## 1 Introduction  
Structural modeling is related to the simulation of the behavior of structures. It is achieved based on processes in which physical problems are explicated into mathematical ones with the use of a series of assumptions. In the literature, several studies dealing with the modeling of the structural behavior of beams can be found [1, 2]. Most of these studies are based on [PERSON]'s theory [3, 4, 5, 6, 7]. This theory is based on the plane section deformation hypothesis, which assumes that plane sections remain in-plane and perpendicular to the beam's neutral axis after the bending deformation [8]. Therefore, shear strains producing non-planar deformations are neglected. Strictly speaking, this theory fails for most loading cases, as shear force diagrams are usually coupled with the bending moment ones. In fact, the only loading case with zero shear forces corresponds with that of a constant bending moment alongside the beam. [PERSON]'s beam theory is traditionally used in slender beams, where shear deformations are significantly smaller than the flexural ones, and therefore, their effect can be neglected. Nevertheless, in structures lacking a unidimensional geometry (such as deep beams, laminated composite walls or sandwich structures), shear deformation might play an important role and its effects should be introduced into the formulation [9]. [PERSON] [10,11] was the first to deal with the shear effects in beams and his formulation was widely used in several works in the literature [12, 13, 14]. Unlike [PERSON]'s beam theory, which only considers bending rotations (\(w_{b}\)), in this approach (known as [PERSON]'s beam theory or first-order shear deformation theory), the total rotation is composed of two different rotations: rotation due to the bending \(w_{b}\) and rotation due to the shear \(w_{s}\). The rotations considered in the [PERSON]'s and [PERSON]'s beam theories in a support are illustrated in Figure 1a,b, respectively. [PERSON]'s theory was improved by [PERSON] [15], who simplified the transverse shear strain as a constant distribution through the beam thickness. In this approach, a shear coefficient is used to appropriately represent the strain energy in the structures. This coefficient is used to include the effect of the non-constant shear stresses and strains in the members' cross-sections. The values of the shear coefficient for different cross-sections were proposed by [PERSON] [16]. In order to model the variation in shear stresses without the need for these coefficients, the finite element model (FEM) [17,18] was widely used in the literature. [PERSON] et al. [19] described many of the early models that deal with shear. According to these authors, the elements of these models can be classified into the following two classes: (1) simple, with two degrees of freedom at each of its two nodes, or (2) complex, with more than two degrees of freedom per node or more than two nodes per element. The first FEM for [PERSON]'s beam theory was proposed by [PERSON] [20], who developed a two-node, four-degree-of-freedom element (transverse displacement and cross-section rotation at each node). This FEM was extended to a tapered beam by [PERSON] [21]. [PERSON] [22] found that in clamped-end elements, [PERSON]'s formulation could not represent the exact boundary conditions (as the rotation due to shear was constrained to be zero). To deal with this problem, he developed a new element based on a cubic displacement function for both the bending and the shear displacements. In this formulation, the effects of shear rotations and bending rotations were analyzed separately, resulting in an eight-degree-of-freedom beam element. The FEM presented by [PERSON] works properly in simply supported structures (such as in simply supported beams and cantilevers). However, in more complex structures, such as statically redundant beams, [PERSON]'s element does not work properly. Different authors used slightly different approaches for their FEMs and achieved almost the same results as [PERSON].