## 5 Cross-Section Model Based on Polygonal Discretization  
The main cross-section-analysis problem of a typical RC member subjected to a known set of bending moments combined with axial force is to find the correct position and orientation of the neutral axis of the cross-section and of its curvature, adopting the classical Navier-Bernoulli assumption about plain sections after bending. Figure 8: Section forces and deformations. In these geometric computations, calculation of the concrete-compressive force is involved, which is performed by integrating the concrete compressive-stress distribution (usually a second-order polynomial) that acts on a random geometric shape of the compressive zone of the cross-section. In classical fiber models, the cross-section is divided into fibers, which are usually square, rectangular, or in some cases, triangular. Each fiber is associated with a material, which is either concrete or steel, with the appropriate stress-strain constitutive law, considering only its uniaxial response. The control point of each fiber, i.e., the geometric point at which the stresses are calculated according to the relevant constitutive law, is defined at its mass center. The circular reinforcing bars are placed in their predetermined positions, and their control points are placed at their geometric center. An example of discretization of a cross-section with random shape into rectangular or square fibers is shown in Figure 9a,b. Figure 9a shows a coarse-fiber mesh, in which the reinforcement bars are not inscribed in the predefined fibers, which are initially characterized as concrete fibers only. For this reason, the compressive force \(F_{s,\text{bar}}\) of a particular steel bar of area \(A_{s,\text{bar}}\) and centroid strain \(\varepsilon_{s,\text{bar}}\) is computed as  
\[F_{s,\text{bar}}=A_{s,\text{bar}}\cdot[\sigma_{s}(\varepsilon_{s,\text{bar} })-\sigma_{c}(\varepsilon_{s,\text{bar}})] \tag{56}\]  
where \(\sigma_{s}\) and \(\sigma_{c}\) are the stress functions of steel and concrete, respectively. Figure 9b shows a fine mesh of fibers, with fully inscribed reinforcement bars in the predefined fibers. Thus, the fiber mesh consists of discrete concrete or steel fibers. In both of the meshes above, a basic computational problem of a geometric nature arises from the use of the classical fiber model, as described above, which in turn can lead to a loss of accuracy in subsequent calculations. The problem is related to the geometrical shape of the fibers (mainly rectangular) that can lead to a loss of computational accuracy when the cross-section edges and/or the neutral axis, \(n\)-\(n\), are not parallel to the local \(y\) and \(z\) coordinate axes. Figure 9: Approximation of cross-section boundaries based on discretization with the classical fiber method. (a) mesh of coarse fibers, (b) mesh of fine fibers. In this case, any inclined line (edge or neutral axis) is approximated in a \"scalable\" form, as shown in Figure 9. Thus, in cases where an inclined line intersects a fiber, the latter is considered to be belonging entirely to the cross-section (for an edge) or in its compressive or tensile part (for the neutral axis), depending on which side of the line its geometric center lies on, as illustrated in Figure 9a,b. For the special case of finding the exact location of an inclined neutral axis, this control leads to enormous convergence problems of the iterative process of searching for this location, especially when this axis is at finite normal distances from the fiber centers. This happens because of the different fiber areas from each side of the inclined neutral axis in combination with their different strains and consequently stresses. For the classical fiber method, it is known that the accurate computation of the neutral axis position governs in turn the accuracy of the computation of the cross-section's internal force magnitudes. This accuracy and the iterative stability of the algorithm depend on the density of the fiber mesh. The finer the mesh, the more accurate the results received.