###### Abstract  
A mathematical model for RC beam elements is presented that falls into the category of distributed inelasticity models discretizing the cross-section in polygons (trapezoids, triangles). The models falling into these categories are considered to be able to describe in the best manner the inelastic behavior of the element across its whole clear length, since its response results from the numerical integration of the stiffnesses of its cross-sections, while presenting an ideal combination of accuracy, simplicity, and computational cost. The behavior of the cross-section is described through the constitutive relationships \(\sigma\)-\(\varepsilon\) of its materials for cyclic loading. The main objectives for the development of the proposed mathematical model are as follows: (a) the increased accuracy of the results compared to existing experimental ones; (b) the limitless generalization of its application, regarding of the cross-section shape; and (c) the elimination of the numerical problems presented by the application of other related models, a fact that leads to their impractical use in real three-dimensional structures. The proposed model falls under the category of distributed inelasticity models. This paper focuses on its initial version, which targets slender beam elements with negligible shear and bond-slip effects (i.e., with ribbed bars, sufficiently anchored). Thus, it is applicable to 2D and 3D framed structures that fulfill these conditions, while its modular structure allows for future adjustments for the inclusion of other effects. b 2024 14 2188  
Coming to the cross-section's inelastic behavior, according to the well-known fiber model approach, a section is usually discretized into a large number of material fibers each adopting uniaxial behavior. The response of each material fiber is calculated in terms of its uniaxial constitutive stress-strain law, \(\sigma\)-\(\varepsilon\), using [PERSON]'s classic hypothesis of plane cross-sections after bending. The resultant internal forces of the cross-section are derived from the integration of the stresses of all its fibers. An alternative approach to that of the fiber models, which is used in the proposed model, is to discretize the section into a number of polygons, mainly trapezoids and triangles, and to compute their inelastic responses in a similar manner to that of the fibers, as will be described in the following sections. Historically, two categories of finite element formulations have been used: the stiffness method and the flexibility method. In the flexibility method (also known as the force method), flexibilities appear in the equations of compatibility and forces are the unknown quantities. On the other hand, in the stiffness method (also known as displacement method), displacements are the unknown quantities. Typical models using the stiffness method are those of [PERSON] and [PERSON] [1] and [PERSON] and [PERSON] [2], which use cubic polynomials as interpolation functions to approach the distribution of displacements along the length of the member. The flexibility method was first followed by [PERSON] and [PERSON] [3], using interpolation functions that are updated at each step, depending on the degree of inelasticity, for cross-sections under uniaxial bending. [PERSON] and [PERSON] [4] created a similar model for biaxial bending where the sections' stiffnesses also change and adapt according to the degree of inelasticity in the current step of the analysis. Moreover, they applied some improvements to the member-equilibrium calculation process in order to improve convergence. [PERSON] et al. [5] and [PERSON] et al. [6; 7; 8] developed a mixed-formulation method combining stiffness and flexibility methods. The element-state determination process is based on the stiffness method, while the element-flexibility matrix is calculated by integrating the respective flexibility matrices of the control sections along the element length.