# Structural Analysis and Design of Reinforced Concrete Bridge Corbels
## 3 Discussion
### Numerical Analyses  
A finite element model was developed within the software MIDAS FEA [32] to analyze the corbel behavior to conduct a structural analysis and compare the numerical prediction with the experimental evidence in terms of load-displacement curves. The concrete mechanical properties were chosen in accordance with Eurocode 2 [17] for a concrete class C20/25 (elastic modulus E\({}_{\text{c}}\) = 30 GPa, average compressive strength f\({}_{\text{cm}}\) = 28 MPa) and the steel was assumed to behave elastically (elastic modulus E\({}_{\text{s}}\) = 200 GPa). The tensile behavior of the concrete was modeled using a linear total strain crack law [32] characterized by a tensile strength f\({}_{\text{ct}}\) = 2.2 MPa and a fracture energy G\({}_{\text{f}}\) = 73 f\({}_{\text{cm}}\)\({}^{0.18}\) = 132 N/m. Following the classical assumption of reinforced concrete structures, a perfect bond between steel and concrete interface was assumed. The maximum element mesh size of the model ranged between 10 mm for the concrete elements close to the bars (which were modeled using steel beam elements of equivalent size) and 50 mm for the elements far from the bars. These dimensions were selected after completion of a mesh-size sensitivity analysis. The geometry of the corbel presents a cold joint between the upper face of the slab and the lower face of the curb, where the safety barrier is anchored. The bars, as well as the U-shaped rebars, pass through this cold joint. This joint was modeled as a thin layer of concrete elements (about 1 cm in size) with a reduced tensile strength (1.2 MPa [33]) compared to the typical 2.2 MPa of C20/25 concrete. The ordinary reinforcement located in the slab was modeled with 1D reinforcement elements of B450C steel while, to model the possibility of the plate to uplift at the interface between the steel plate and concrete, 2D elements were selected. To compare the numerical and experimental results, the constraints adopted in the test setup were introduced and the load was increased up to 84 kN (maximum load applied). The complete geometry of the FEM is shown in Figure 6a-e.  
To validate the model, the experimental and numerical results were compared in terms of the displacements of five control points monitored with LVDTs (vertical displacements of the heads of the three bars) and the horizontal sliding of the upper part of the corbel measured on its top and bottom edges (Figure 6e). The comparison between experimental (dashed line) and numerical (continuous line) displacements is shown in Figure 7. As it can be seen, the results are in good agreement (load average difference below 5%). Considering the control points of the bars (Arrows 1-3, Figure 6e), it can be observed that the calculation ended close to the displacements experimentally detected. As the aim of the Finite Element Model (FEM) was the identification of the load path required to identify the consequent strut-and-tie model, the analyses were focused only on the nonlinear behavior up to the maximum applied load. In order to determine the ascending branch of the load-displacement curves, load controlled nonlinear analyses were conducted up to the predefined value of 84 kN. Displacement-driven nonlinear analyses could also be used to describe the post-peak behavior, with the consequent big effort in terms of computational cost due to the difficulties of solving the numerical instabilities related to softening, without any control on the results owing to the post-peak behavior being monitored  
Figure 6: FEM (finite element model) of the tested specimen (**a**) model, (**b**) load and boundary conditions, (**c**) reinforcement layout, (**d**) anchorage system, and (**e**) displacement control points. during tests. As well, the unloading phase of the specimen, represented by the descending branches in the experimental curves in Figure 7, was not simulated.