## 5 Cross-Section Model Based on Polygonal Discretization
### Concrete Compressive Resultant Force \(F_{c}\)  
In the proposed model, for every neutral axis position and orientation in a particular control cross-section, the process of calculating the concrete-resultant-compressive force \(F_{c}\) over the compressive zone of this cross-section significantly differs from that followed in the classical fiber models. In the proposed model, the 3D shell shape of concrete-compressive stresses \(\sigma_{c}\) is obtained point-by-point from the determination of the corresponding stresses at each vertex (control point) of the Delaunay triangles. Thus, the total resultant compressive force \(F_{c}\) of concrete is calculated as the volume enclosed by this 3D shell. The general case of a cross-section under bending with combined axial load considers two parts of the total compressive zone of a cross-section: the unconfined part, consisting of the concrete cover, and the confined one, consisting of the concrete core inside the stirrups. Each of these parts obeys its own 3D shell of concrete-compressive stresses due to the different strengths of these two concrete parts. The subcase where the two parts obey a unique continuous 3D shell of stresses is that in which the confinement by the stirrups is extremely small and is thus neglected. This happens in old RC structures with insufficient stirrup confinement. Figure 14 shows the 3D-shell shape of concrete-compressive stresses \(\sigma_{c}\) for the random cross-section shape of Figures 8 and 9. For presentation simplicity, insufficient stirrup confinement is assumed so both the concrete cover and the core of the compressive zone are affected by a continuous 3D shell of stresses. Due to biaxial bending conditions, an inclined neutral axis \(n\)-\(n\) is shown, at angle \(\theta_{n}\) with respect to the centroidal axis \(y\). It is noted that angle \(\theta_{n}\) differs in general from the angle \(a\) of the bending moment components \(M_{y}\) and \(M_{z}\) of the external imposed-bending moment vector \(M\), as explained in Sfakianakis [22]. A typical triangle with vertices as the control points, \(cp\),\(i\), is shown in the compression zone of the cross-section, in correspondence with Figure 11d, as resulted from the  
Figure 13: Control cross-sections along element length and neutral axis \(n\)-\(n\). Each vertex \(cp\),\(i\) has a strain \(l_{cp,i}\), and a concrete-compressive stress \(\sigma_{cp,i}\). Adopting the Navier-Bernoulli assumption about plain cross-sections after bending, the strain \(\epsilon_{cp,i}\) is computed as  
\[\epsilon_{cp,i}\big{(}y_{cp,i},z_{cp,i}\big{)}=\epsilon_{\rm o}-y_{cp,i}\cdot \varphi_{z}+z_{cp,i}\cdot\varphi_{y} \tag{57}\]  
whereas the concrete-compressive stress \(\sigma_{cp,i}(\epsilon_{cp,i})\) is computed from the constitutive stress-strain law of Section 2. Thus, the shape of the 3D shell of the concrete-compressive stresses is approximated by drawing the values of the stresses \(\sigma_{c}=\sigma_{cp,i}\) of each control point \(c_{p,i}\) of the cross-section, corresponding to a given deformation \(\epsilon_{c}=\epsilon_{cp,i}\) that is calculated using Equation (57). As a result, the 3D shell of the stress space is composed of triangular prisms. The bases of these triangular prisms, on the cross-section plane, correspond to the Delaunay triangulation, while their tops form the 3D shell of the concrete-compressive stresses on the compression zone of the cross-section. An example of the formed 3D stress surface is given in  
Figure 14: 3D shell of concrete compressive stresses under biaxial bending conditions combined with compressive axial load. Figure 15a for a classic rectangular cross-section, considering biaxial bending (inclined neutral axis _n-n_) and an unconfined concrete core. Figure 15b shows the detail of Delaunay triangles intersected by the neutral axis _n-n_. Each of these triangles, and their corresponding triangular prisms, is split into a compressive part (yellow color) and a tensile part (cyan color), the latter with zero stress values.