timoshenko beams are used for both the steel and concrete elements, and pointed out situations in which the interaction between bending and shear plays a substantial role. [PERSON] et al.