## 7 Numerical Investigation
### Numerical Fiber Models and Section Analisys  
The section with the lower flexural strength was selected for each tested column where the applied moments were larger at the base of the column; the section without the turned rebar segments for the ASB column (AA, Table 7) and that with the turned rebar segments (BB, Table 7) for the RR columns. These sections were selected because the maximum column base shear was associated with the maximum flexural strength of the weaker section (flexural failure in Section 6.7) and the column damage was distributed along the more stressed column part with the weaker sections.  
Three section models were built with two concrete patches (P1, P2) and 14 steel fibers. The concrete patches were a ring region (P2) and a circular region (P1) divided in fibers. The 14 steel fibers were used  
Figure 17: Damage at the end of the cyclic tests on: (**a**) The ASB column and (**b**) the RR columns.  
to simulate the longitudinal rebar in each section (Table 7) with the diameter and the configuration outlined in Table 7. P1 simulated the column core, whereas P2 was used to model the unconfined concrete cover of the ASB column or the HPFRC jacket of the RR columns with steel or polymer fibers (Table 7). The nonlinear behavior of the sections was modelled by the nonlinear stress-strain behavior of the fibers. Four multilinear models were simulated:  
1. The stress-strain model for the concrete confined by a few stirrups (region P1 and P2 of the AA section, Table 7) on the base of previously reported models [70];
2. The experimental stress-strain curve for the UHS concrete used to repair the column R16-UHS (region P2 of the BB section, Table 7);
3. The experimental stress-strain curve for the UHP concrete used to repair the column R16-UHP (region P2 of the BB section, Table 7);
4. The skeleton curve that approximates the experimental curves obtained for the turned and unturned steel rebar in Figure 8.  
The concrete behavior of the P1 part of the RR columns sections (original ASB column core) was modelled by the same model used for the P1 part of the ASB column.  
The stress-strain curves of the concrete and the steel fibers were simulated in OpenSees by the uniaxial hysteretic material model, which is a multilinear model defined by three stress-strain points. The numerical analyses were performed by applying the experimental vertical axial loads section that was applied on the column (266 kN), and by increasing the curvatures of the section monotonically.  
A zero-length rotational-spring element was created in OpenSees to perform the moment-curvature analysis of the section. After the application of a user-defined constant axial load (column vertical load), the section was subjected to a linearly-increasing moment up to a user-defined maximum curvature. The Displacement Control integrator was used to determine the necessary load factor to apply to create the imposed displacement. The energy convergence test was used, and a tool command language (TCL) procedure permitted us to change the values of the tolerance, the solution algorithm (Newton, Newton with Initial Tangent, Broyden, Newton With Line Search), and the number of the iteration steps until the solution convergence was attained.